A preview-safe 300-case reconstruction of John Torrence Tate’s mathematical method across adèles and idèles, Tate’s thesis, Hecke L-functions, class field theory, Galois cohomology, Tate duality, abelian varieties, Tate modules, heights, elliptic curves, Lubin–Tate formal groups, rigid analytic spaces, Hodge–Tate theory, p-divisible groups, the Tate conjecture, and arithmetic-geometric correspondence culture.
This is a bibliographic and methodological reconstruction, not a reproduction of Tate’s papers, books, letters, or archival materials. The 300 cases are lecture-style units organized from public source spines and thematic clusters. Strategy tags overlap; percentages are method-frequency indicators, not a probability distribution.
Tate’s method is a discipline of reformulation: move the problem to its natural carrier space, normalize local data precisely, encode the structure functorially, and then let duality, cohomology, representation, or analytic transform produce the theorem.
Each case is read as a research pipeline: arithmetic object → local/global carrier → normalization → functorial encoding → pairing, transform, or exact sequence → theorem, algorithm, conjecture, or program.
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Is the arithmetic object better understood after being moved to adeles, ideles, test functions, and local zeta integrals?
Move a global arithmetic problem to the adele or idele group, where local factors become visible and recombinable.
Move. Replace scattered congruence and ideal data by one locally compact object carrying all completions at once.
Diagnostic. Can the global object be written as a quotient or product over all places?
Artifact. idele quotient, local-global decomposition, product formula
Failure mode. Treating adeles as notation instead of as the working space where analysis happens.
Use Fourier transform and Poisson summation as the structural hinge between a function and its functional equation.
Move. Pick the self-dual normalization so the transform becomes the proof rather than an auxiliary computation.
Diagnostic. Is the desired symmetry just Fourier inversion on the right locally compact group?
Artifact. Poisson summation argument, transformed test function, functional equation
Failure mode. Forgetting that the normalization of Haar measure is part of the theorem.
Make characters, conductors, twists, and normalizations explicit before analysis begins.
Move. Treat a character as structured data: local components, ramification, conductor, central character, and absolute value twist.
Diagnostic. Which character data changes the factor, sign, or pole?
Artifact. character table, conductor bookkeeping, twist convention
Failure mode. Changing normalizations mid-proof and losing compatibility between local and global conventions.
Convert an L-function into an integral whose analytic properties can be proved by harmonic analysis.
Move. Encode Euler factors through a test function and a character, then let the integral produce the analytic object.
Diagnostic. Which arithmetic series becomes tractable after being represented as a zeta integral?
Artifact. zeta integral, local factors, Euler product reconstruction
Failure mode. Writing the integral without controlling measures, convergence, and local normalizations.
Factor a global construction into precise archimedean and nonarchimedean local computations.
Move. Break the global integral into local integrals, compute each one, and reassemble the Euler product.
Diagnostic. What is the local computation at every place, including bad and archimedean places?
Artifact. table of local factors, gamma factors, epsilon factors
Failure mode. Solving the unramified case and silently ignoring the places where the theorem actually records information.
Package analytic continuation, factors, poles, and symmetry into a single invariant statement.
Move. Do not merely prove continuation; identify the completed object whose symmetry is natural.
Diagnostic. What completion makes the equation symmetric and functorial?
Artifact. completed L-function statement and epsilon factor
Failure mode. A functional equation with unspecified completions, measures, or exceptional terms.
Can a reciprocity law, obstruction, or classification be made functorial through Galois cohomology and duality?
Make Galois actions functorial so arithmetic objects can be compared through cohomology.
Move. Choose the module, identify its Galois action, then study invariants, co-invariants, extensions, and pairings.
Diagnostic. Which module carries the arithmetic problem most faithfully?
Artifact. Galois module and cohomological comparison
Failure mode. Computing groups without making the functorial input explicit.
Expose hidden arithmetic by forcing it through exact sequences, connecting maps, and boundary terms.
Move. Make the obstruction appear as the term that prevents exactness from being trivial.
Diagnostic. Which boundary map carries the arithmetic content?
Artifact. long exact sequence, obstruction group, connecting homomorphism
Failure mode. Using exact sequences descriptively rather than as a pressure device.
Track exactly where local solvability fails to imply global solvability.
Move. Assemble local data, compare it globally, and name the failure as an arithmetic object rather than an anomaly.
Diagnostic. Which group measures the difference between local and global truth?
Artifact. Selmer group, Sha group, Poitou–Tate sequence
Failure mode. Treating local-global failure as exception rather than as the central invariant.
Use duality to turn class formation into computable pairings and reciprocal classifications.
Move. Find the dual object, identify the fundamental class, and let cup product produce the classification.
Diagnostic. What pairing makes the two sides of the theorem determine one another?
Artifact. duality theorem, fundamental class, pairing diagram
Failure mode. Stating duality as a slogan without the exact modules and degrees.
Construct maps that convert field-theoretic symmetry into multiplicative or cohomological data.
Move. Define the reciprocity map by local normalization and global compatibility, then test it on uniformizers and norms.
Diagnostic. What element should map to Frobenius, and under which convention?
Artifact. local/global reciprocity map and compatibility checks
Failure mode. Mixing arithmetic and geometric Frobenius conventions without a visible warning.
Translate reciprocity laws into cohomological statements about formations and exact sequences.
Move. Replace explicit reciprocity formulas by a functorial cohomological machine whose output is class field theory.
Diagnostic. Can the reciprocity statement be read as a vanishing, connecting map, or comparison theorem?
Artifact. class formation, reciprocity map, cohomology groups
Failure mode. Preserving classical terminology while missing the cohomological mechanism that makes it portable.
Can torsion, height, reduction, or isogeny data linearize the geometry enough to expose arithmetic structure?
Classify abelian varieties over finite fields by Frobenius eigenvalues and isogeny data.
Move. Let Frobenius act as the fingerprint of the variety, then classify isogeny classes by its characteristic data.
Diagnostic. Which Weil number encodes the isogeny class?
Artifact. Honda–Tate classification table, Frobenius polynomial
Failure mode. Ignoring integrality and slope constraints while manipulating eigenvalues formally.
Convert arithmetic size into a canonical quadratic pairing.
Move. Separate naive height from canonical height and use the limit to obtain functorial arithmetic geometry.
Diagnostic. Which normalization turns height into a bilinear or quadratic invariant?
Artifact. Néron–Tate height, regulator, Mordell–Weil lattice
Failure mode. Mixing local height contributions without proving the global normalization.
Use Selmer groups to mediate between computable local conditions and elusive global rational points.
Move. Compute the accessible group, isolate the hidden obstruction, and relate both to conjectural invariants.
Diagnostic. What part is visible by descent, and what remains hidden in Sha?
Artifact. descent exact sequence, Selmer computation, BSD interface
Failure mode. Reporting a Selmer group without explaining what it does and does not prove about rational points.
Turn local equations of elliptic curves into reduction data, conductors, and arithmetic invariants.
Move. Inspect the local Weierstrass model, transform it, and extract reduction type and conductor contribution.
Diagnostic. Which local transformation reveals the minimal model?
Artifact. Tate algorithm output, reduction type, conductor exponent
Failure mode. Treating a global Weierstrass equation as if it were already minimal at every prime.
Recover morphisms of abelian varieties from compatible maps of Tate modules.
Move. Turn geometric morphisms into linear Galois-equivariant maps, then prove the comparison is exact enough.
Diagnostic. Can a geometric map be recognized purely from l-adic representation data?
Artifact. isogeny theorem, Hom comparison, semisimplicity statement
Failure mode. Confusing a representation-theoretic shadow with a proven geometric morphism.
Linearize torsion in abelian varieties into a Galois representation.
Move. Replace torsion points at all powers of a prime by a single module with Galois action.
Diagnostic. What does the Galois action on torsion know about the variety?
Artifact. Tate module, l-adic representation, endomorphism comparison
Failure mode. Studying torsion pointwise instead of using the inverse-limit representation.
Is the problem local, p-adic, infinitesimal, or nonarchimedean enough to require formal groups, p-divisible groups, or rigid spaces?
Use p-divisible groups to separate deformation, height, dimension, and slope phenomena.
Move. Replace a sequence of finite group schemes by the inductive object whose deformation theory is stable.
Diagnostic. What is controlled by height, dimension, slope, or connected-étale decomposition?
Artifact. Barsotti–Tate group, deformation space, Newton data
Failure mode. Studying each p-power layer separately and missing the stable deformation object.
Use formal groups to encode local deformation and local reciprocity.
Move. Study arithmetic near the identity through power series laws rather than global algebraic coordinates.
Diagnostic. Which local phenomenon becomes linear or functorial in the formal group?
Artifact. formal group law, deformation ring, local parameter
Failure mode. Applying global intuition where the formal neighborhood carries the actual structure.
Compare p-adic étale cohomology with Hodge-theoretic pieces.
Move. Identify how Galois representations remember differential-geometric data after p-adic extension.
Diagnostic. Which weights and twists make the decomposition visible?
Artifact. Hodge–Tate weights, comparison statement, p-adic period structure
Failure mode. Forgetting the twist and field extension needed for the comparison to make sense.
Replace insufficient p-adic topology by a geometry of affinoids and admissible coverings.
Move. Build the space whose functions behave analytically over nonarchimedean fields.
Diagnostic. What classical analytic argument fails p-adically, and what rigid object repairs it?
Artifact. affinoid algebra, rigid space, admissible cover
Failure mode. Using naive p-adic open sets where admissibility is the true topology.
Construct local class field theory through formal modules and torsion points.
Move. Let a chosen uniformizer act on a formal group, then read abelian extensions from its torsion.
Diagnostic. Can local abelian extensions be generated by structured torsion?
Artifact. Lubin–Tate formal group, tower, reciprocity law
Failure mode. Forgetting that the construction depends on choices while the final reciprocity law is canonical.
Uniformize certain elliptic curves p-adically by multiplicative quotients.
Move. Replace a degenerating elliptic curve by a quotient that makes its p-adic analytic structure explicit.
Diagnostic. When does multiplicative uniformization reveal the curve better than its equation?
Artifact. Tate curve, q-parameter, p-adic uniformization
Failure mode. Treating q as a formal symbol rather than a parameter encoding geometry and arithmetic.
Is the correct result a named bridge between cycles, motives, l-adic representations, and arithmetic invariants?
Solve by changing the language so thoroughly that old theorems become instances of a broader machine.
Move. Find the formulation that makes many formerly separate results look inevitable.
Diagnostic. What reformulation makes the theorem portable?
Artifact. new dictionary, unified framework, transportable proof method
Failure mode. Rephrasing without increasing explanatory or technical power.
Treat Galois representations as the arithmetic shadow of geometry and motives.
Move. Pass from varieties to cohomology to representations, then ask which geometric cycles are visible in the representation.
Diagnostic. What part of geometry can be recovered from the Galois action?
Artifact. l-adic representation, motive-level dictionary, comparison map
Failure mode. Assuming every representation has a geometric origin without checking the motivic constraints.
Test algebraic cycles by their fixed classes in l-adic cohomology.
Move. Map cycles to cohomology, isolate Galois-invariant classes, and ask whether every such class is algebraic.
Diagnostic. Which invariant cohomology classes should come from algebraic cycles?
Artifact. cycle class map, fixed subspace, Tate conjecture formulation
Failure mode. Equating invariant classes with cycles without proving surjectivity of the cycle map.
Stabilize an emerging phenomenon by naming the conjecture or object precisely enough to guide decades of work.
Move. Identify the invariant, formulate the relation, and give the field a target object to attack.
Diagnostic. What name turns a pattern into a program?
Artifact. Tate conjecture, Sato–Tate problem, Tate motive, named invariant
Failure mode. Naming without a precise testable mathematical statement.
Is the working method visible in letters, lectures, comments, examples, and the discipline of concise writing?
Use explicit examples to force the correct abstraction rather than imposing formalism prematurely.
Move. Compute or explain the small case until the general definition becomes unavoidable.
Diagnostic. Which example makes the abstraction necessary?
Artifact. worked example, table, local computation, motivating diagram
Failure mode. Opening with the generality before the reader has seen why it exists.
Let ideas mature through letters, conversations, lectures, comments, and circulating manuscripts.
Move. Use informal exchange as a controlled laboratory before formal publication.
Diagnostic. Which part of the argument needs conversation before it is ready for print?
Artifact. letter, seminar note, circulating manuscript, commentary
Failure mode. Letting informal brilliance become inaccessible or uncheckable.
Treat correspondence and commentary as part of the mathematical record of formation.
Move. Read letters and author comments not as anecdotes but as evidence of the method by which ideas crystallized.
Diagnostic. Which informal trace explains why the final formulation has its shape?
Artifact. correspondence map, author commentary, source-lineage note
Failure mode. Separating finished papers from the working process that made them intelligible.
Write only after the architecture is clean enough for a concise proof.
Move. Compress the result to the essential object, map, pairing, or sequence that carries the theorem.
Diagnostic. What can be removed without weakening the proof?
Artifact. short paper, clean proof, decisive definition
Failure mode. Conciseness that hides prerequisites or leaves normalization implicit.
Calibrate problems to a learner’s exact stage while preserving genuine research depth.
Move. Diagnose what the student can already see, then assign the next exercise or problem that opens the field.
Diagnostic. What problem is hard enough to form judgment but not so vague that it becomes noise?
Artifact. problem ladder, reading path, research apprenticeship
Failure mode. Giving either routine exercises with no horizon or impossible problems with no entry path.
Bars show count divided by 300 cases. Since a case carries multiple strategy tags, the displayed prevalence is a method-frequency map rather than a probability distribution.
Move from classical notation to the object where the theorem is natural: adele ring, idele class group, cohomology group, Tate module, formal group, or rigid space.
Compute at each completion and preserve the product, reciprocity, or compatibility condition that makes the global object reappear.
Fix Haar measures, Frobenius conventions, twists, conductors, local factors, and exact-sequence signs before proving anything substantial.
Replace hard geometry by torsion modules, cohomology groups, Galois representations, height pairings, or cycle classes.
When the object resists direct computation, find the dual object and the pairing that forces the answer.
Do not hide failure of local-global principles; make it a Selmer group, Sha group, obstruction term, or boundary map.
Use formal groups, Lubin-Tate towers, minimal models, and rigid analytic objects to compute what global language cannot see directly.
Use explicit finite fields, local fields, elliptic curves, and bad primes as pressure tests for the abstraction.
A Tate-style result often creates a new language that later theorems can use; judge the work by how much it makes inevitable.
Read letters, comments, and concise notes as part of the method: they show which choices produced the final architecture.
| # | Case | Family | Deep reading move | Strategies |
|---|---|---|---|---|
| 1 | From number field to adele ring Tate thesis and harmonic analysis on number fields: Lecture-case 01 — From number field to adele ring |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “From number field to adele ring” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes from number field to adele ring natural, computable, functorial, and stable under passage between local and global fields? |
S01 S03 S05 |
| 2 | Ideles as the carrier space of Hecke characters Tate thesis and harmonic analysis on number fields: Lecture-case 02 — Ideles as the carrier space of Hecke characters |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Ideles as the carrier space of Hecke characters” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ideles as the carrier space of hecke characters natural, computable, functorial, and stable under passage between local and global fields? |
S02 S04 S06 |
| 3 | Schwartz-Bruhat functions as test objects Tate thesis and harmonic analysis on number fields: Lecture-case 03 — Schwartz-Bruhat functions as test objects |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Schwartz-Bruhat functions as test objects” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes schwartz-bruhat functions as test objects natural, computable, functorial, and stable under passage between local and global fields? |
S03 S05 S01 |
| 4 | Local zeta integrals Tate thesis and harmonic analysis on number fields: Lecture-case 04 — Local zeta integrals |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Local zeta integrals” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local zeta integrals natural, computable, functorial, and stable under passage between local and global fields? |
S04 S06 S02 |
| 5 | Euler product recovery from local data Tate thesis and harmonic analysis on number fields: Lecture-case 05 — Euler product recovery from local data |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Euler product recovery from local data” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes euler product recovery from local data natural, computable, functorial, and stable under passage between local and global fields? |
S05 S01 S03 |
| 6 | Self-dual Haar measure normalization Tate thesis and harmonic analysis on number fields: Lecture-case 06 — Self-dual Haar measure normalization |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Self-dual Haar measure normalization” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes self-dual haar measure normalization natural, computable, functorial, and stable under passage between local and global fields? |
S06 S02 S04 |
| 7 | Fourier transform on locally compact groups Tate thesis and harmonic analysis on number fields: Lecture-case 07 — Fourier transform on locally compact groups |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Fourier transform on locally compact groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes fourier transform on locally compact groups natural, computable, functorial, and stable under passage between local and global fields? |
S01 S03 S05 |
| 8 | Poisson summation as global hinge Tate thesis and harmonic analysis on number fields: Lecture-case 08 — Poisson summation as global hinge |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Poisson summation as global hinge” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes poisson summation as global hinge natural, computable, functorial, and stable under passage between local and global fields? |
S02 S04 S06 |
| 9 | Archimedean gamma factor extraction Tate thesis and harmonic analysis on number fields: Lecture-case 09 — Archimedean gamma factor extraction |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Archimedean gamma factor extraction” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes archimedean gamma factor extraction natural, computable, functorial, and stable under passage between local and global fields? |
S03 S05 S01 |
| 10 | Nonarchimedean factor computation Tate thesis and harmonic analysis on number fields: Lecture-case 10 — Nonarchimedean factor computation |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Nonarchimedean factor computation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes nonarchimedean factor computation natural, computable, functorial, and stable under passage between local and global fields? |
S04 S06 S02 |
| 11 | Analytic continuation of Hecke L-functions Tate thesis and harmonic analysis on number fields: Lecture-case 11 — Analytic continuation of Hecke L-functions |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Analytic continuation of Hecke L-functions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes analytic continuation of hecke l-functions natural, computable, functorial, and stable under passage between local and global fields? |
S05 S01 S03 |
| 12 | Functional equation by transform Tate thesis and harmonic analysis on number fields: Lecture-case 12 — Functional equation by transform |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Functional equation by transform” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes functional equation by transform natural, computable, functorial, and stable under passage between local and global fields? |
S06 S02 S04 |
| 13 | Pole accounting for twisted zeta integrals Tate thesis and harmonic analysis on number fields: Lecture-case 13 — Pole accounting for twisted zeta integrals |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Pole accounting for twisted zeta integrals” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes pole accounting for twisted zeta integrals natural, computable, functorial, and stable under passage between local and global fields? |
S01 S03 S05 |
| 14 | Classical Hecke theory rephrased adelically Tate thesis and harmonic analysis on number fields: Lecture-case 14 — Classical Hecke theory rephrased adelically |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Classical Hecke theory rephrased adelically” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes classical hecke theory rephrased adelically natural, computable, functorial, and stable under passage between local and global fields? |
S02 S04 S06 |
| 15 | GL(1) as automorphic prototype Tate thesis and harmonic analysis on number fields: Lecture-case 15 — GL(1) as automorphic prototype |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “GL(1) as automorphic prototype” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes gl(1) as automorphic prototype natural, computable, functorial, and stable under passage between local and global fields? |
S03 S05 S01 |
| 16 | Characters on ideles modulo units Tate thesis and harmonic analysis on number fields: Lecture-case 16 — Characters on ideles modulo units |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Characters on ideles modulo units” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes characters on ideles modulo units natural, computable, functorial, and stable under passage between local and global fields? |
S04 S06 S02 |
| 17 | Local epsilon factors as diagnostics Tate thesis and harmonic analysis on number fields: Lecture-case 17 — Local epsilon factors as diagnostics |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Local epsilon factors as diagnostics” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local epsilon factors as diagnostics natural, computable, functorial, and stable under passage between local and global fields? |
S05 S01 S03 |
| 18 | Global product formula in analysis Tate thesis and harmonic analysis on number fields: Lecture-case 18 — Global product formula in analysis |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Global product formula in analysis” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes global product formula in analysis natural, computable, functorial, and stable under passage between local and global fields? |
S06 S02 S04 |
| 19 | Theta series avoided by reformulation Tate thesis and harmonic analysis on number fields: Lecture-case 19 — Theta series avoided by reformulation |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Theta series avoided by reformulation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes theta series avoided by reformulation natural, computable, functorial, and stable under passage between local and global fields? |
S01 S03 S05 |
| 20 | Distributional proof architecture Tate thesis and harmonic analysis on number fields: Lecture-case 20 — Distributional proof architecture |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Distributional proof architecture” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes distributional proof architecture natural, computable, functorial, and stable under passage between local and global fields? |
S02 S04 S06 |
| 21 | Adelic compactness and quotient measures Tate thesis and harmonic analysis on number fields: Lecture-case 21 — Adelic compactness and quotient measures |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Adelic compactness and quotient measures” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes adelic compactness and quotient measures natural, computable, functorial, and stable under passage between local and global fields? |
S03 S05 S01 |
| 22 | Hecke characters as representation data Tate thesis and harmonic analysis on number fields: Lecture-case 22 — Hecke characters as representation data |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Hecke characters as representation data” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes hecke characters as representation data natural, computable, functorial, and stable under passage between local and global fields? |
S04 S06 S02 |
| 23 | Function-field variant Tate thesis and harmonic analysis on number fields: Lecture-case 23 — Function-field variant |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Function-field variant” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes function-field variant natural, computable, functorial, and stable under passage between local and global fields? |
S05 S01 S03 |
| 24 | Tate thesis as language converter Tate thesis and harmonic analysis on number fields: Lecture-case 24 — Tate thesis as language converter |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Tate thesis as language converter” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate thesis as language converter natural, computable, functorial, and stable under passage between local and global fields? |
S06 S02 S04 |
| 25 | Lecture-note afterlife of the thesis Tate thesis and harmonic analysis on number fields: Lecture-case 25 — Lecture-note afterlife of the thesis |
Tate thesis and harmonic analysis on number fields 1950–1967 |
Read “Lecture-note afterlife of the thesis” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes lecture-note afterlife of the thesis natural, computable, functorial, and stable under passage between local and global fields? |
S01 S03 S05 |
| 26 | Class formations as reusable infrastructure Global class field theory and Artin-Tate architecture: Lecture-case 01 — Class formations as reusable infrastructure |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Class formations as reusable infrastructure” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes class formations as reusable infrastructure natural, computable, functorial, and stable under passage between local and global fields? |
S07 S09 S11 |
| 27 | Fundamental class as governing object Global class field theory and Artin-Tate architecture: Lecture-case 02 — Fundamental class as governing object |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Fundamental class as governing object” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes fundamental class as governing object natural, computable, functorial, and stable under passage between local and global fields? |
S08 S10 S12 |
| 28 | Artin reciprocity in cohomological language Global class field theory and Artin-Tate architecture: Lecture-case 03 — Artin reciprocity in cohomological language |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Artin reciprocity in cohomological language” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes artin reciprocity in cohomological language natural, computable, functorial, and stable under passage between local and global fields? |
S09 S11 S07 |
| 29 | Idele class groups and Galois groups Global class field theory and Artin-Tate architecture: Lecture-case 04 — Idele class groups and Galois groups |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Idele class groups and Galois groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes idele class groups and galois groups natural, computable, functorial, and stable under passage between local and global fields? |
S10 S12 S08 |
| 30 | Norm maps as visible descent Global class field theory and Artin-Tate architecture: Lecture-case 05 — Norm maps as visible descent |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Norm maps as visible descent” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes norm maps as visible descent natural, computable, functorial, and stable under passage between local and global fields? |
S11 S07 S09 |
| 31 | Tate-Nakayama comparison Global class field theory and Artin-Tate architecture: Lecture-case 06 — Tate-Nakayama comparison |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Tate-Nakayama comparison” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate-nakayama comparison natural, computable, functorial, and stable under passage between local and global fields? |
S12 S08 S10 |
| 32 | Cup products as reciprocity carriers Global class field theory and Artin-Tate architecture: Lecture-case 07 — Cup products as reciprocity carriers |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Cup products as reciprocity carriers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cup products as reciprocity carriers natural, computable, functorial, and stable under passage between local and global fields? |
S07 S09 S11 |
| 33 | Local and global formations Global class field theory and Artin-Tate architecture: Lecture-case 08 — Local and global formations |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Local and global formations” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local and global formations natural, computable, functorial, and stable under passage between local and global fields? |
S08 S10 S12 |
| 34 | Connecting maps and boundary terms Global class field theory and Artin-Tate architecture: Lecture-case 09 — Connecting maps and boundary terms |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Connecting maps and boundary terms” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes connecting maps and boundary terms natural, computable, functorial, and stable under passage between local and global fields? |
S09 S11 S07 |
| 35 | Cohomology of finite Galois groups Global class field theory and Artin-Tate architecture: Lecture-case 10 — Cohomology of finite Galois groups |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Cohomology of finite Galois groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomology of finite galois groups natural, computable, functorial, and stable under passage between local and global fields? |
S10 S12 S08 |
| 36 | Global class field theory with Artin Global class field theory and Artin-Tate architecture: Lecture-case 11 — Global class field theory with Artin |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Global class field theory with Artin” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes global class field theory with artin natural, computable, functorial, and stable under passage between local and global fields? |
S11 S07 S09 |
| 37 | Compatibility with local reciprocity Global class field theory and Artin-Tate architecture: Lecture-case 12 — Compatibility with local reciprocity |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Compatibility with local reciprocity” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes compatibility with local reciprocity natural, computable, functorial, and stable under passage between local and global fields? |
S12 S08 S10 |
| 38 | Frobenius convention discipline Global class field theory and Artin-Tate architecture: Lecture-case 13 — Frobenius convention discipline |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Frobenius convention discipline” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes frobenius convention discipline natural, computable, functorial, and stable under passage between local and global fields? |
S07 S09 S11 |
| 39 | The role of H^2 in reciprocity Global class field theory and Artin-Tate architecture: Lecture-case 14 — The role of H^2 in reciprocity |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “The role of H^2 in reciprocity” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes the role of h^2 in reciprocity natural, computable, functorial, and stable under passage between local and global fields? |
S08 S10 S12 |
| 40 | Exactness as theorem detector Global class field theory and Artin-Tate architecture: Lecture-case 15 — Exactness as theorem detector |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Exactness as theorem detector” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes exactness as theorem detector natural, computable, functorial, and stable under passage between local and global fields? |
S09 S11 S07 |
| 41 | Norm residue symbols Global class field theory and Artin-Tate architecture: Lecture-case 16 — Norm residue symbols |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Norm residue symbols” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes norm residue symbols natural, computable, functorial, and stable under passage between local and global fields? |
S10 S12 S08 |
| 42 | Brauer group comparison Global class field theory and Artin-Tate architecture: Lecture-case 17 — Brauer group comparison |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Brauer group comparison” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes brauer group comparison natural, computable, functorial, and stable under passage between local and global fields? |
S11 S07 S09 |
| 43 | Restriction and corestriction calculus Global class field theory and Artin-Tate architecture: Lecture-case 18 — Restriction and corestriction calculus |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Restriction and corestriction calculus” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes restriction and corestriction calculus natural, computable, functorial, and stable under passage between local and global fields? |
S12 S08 S10 |
| 44 | Cohomological proof compression Global class field theory and Artin-Tate architecture: Lecture-case 19 — Cohomological proof compression |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Cohomological proof compression” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomological proof compression natural, computable, functorial, and stable under passage between local and global fields? |
S07 S09 S11 |
| 45 | Functoriality across extensions Global class field theory and Artin-Tate architecture: Lecture-case 20 — Functoriality across extensions |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Functoriality across extensions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes functoriality across extensions natural, computable, functorial, and stable under passage between local and global fields? |
S08 S10 S12 |
| 46 | The class field axiom as machine Global class field theory and Artin-Tate architecture: Lecture-case 21 — The class field axiom as machine |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “The class field axiom as machine” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes the class field axiom as machine natural, computable, functorial, and stable under passage between local and global fields? |
S09 S11 S07 |
| 47 | Explicit reciprocity from abstract formation Global class field theory and Artin-Tate architecture: Lecture-case 22 — Explicit reciprocity from abstract formation |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Explicit reciprocity from abstract formation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes explicit reciprocity from abstract formation natural, computable, functorial, and stable under passage between local and global fields? |
S10 S12 S08 |
| 48 | Unramified extensions as test case Global class field theory and Artin-Tate architecture: Lecture-case 23 — Unramified extensions as test case |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Unramified extensions as test case” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes unramified extensions as test case natural, computable, functorial, and stable under passage between local and global fields? |
S11 S07 S09 |
| 49 | Ramified local terms Global class field theory and Artin-Tate architecture: Lecture-case 24 — Ramified local terms |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Ramified local terms” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ramified local terms natural, computable, functorial, and stable under passage between local and global fields? |
S12 S08 S10 |
| 50 | Class field theory as reformulated arithmetic Global class field theory and Artin-Tate architecture: Lecture-case 25 — Class field theory as reformulated arithmetic |
Global class field theory and Artin-Tate architecture 1950s–1960s |
Read “Class field theory as reformulated arithmetic” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes class field theory as reformulated arithmetic natural, computable, functorial, and stable under passage between local and global fields? |
S07 S09 S11 |
| 51 | Duality pairings for arithmetic modules Galois cohomology, duality and local-global obstructions: Lecture-case 01 — Duality pairings for arithmetic modules |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Duality pairings for arithmetic modules” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes duality pairings for arithmetic modules natural, computable, functorial, and stable under passage between local and global fields? |
S08 S11 S17 |
| 52 | Tate cohomology as periodic machine Galois cohomology, duality and local-global obstructions: Lecture-case 02 — Tate cohomology as periodic machine |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Tate cohomology as periodic machine” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate cohomology as periodic machine natural, computable, functorial, and stable under passage between local and global fields? |
S09 S12 S27 |
| 53 | Local duality theorem as pairing architecture Galois cohomology, duality and local-global obstructions: Lecture-case 03 — Local duality theorem as pairing architecture |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Local duality theorem as pairing architecture” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local duality theorem as pairing architecture natural, computable, functorial, and stable under passage between local and global fields? |
S11 S17 S08 |
| 54 | Global duality and exact sequence control Galois cohomology, duality and local-global obstructions: Lecture-case 04 — Global duality and exact sequence control |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Global duality and exact sequence control” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes global duality and exact sequence control natural, computable, functorial, and stable under passage between local and global fields? |
S12 S27 S09 |
| 55 | Poitou-Tate style obstruction accounting Galois cohomology, duality and local-global obstructions: Lecture-case 05 — Poitou-Tate style obstruction accounting |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Poitou-Tate style obstruction accounting” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes poitou-tate style obstruction accounting natural, computable, functorial, and stable under passage between local and global fields? |
S17 S08 S11 |
| 56 | Selmer groups as local-condition filters Galois cohomology, duality and local-global obstructions: Lecture-case 06 — Selmer groups as local-condition filters |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Selmer groups as local-condition filters” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes selmer groups as local-condition filters natural, computable, functorial, and stable under passage between local and global fields? |
S27 S09 S12 |
| 57 | Sha as defect of local-global principle Galois cohomology, duality and local-global obstructions: Lecture-case 07 — Sha as defect of local-global principle |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Sha as defect of local-global principle” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes sha as defect of local-global principle natural, computable, functorial, and stable under passage between local and global fields? |
S08 S11 S17 |
| 58 | Cohomology with finite modules Galois cohomology, duality and local-global obstructions: Lecture-case 08 — Cohomology with finite modules |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Cohomology with finite modules” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomology with finite modules natural, computable, functorial, and stable under passage between local and global fields? |
S09 S12 S27 |
| 59 | Cohomology with tori Galois cohomology, duality and local-global obstructions: Lecture-case 09 — Cohomology with tori |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Cohomology with tori” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomology with tori natural, computable, functorial, and stable under passage between local and global fields? |
S11 S17 S08 |
| 60 | Boundary maps in descent Galois cohomology, duality and local-global obstructions: Lecture-case 10 — Boundary maps in descent |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Boundary maps in descent” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes boundary maps in descent natural, computable, functorial, and stable under passage between local and global fields? |
S12 S27 S09 |
| 61 | Obstruction groups in rational point problems Galois cohomology, duality and local-global obstructions: Lecture-case 11 — Obstruction groups in rational point problems |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Obstruction groups in rational point problems” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes obstruction groups in rational point problems natural, computable, functorial, and stable under passage between local and global fields? |
S17 S08 S11 |
| 62 | Compatibility of local pairings Galois cohomology, duality and local-global obstructions: Lecture-case 12 — Compatibility of local pairings |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Compatibility of local pairings” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes compatibility of local pairings natural, computable, functorial, and stable under passage between local and global fields? |
S27 S09 S12 |
| 63 | Global reciprocity from local sums Galois cohomology, duality and local-global obstructions: Lecture-case 13 — Global reciprocity from local sums |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Global reciprocity from local sums” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes global reciprocity from local sums natural, computable, functorial, and stable under passage between local and global fields? |
S08 S11 S17 |
| 64 | Exactness across all places Galois cohomology, duality and local-global obstructions: Lecture-case 14 — Exactness across all places |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Exactness across all places” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes exactness across all places natural, computable, functorial, and stable under passage between local and global fields? |
S09 S12 S27 |
| 65 | Dual modules and character groups Galois cohomology, duality and local-global obstructions: Lecture-case 15 — Dual modules and character groups |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Dual modules and character groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes dual modules and character groups natural, computable, functorial, and stable under passage between local and global fields? |
S11 S17 S08 |
| 66 | Finite support conditions Galois cohomology, duality and local-global obstructions: Lecture-case 16 — Finite support conditions |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Finite support conditions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finite support conditions natural, computable, functorial, and stable under passage between local and global fields? |
S12 S27 S09 |
| 67 | Cassels-Tate style pairing intuition Galois cohomology, duality and local-global obstructions: Lecture-case 17 — Cassels-Tate style pairing intuition |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Cassels-Tate style pairing intuition” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cassels-tate style pairing intuition natural, computable, functorial, and stable under passage between local and global fields? |
S17 S08 S11 |
| 68 | Cohomological descent for curves Galois cohomology, duality and local-global obstructions: Lecture-case 18 — Cohomological descent for curves |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Cohomological descent for curves” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomological descent for curves natural, computable, functorial, and stable under passage between local and global fields? |
S27 S09 S12 |
| 69 | Galois modules from torsion Galois cohomology, duality and local-global obstructions: Lecture-case 19 — Galois modules from torsion |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Galois modules from torsion” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes galois modules from torsion natural, computable, functorial, and stable under passage between local and global fields? |
S08 S11 S17 |
| 70 | Spectral-sequence avoidance by clean maps Galois cohomology, duality and local-global obstructions: Lecture-case 20 — Spectral-sequence avoidance by clean maps |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Spectral-sequence avoidance by clean maps” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes spectral-sequence avoidance by clean maps natural, computable, functorial, and stable under passage between local and global fields? |
S09 S12 S27 |
| 71 | Arithmetic meaning of H^1 Galois cohomology, duality and local-global obstructions: Lecture-case 21 — Arithmetic meaning of H^1 |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Arithmetic meaning of H^1” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes arithmetic meaning of h^1 natural, computable, functorial, and stable under passage between local and global fields? |
S11 S17 S08 |
| 72 | Arithmetic meaning of H^2 Galois cohomology, duality and local-global obstructions: Lecture-case 22 — Arithmetic meaning of H^2 |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Arithmetic meaning of H^2” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes arithmetic meaning of h^2 natural, computable, functorial, and stable under passage between local and global fields? |
S12 S27 S09 |
| 73 | Vanishing as reciprocity Galois cohomology, duality and local-global obstructions: Lecture-case 23 — Vanishing as reciprocity |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Vanishing as reciprocity” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes vanishing as reciprocity natural, computable, functorial, and stable under passage between local and global fields? |
S17 S08 S11 |
| 74 | Finite-level approximation of global truth Galois cohomology, duality and local-global obstructions: Lecture-case 24 — Finite-level approximation of global truth |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Finite-level approximation of global truth” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finite-level approximation of global truth natural, computable, functorial, and stable under passage between local and global fields? |
S27 S09 S12 |
| 75 | Duality as classification method Galois cohomology, duality and local-global obstructions: Lecture-case 25 — Duality as classification method |
Galois cohomology, duality and local-global obstructions 1950s–1970s |
Read “Duality as classification method” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes duality as classification method natural, computable, functorial, and stable under passage between local and global fields? |
S08 S11 S17 |
| 76 | Tate modules of abelian varieties Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 01 — Tate modules of abelian varieties |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Tate modules of abelian varieties” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate modules of abelian varieties natural, computable, functorial, and stable under passage between local and global fields? |
S13 S15 S27 |
| 77 | Galois action on torsion points Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 02 — Galois action on torsion points |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Galois action on torsion points” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes galois action on torsion points natural, computable, functorial, and stable under passage between local and global fields? |
S14 S26 S32 |
| 78 | Endomorphisms through l-adic representations Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 03 — Endomorphisms through l-adic representations |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Endomorphisms through l-adic representations” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes endomorphisms through l-adic representations natural, computable, functorial, and stable under passage between local and global fields? |
S15 S27 S13 |
| 79 | The isogeny theorem over finite fields Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 04 — The isogeny theorem over finite fields |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “The isogeny theorem over finite fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes the isogeny theorem over finite fields natural, computable, functorial, and stable under passage between local and global fields? |
S26 S32 S14 |
| 80 | Frobenius as arithmetic fingerprint Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 05 — Frobenius as arithmetic fingerprint |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Frobenius as arithmetic fingerprint” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes frobenius as arithmetic fingerprint natural, computable, functorial, and stable under passage between local and global fields? |
S27 S13 S15 |
| 81 | Honda-Tate classification route Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 06 — Honda-Tate classification route |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Honda-Tate classification route” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes honda-tate classification route natural, computable, functorial, and stable under passage between local and global fields? |
S32 S14 S26 |
| 82 | Weil numbers and isogeny classes Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 07 — Weil numbers and isogeny classes |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Weil numbers and isogeny classes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes weil numbers and isogeny classes natural, computable, functorial, and stable under passage between local and global fields? |
S13 S15 S27 |
| 83 | Semisimplicity questions Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 08 — Semisimplicity questions |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Semisimplicity questions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes semisimplicity questions natural, computable, functorial, and stable under passage between local and global fields? |
S14 S26 S32 |
| 84 | Hom comparison theorem Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 09 — Hom comparison theorem |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Hom comparison theorem” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes hom comparison theorem natural, computable, functorial, and stable under passage between local and global fields? |
S15 S27 S13 |
| 85 | Tate classes in divisor theory Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 10 — Tate classes in divisor theory |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Tate classes in divisor theory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate classes in divisor theory natural, computable, functorial, and stable under passage between local and global fields? |
S26 S32 S14 |
| 86 | Abelian varieties over finite fields Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 11 — Abelian varieties over finite fields |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Abelian varieties over finite fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes abelian varieties over finite fields natural, computable, functorial, and stable under passage between local and global fields? |
S27 S13 S15 |
| 87 | Picard and Néron-Severi groups Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 12 — Picard and Néron-Severi groups |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Picard and Néron-Severi groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes picard and néron-severi groups natural, computable, functorial, and stable under passage between local and global fields? |
S32 S14 S26 |
| 88 | Finiteness through representation theory Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 13 — Finiteness through representation theory |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Finiteness through representation theory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finiteness through representation theory natural, computable, functorial, and stable under passage between local and global fields? |
S13 S15 S27 |
| 89 | Frobenius polynomial as classifier Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 14 — Frobenius polynomial as classifier |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Frobenius polynomial as classifier” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes frobenius polynomial as classifier natural, computable, functorial, and stable under passage between local and global fields? |
S14 S26 S32 |
| 90 | Isogeny categories Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 15 — Isogeny categories |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Isogeny categories” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes isogeny categories natural, computable, functorial, and stable under passage between local and global fields? |
S15 S27 S13 |
| 91 | Polarizations and dual abelian varieties Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 16 — Polarizations and dual abelian varieties |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Polarizations and dual abelian varieties” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes polarizations and dual abelian varieties natural, computable, functorial, and stable under passage between local and global fields? |
S26 S32 S14 |
| 92 | Algebraic cycles and fixed classes Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 17 — Algebraic cycles and fixed classes |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Algebraic cycles and fixed classes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algebraic cycles and fixed classes natural, computable, functorial, and stable under passage between local and global fields? |
S27 S13 S15 |
| 93 | Tate conjecture for abelian varieties Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 18 — Tate conjecture for abelian varieties |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Tate conjecture for abelian varieties” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate conjecture for abelian varieties natural, computable, functorial, and stable under passage between local and global fields? |
S32 S14 S26 |
| 94 | Reduction mod p as method Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 19 — Reduction mod p as method |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Reduction mod p as method” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes reduction mod p as method natural, computable, functorial, and stable under passage between local and global fields? |
S13 S15 S27 |
| 95 | l-adic compatibility across primes Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 20 — l-adic compatibility across primes |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “l-adic compatibility across primes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes l-adic compatibility across primes natural, computable, functorial, and stable under passage between local and global fields? |
S14 S26 S32 |
| 96 | Finite-field examples before motives Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 21 — Finite-field examples before motives |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Finite-field examples before motives” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finite-field examples before motives natural, computable, functorial, and stable under passage between local and global fields? |
S15 S27 S13 |
| 97 | Endomorphism algebra recovery Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 22 — Endomorphism algebra recovery |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Endomorphism algebra recovery” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes endomorphism algebra recovery natural, computable, functorial, and stable under passage between local and global fields? |
S26 S32 S14 |
| 98 | Ordinary versus supersingular examples Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 23 — Ordinary versus supersingular examples |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Ordinary versus supersingular examples” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ordinary versus supersingular examples natural, computable, functorial, and stable under passage between local and global fields? |
S27 S13 S15 |
| 99 | Correspondence with Honda theory Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 24 — Correspondence with Honda theory |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Correspondence with Honda theory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes correspondence with honda theory natural, computable, functorial, and stable under passage between local and global fields? |
S32 S14 S26 |
| 100 | Isogeny as linearized geometry Abelian varieties, Tate modules and finite-field isogeny: Lecture-case 25 — Isogeny as linearized geometry |
Abelian varieties, Tate modules and finite-field isogeny 1960s–1970s |
Read “Isogeny as linearized geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes isogeny as linearized geometry natural, computable, functorial, and stable under passage between local and global fields? |
S13 S15 S27 |
| 101 | Canonical height as quadratic form Elliptic curves, heights, descent and BSD interface: Lecture-case 01 — Canonical height as quadratic form |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Canonical height as quadratic form” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes canonical height as quadratic form natural, computable, functorial, and stable under passage between local and global fields? |
S16 S18 S32 |
| 102 | Néron-Tate pairing on Mordell-Weil groups Elliptic curves, heights, descent and BSD interface: Lecture-case 02 — Néron-Tate pairing on Mordell-Weil groups |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Néron-Tate pairing on Mordell-Weil groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes néron-tate pairing on mordell-weil groups natural, computable, functorial, and stable under passage between local and global fields? |
S17 S11 S25 |
| 103 | Local heights and global normalization Elliptic curves, heights, descent and BSD interface: Lecture-case 03 — Local heights and global normalization |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Local heights and global normalization” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local heights and global normalization natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S16 |
| 104 | Regulators as arithmetic volume Elliptic curves, heights, descent and BSD interface: Lecture-case 04 — Regulators as arithmetic volume |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Regulators as arithmetic volume” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes regulators as arithmetic volume natural, computable, functorial, and stable under passage between local and global fields? |
S11 S25 S17 |
| 105 | Tate algorithm for elliptic curves Elliptic curves, heights, descent and BSD interface: Lecture-case 05 — Tate algorithm for elliptic curves |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Tate algorithm for elliptic curves” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate algorithm for elliptic curves natural, computable, functorial, and stable under passage between local and global fields? |
S32 S16 S18 |
| 106 | Minimal Weierstrass models Elliptic curves, heights, descent and BSD interface: Lecture-case 06 — Minimal Weierstrass models |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Minimal Weierstrass models” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes minimal weierstrass models natural, computable, functorial, and stable under passage between local and global fields? |
S25 S17 S11 |
| 107 | Kodaira reduction types Elliptic curves, heights, descent and BSD interface: Lecture-case 07 — Kodaira reduction types |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Kodaira reduction types” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes kodaira reduction types natural, computable, functorial, and stable under passage between local and global fields? |
S16 S18 S32 |
| 108 | Conductor exponent extraction Elliptic curves, heights, descent and BSD interface: Lecture-case 08 — Conductor exponent extraction |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Conductor exponent extraction” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes conductor exponent extraction natural, computable, functorial, and stable under passage between local and global fields? |
S17 S11 S25 |
| 109 | Component groups and Tamagawa numbers Elliptic curves, heights, descent and BSD interface: Lecture-case 09 — Component groups and Tamagawa numbers |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Component groups and Tamagawa numbers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes component groups and tamagawa numbers natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S16 |
| 110 | Descent and Selmer exact sequence Elliptic curves, heights, descent and BSD interface: Lecture-case 10 — Descent and Selmer exact sequence |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Descent and Selmer exact sequence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes descent and selmer exact sequence natural, computable, functorial, and stable under passage between local and global fields? |
S11 S25 S17 |
| 111 | Sha as missing global information Elliptic curves, heights, descent and BSD interface: Lecture-case 11 — Sha as missing global information |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Sha as missing global information” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes sha as missing global information natural, computable, functorial, and stable under passage between local and global fields? |
S32 S16 S18 |
| 112 | Birch and Swinnerton-Dyer interface Elliptic curves, heights, descent and BSD interface: Lecture-case 12 — Birch and Swinnerton-Dyer interface |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Birch and Swinnerton-Dyer interface” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes birch and swinnerton-dyer interface natural, computable, functorial, and stable under passage between local and global fields? |
S25 S17 S11 |
| 113 | Elliptic curve local computations Elliptic curves, heights, descent and BSD interface: Lecture-case 13 — Elliptic curve local computations |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Elliptic curve local computations” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes elliptic curve local computations natural, computable, functorial, and stable under passage between local and global fields? |
S16 S18 S32 |
| 114 | Bad primes as information-rich places Elliptic curves, heights, descent and BSD interface: Lecture-case 14 — Bad primes as information-rich places |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Bad primes as information-rich places” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes bad primes as information-rich places natural, computable, functorial, and stable under passage between local and global fields? |
S17 S11 S25 |
| 115 | Height pairing in examples Elliptic curves, heights, descent and BSD interface: Lecture-case 15 — Height pairing in examples |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Height pairing in examples” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes height pairing in examples natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S16 |
| 116 | Local solubility filters Elliptic curves, heights, descent and BSD interface: Lecture-case 16 — Local solubility filters |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Local solubility filters” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local solubility filters natural, computable, functorial, and stable under passage between local and global fields? |
S11 S25 S17 |
| 117 | Mordell-Weil lattice formation Elliptic curves, heights, descent and BSD interface: Lecture-case 17 — Mordell-Weil lattice formation |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Mordell-Weil lattice formation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes mordell-weil lattice formation natural, computable, functorial, and stable under passage between local and global fields? |
S32 S16 S18 |
| 118 | Explicit reduction at small primes Elliptic curves, heights, descent and BSD interface: Lecture-case 18 — Explicit reduction at small primes |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Explicit reduction at small primes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes explicit reduction at small primes natural, computable, functorial, and stable under passage between local and global fields? |
S25 S17 S11 |
| 119 | q-expansion intuition for degenerating curves Elliptic curves, heights, descent and BSD interface: Lecture-case 19 — q-expansion intuition for degenerating curves |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “q-expansion intuition for degenerating curves” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes q-expansion intuition for degenerating curves natural, computable, functorial, and stable under passage between local and global fields? |
S16 S18 S32 |
| 120 | Tate-Shafarevich group notation as program Elliptic curves, heights, descent and BSD interface: Lecture-case 20 — Tate-Shafarevich group notation as program |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Tate-Shafarevich group notation as program” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate-shafarevich group notation as program natural, computable, functorial, and stable under passage between local and global fields? |
S17 S11 S25 |
| 121 | Rank computation heuristics Elliptic curves, heights, descent and BSD interface: Lecture-case 21 — Rank computation heuristics |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Rank computation heuristics” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes rank computation heuristics natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S16 |
| 122 | Regulator and leading coefficient Elliptic curves, heights, descent and BSD interface: Lecture-case 22 — Regulator and leading coefficient |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Regulator and leading coefficient” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes regulator and leading coefficient natural, computable, functorial, and stable under passage between local and global fields? |
S11 S25 S17 |
| 123 | Finiteness conjectures around Sha Elliptic curves, heights, descent and BSD interface: Lecture-case 23 — Finiteness conjectures around Sha |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Finiteness conjectures around Sha” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finiteness conjectures around sha natural, computable, functorial, and stable under passage between local and global fields? |
S32 S16 S18 |
| 124 | Elliptic curves as arithmetic laboratory Elliptic curves, heights, descent and BSD interface: Lecture-case 24 — Elliptic curves as arithmetic laboratory |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Elliptic curves as arithmetic laboratory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes elliptic curves as arithmetic laboratory natural, computable, functorial, and stable under passage between local and global fields? |
S25 S17 S11 |
| 125 | Algorithmic arithmetic made canonical Elliptic curves, heights, descent and BSD interface: Lecture-case 25 — Algorithmic arithmetic made canonical |
Elliptic curves, heights, descent and BSD interface 1960s–1980s |
Read “Algorithmic arithmetic made canonical” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algorithmic arithmetic made canonical natural, computable, functorial, and stable under passage between local and global fields? |
S16 S18 S32 |
| 126 | Formal group laws over local rings Formal groups, Lubin-Tate theory and local fields: Lecture-case 01 — Formal group laws over local rings |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Formal group laws over local rings” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes formal group laws over local rings natural, computable, functorial, and stable under passage between local and global fields? |
S19 S10 S24 |
| 127 | Uniformizer action on formal modules Formal groups, Lubin-Tate theory and local fields: Lecture-case 02 — Uniformizer action on formal modules |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Uniformizer action on formal modules” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes uniformizer action on formal modules natural, computable, functorial, and stable under passage between local and global fields? |
S20 S12 S32 |
| 128 | Lubin-Tate extensions from torsion Formal groups, Lubin-Tate theory and local fields: Lecture-case 03 — Lubin-Tate extensions from torsion |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Lubin-Tate extensions from torsion” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes lubin-tate extensions from torsion natural, computable, functorial, and stable under passage between local and global fields? |
S10 S24 S19 |
| 129 | Local reciprocity via formal groups Formal groups, Lubin-Tate theory and local fields: Lecture-case 04 — Local reciprocity via formal groups |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Local reciprocity via formal groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local reciprocity via formal groups natural, computable, functorial, and stable under passage between local and global fields? |
S12 S32 S20 |
| 130 | Endomorphism rings and local fields Formal groups, Lubin-Tate theory and local fields: Lecture-case 05 — Endomorphism rings and local fields |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Endomorphism rings and local fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes endomorphism rings and local fields natural, computable, functorial, and stable under passage between local and global fields? |
S24 S19 S10 |
| 131 | Power series as field generators Formal groups, Lubin-Tate theory and local fields: Lecture-case 06 — Power series as field generators |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Power series as field generators” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes power series as field generators natural, computable, functorial, and stable under passage between local and global fields? |
S32 S20 S12 |
| 132 | Ramification in formal group towers Formal groups, Lubin-Tate theory and local fields: Lecture-case 07 — Ramification in formal group towers |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Ramification in formal group towers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ramification in formal group towers natural, computable, functorial, and stable under passage between local and global fields? |
S19 S10 S24 |
| 133 | Torsion points as abelian extensions Formal groups, Lubin-Tate theory and local fields: Lecture-case 08 — Torsion points as abelian extensions |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Torsion points as abelian extensions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes torsion points as abelian extensions natural, computable, functorial, and stable under passage between local and global fields? |
S20 S12 S32 |
| 134 | Local class field theory without global detour Formal groups, Lubin-Tate theory and local fields: Lecture-case 09 — Local class field theory without global detour |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Local class field theory without global detour” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local class field theory without global detour natural, computable, functorial, and stable under passage between local and global fields? |
S10 S24 S19 |
| 135 | Choice of uniformizer and canonical output Formal groups, Lubin-Tate theory and local fields: Lecture-case 10 — Choice of uniformizer and canonical output |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Choice of uniformizer and canonical output” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes choice of uniformizer and canonical output natural, computable, functorial, and stable under passage between local and global fields? |
S12 S32 S20 |
| 136 | Deformation of formal modules Formal groups, Lubin-Tate theory and local fields: Lecture-case 11 — Deformation of formal modules |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Deformation of formal modules” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes deformation of formal modules natural, computable, functorial, and stable under passage between local and global fields? |
S24 S19 S10 |
| 137 | Height of formal groups Formal groups, Lubin-Tate theory and local fields: Lecture-case 12 — Height of formal groups |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Height of formal groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes height of formal groups natural, computable, functorial, and stable under passage between local and global fields? |
S32 S20 S12 |
| 138 | One-dimensional formal modules Formal groups, Lubin-Tate theory and local fields: Lecture-case 13 — One-dimensional formal modules |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “One-dimensional formal modules” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes one-dimensional formal modules natural, computable, functorial, and stable under passage between local and global fields? |
S19 S10 S24 |
| 139 | Universal deformation rings Formal groups, Lubin-Tate theory and local fields: Lecture-case 14 — Universal deformation rings |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Universal deformation rings” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes universal deformation rings natural, computable, functorial, and stable under passage between local and global fields? |
S20 S12 S32 |
| 140 | Division points and reciprocity Formal groups, Lubin-Tate theory and local fields: Lecture-case 15 — Division points and reciprocity |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Division points and reciprocity” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes division points and reciprocity natural, computable, functorial, and stable under passage between local and global fields? |
S10 S24 S19 |
| 141 | Local Galois action on formal torsion Formal groups, Lubin-Tate theory and local fields: Lecture-case 16 — Local Galois action on formal torsion |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Local Galois action on formal torsion” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local galois action on formal torsion natural, computable, functorial, and stable under passage between local and global fields? |
S12 S32 S20 |
| 142 | Norm compatibility in towers Formal groups, Lubin-Tate theory and local fields: Lecture-case 17 — Norm compatibility in towers |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Norm compatibility in towers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes norm compatibility in towers natural, computable, functorial, and stable under passage between local and global fields? |
S24 S19 S10 |
| 143 | Explicit examples over p-adic fields Formal groups, Lubin-Tate theory and local fields: Lecture-case 18 — Explicit examples over p-adic fields |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Explicit examples over p-adic fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes explicit examples over p-adic fields natural, computable, functorial, and stable under passage between local and global fields? |
S32 S20 S12 |
| 144 | Ramification filtration evidence Formal groups, Lubin-Tate theory and local fields: Lecture-case 19 — Ramification filtration evidence |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Ramification filtration evidence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ramification filtration evidence natural, computable, functorial, and stable under passage between local and global fields? |
S19 S10 S24 |
| 145 | Lubin-Tate character Formal groups, Lubin-Tate theory and local fields: Lecture-case 20 — Lubin-Tate character |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Lubin-Tate character” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes lubin-tate character natural, computable, functorial, and stable under passage between local and global fields? |
S20 S12 S32 |
| 146 | Formal logarithm and exponential Formal groups, Lubin-Tate theory and local fields: Lecture-case 21 — Formal logarithm and exponential |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Formal logarithm and exponential” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes formal logarithm and exponential natural, computable, functorial, and stable under passage between local and global fields? |
S10 S24 S19 |
| 147 | Nonarchimedean analytic intuition Formal groups, Lubin-Tate theory and local fields: Lecture-case 22 — Nonarchimedean analytic intuition |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Nonarchimedean analytic intuition” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes nonarchimedean analytic intuition natural, computable, functorial, and stable under passage between local and global fields? |
S12 S32 S20 |
| 148 | Formal groups as local machines Formal groups, Lubin-Tate theory and local fields: Lecture-case 23 — Formal groups as local machines |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Formal groups as local machines” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes formal groups as local machines natural, computable, functorial, and stable under passage between local and global fields? |
S24 S19 S10 |
| 149 | Bridge to p-divisible groups Formal groups, Lubin-Tate theory and local fields: Lecture-case 24 — Bridge to p-divisible groups |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Bridge to p-divisible groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes bridge to p-divisible groups natural, computable, functorial, and stable under passage between local and global fields? |
S32 S20 S12 |
| 150 | Local theory as constructive arithmetic Formal groups, Lubin-Tate theory and local fields: Lecture-case 25 — Local theory as constructive arithmetic |
Formal groups, Lubin-Tate theory and local fields 1960s–1970s |
Read “Local theory as constructive arithmetic” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local theory as constructive arithmetic natural, computable, functorial, and stable under passage between local and global fields? |
S19 S10 S24 |
| 151 | Affinoid algebras as coordinate rings Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 01 — Affinoid algebras as coordinate rings |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Affinoid algebras as coordinate rings” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes affinoid algebras as coordinate rings natural, computable, functorial, and stable under passage between local and global fields? |
S21 S23 S28 |
| 152 | Admissible coverings replace naive topology Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 02 — Admissible coverings replace naive topology |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Admissible coverings replace naive topology” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes admissible coverings replace naive topology natural, computable, functorial, and stable under passage between local and global fields? |
S22 S24 S32 |
| 153 | Rigid analytic spaces as p-adic repair Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 03 — Rigid analytic spaces as p-adic repair |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Rigid analytic spaces as p-adic repair” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes rigid analytic spaces as p-adic repair natural, computable, functorial, and stable under passage between local and global fields? |
S23 S28 S21 |
| 154 | Tate acyclicity as structural theorem Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 04 — Tate acyclicity as structural theorem |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Tate acyclicity as structural theorem” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate acyclicity as structural theorem natural, computable, functorial, and stable under passage between local and global fields? |
S24 S32 S22 |
| 155 | The Tate curve and multiplicative uniformization Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 05 — The Tate curve and multiplicative uniformization |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “The Tate curve and multiplicative uniformization” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes the tate curve and multiplicative uniformization natural, computable, functorial, and stable under passage between local and global fields? |
S28 S21 S23 |
| 156 | q-parameter as arithmetic geometry Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 06 — q-parameter as arithmetic geometry |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “q-parameter as arithmetic geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes q-parameter as arithmetic geometry natural, computable, functorial, and stable under passage between local and global fields? |
S32 S22 S24 |
| 157 | p-adic annuli and discs Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 07 — p-adic annuli and discs |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “p-adic annuli and discs” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic annuli and discs natural, computable, functorial, and stable under passage between local and global fields? |
S21 S23 S28 |
| 158 | Nonarchimedean maximum principles Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 08 — Nonarchimedean maximum principles |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Nonarchimedean maximum principles” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes nonarchimedean maximum principles natural, computable, functorial, and stable under passage between local and global fields? |
S22 S24 S32 |
| 159 | Analytic continuation in rigid geometry Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 09 — Analytic continuation in rigid geometry |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Analytic continuation in rigid geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes analytic continuation in rigid geometry natural, computable, functorial, and stable under passage between local and global fields? |
S23 S28 S21 |
| 160 | Meromorphic functions on rigid spaces Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 10 — Meromorphic functions on rigid spaces |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Meromorphic functions on rigid spaces” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes meromorphic functions on rigid spaces natural, computable, functorial, and stable under passage between local and global fields? |
S24 S32 S22 |
| 161 | Uniformization of elliptic curves Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 11 — Uniformization of elliptic curves |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Uniformization of elliptic curves” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes uniformization of elliptic curves natural, computable, functorial, and stable under passage between local and global fields? |
S28 S21 S23 |
| 162 | Degeneration as quotient geometry Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 12 — Degeneration as quotient geometry |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Degeneration as quotient geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes degeneration as quotient geometry natural, computable, functorial, and stable under passage between local and global fields? |
S32 S22 S24 |
| 163 | Rigid spaces as language converter Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 13 — Rigid spaces as language converter |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Rigid spaces as language converter” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes rigid spaces as language converter natural, computable, functorial, and stable under passage between local and global fields? |
S21 S23 S28 |
| 164 | Comparison with complex analytic intuition Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 14 — Comparison with complex analytic intuition |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Comparison with complex analytic intuition” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes comparison with complex analytic intuition natural, computable, functorial, and stable under passage between local and global fields? |
S22 S24 S32 |
| 165 | Covering data and gluing discipline Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 15 — Covering data and gluing discipline |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Covering data and gluing discipline” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes covering data and gluing discipline natural, computable, functorial, and stable under passage between local and global fields? |
S23 S28 S21 |
| 166 | p-adic analytic families Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 16 — p-adic analytic families |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “p-adic analytic families” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic analytic families natural, computable, functorial, and stable under passage between local and global fields? |
S24 S32 S22 |
| 167 | Formal schemes and generic fibers Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 17 — Formal schemes and generic fibers |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Formal schemes and generic fibers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes formal schemes and generic fibers natural, computable, functorial, and stable under passage between local and global fields? |
S28 S21 S23 |
| 168 | Analytic functions with Gauss norms Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 18 — Analytic functions with Gauss norms |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Analytic functions with Gauss norms” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes analytic functions with gauss norms natural, computable, functorial, and stable under passage between local and global fields? |
S32 S22 S24 |
| 169 | Tate algebras as basic objects Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 19 — Tate algebras as basic objects |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Tate algebras as basic objects” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate algebras as basic objects natural, computable, functorial, and stable under passage between local and global fields? |
S21 S23 S28 |
| 170 | Bad reduction through rigid eyes Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 20 — Bad reduction through rigid eyes |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Bad reduction through rigid eyes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes bad reduction through rigid eyes natural, computable, functorial, and stable under passage between local and global fields? |
S22 S24 S32 |
| 171 | p-adic periods as analytic evidence Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 21 — p-adic periods as analytic evidence |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “p-adic periods as analytic evidence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic periods as analytic evidence natural, computable, functorial, and stable under passage between local and global fields? |
S23 S28 S21 |
| 172 | From curves to analytic spaces Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 22 — From curves to analytic spaces |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “From curves to analytic spaces” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes from curves to analytic spaces natural, computable, functorial, and stable under passage between local and global fields? |
S24 S32 S22 |
| 173 | Rigid methods in arithmetic geometry Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 23 — Rigid methods in arithmetic geometry |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Rigid methods in arithmetic geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes rigid methods in arithmetic geometry natural, computable, functorial, and stable under passage between local and global fields? |
S28 S21 S23 |
| 174 | Examples that force admissibility Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 24 — Examples that force admissibility |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “Examples that force admissibility” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes examples that force admissibility natural, computable, functorial, and stable under passage between local and global fields? |
S32 S22 S24 |
| 175 | A new geometry from local failure Rigid analytic geometry, Tate curve and p-adic spaces: Lecture-case 25 — A new geometry from local failure |
Rigid analytic geometry, Tate curve and p-adic spaces 1960s–1980s |
Read “A new geometry from local failure” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes a new geometry from local failure natural, computable, functorial, and stable under passage between local and global fields? |
S21 S23 S28 |
| 176 | Hodge-Tate decomposition prototype Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 01 — Hodge-Tate decomposition prototype |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Hodge-Tate decomposition prototype” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes hodge-tate decomposition prototype natural, computable, functorial, and stable under passage between local and global fields? |
S23 S26 S19 |
| 177 | p-adic étale cohomology and differentials Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 02 — p-adic étale cohomology and differentials |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “p-adic étale cohomology and differentials” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic étale cohomology and differentials natural, computable, functorial, and stable under passage between local and global fields? |
S24 S27 S28 |
| 178 | Galois representations with Hodge-Tate weights Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 03 — Galois representations with Hodge-Tate weights |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Galois representations with Hodge-Tate weights” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes galois representations with hodge-tate weights natural, computable, functorial, and stable under passage between local and global fields? |
S26 S19 S23 |
| 179 | Tate twists as weight bookkeeping Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 04 — Tate twists as weight bookkeeping |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Tate twists as weight bookkeeping” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate twists as weight bookkeeping natural, computable, functorial, and stable under passage between local and global fields? |
S27 S28 S24 |
| 180 | p-divisible groups as stable torsion Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 05 — p-divisible groups as stable torsion |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “p-divisible groups as stable torsion” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-divisible groups as stable torsion natural, computable, functorial, and stable under passage between local and global fields? |
S19 S23 S26 |
| 181 | Barsotti-Tate groups and deformation Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 06 — Barsotti-Tate groups and deformation |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Barsotti-Tate groups and deformation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes barsotti-tate groups and deformation natural, computable, functorial, and stable under passage between local and global fields? |
S28 S24 S27 |
| 182 | Connected-etale decomposition Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 07 — Connected-etale decomposition |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Connected-etale decomposition” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes connected-etale decomposition natural, computable, functorial, and stable under passage between local and global fields? |
S23 S26 S19 |
| 183 | Newton slopes and height data Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 08 — Newton slopes and height data |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Newton slopes and height data” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes newton slopes and height data natural, computable, functorial, and stable under passage between local and global fields? |
S24 S27 S28 |
| 184 | Serre-Tate deformation philosophy Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 09 — Serre-Tate deformation philosophy |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Serre-Tate deformation philosophy” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes serre-tate deformation philosophy natural, computable, functorial, and stable under passage between local and global fields? |
S26 S19 S23 |
| 185 | Ordinary abelian varieties through p-divisible groups Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 10 — Ordinary abelian varieties through p-divisible groups |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Ordinary abelian varieties through p-divisible groups” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes ordinary abelian varieties through p-divisible groups natural, computable, functorial, and stable under passage between local and global fields? |
S27 S28 S24 |
| 186 | Comparison maps and period fields Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 11 — Comparison maps and period fields |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Comparison maps and period fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes comparison maps and period fields natural, computable, functorial, and stable under passage between local and global fields? |
S19 S23 S26 |
| 187 | C_p as comparison field Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 12 — C_p as comparison field |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “C_p as comparison field” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes c_p as comparison field natural, computable, functorial, and stable under passage between local and global fields? |
S28 S24 S27 |
| 188 | Weights as arithmetic fingerprints Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 13 — Weights as arithmetic fingerprints |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Weights as arithmetic fingerprints” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes weights as arithmetic fingerprints natural, computable, functorial, and stable under passage between local and global fields? |
S23 S26 S19 |
| 189 | Tangent spaces and formal deformation Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 14 — Tangent spaces and formal deformation |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Tangent spaces and formal deformation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tangent spaces and formal deformation natural, computable, functorial, and stable under passage between local and global fields? |
S24 S27 S28 |
| 190 | Crystalline intuition before full theory Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 15 — Crystalline intuition before full theory |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Crystalline intuition before full theory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes crystalline intuition before full theory natural, computable, functorial, and stable under passage between local and global fields? |
S26 S19 S23 |
| 191 | Local Galois representations from geometry Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 16 — Local Galois representations from geometry |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Local Galois representations from geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local galois representations from geometry natural, computable, functorial, and stable under passage between local and global fields? |
S27 S28 S24 |
| 192 | p-adic Hodge theory seeds Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 17 — p-adic Hodge theory seeds |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “p-adic Hodge theory seeds” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic hodge theory seeds natural, computable, functorial, and stable under passage between local and global fields? |
S19 S23 S26 |
| 193 | Tate modules at p Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 18 — Tate modules at p |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Tate modules at p” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate modules at p natural, computable, functorial, and stable under passage between local and global fields? |
S28 S24 S27 |
| 194 | Filtrations and graded pieces Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 19 — Filtrations and graded pieces |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Filtrations and graded pieces” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes filtrations and graded pieces natural, computable, functorial, and stable under passage between local and global fields? |
S23 S26 S19 |
| 195 | Formal neighborhoods in moduli Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 20 — Formal neighborhoods in moduli |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Formal neighborhoods in moduli” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes formal neighborhoods in moduli natural, computable, functorial, and stable under passage between local and global fields? |
S24 S27 S28 |
| 196 | Deformation parameters from torsion Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 21 — Deformation parameters from torsion |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Deformation parameters from torsion” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes deformation parameters from torsion natural, computable, functorial, and stable under passage between local and global fields? |
S26 S19 S23 |
| 197 | Slope data as stratification Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 22 — Slope data as stratification |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Slope data as stratification” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes slope data as stratification natural, computable, functorial, and stable under passage between local and global fields? |
S27 S28 S24 |
| 198 | Hodge data through Galois action Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 23 — Hodge data through Galois action |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Hodge data through Galois action” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes hodge data through galois action natural, computable, functorial, and stable under passage between local and global fields? |
S19 S23 S26 |
| 199 | Tate twist as notation discipline Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 24 — Tate twist as notation discipline |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “Tate twist as notation discipline” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate twist as notation discipline natural, computable, functorial, and stable under passage between local and global fields? |
S28 S24 S27 |
| 200 | p-adic comparison as reformulation Hodge-Tate theory, p-divisible groups and p-adic comparison: Lecture-case 25 — p-adic comparison as reformulation |
Hodge-Tate theory, p-divisible groups and p-adic comparison 1960s–1990s |
Read “p-adic comparison as reformulation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes p-adic comparison as reformulation natural, computable, functorial, and stable under passage between local and global fields? |
S23 S26 S19 |
| 201 | Tate conjecture as cycle-class question Tate conjecture, motives and algebraic cycles: Lecture-case 01 — Tate conjecture as cycle-class question |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Tate conjecture as cycle-class question” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate conjecture as cycle-class question natural, computable, functorial, and stable under passage between local and global fields? |
S25 S27 S15 |
| 202 | Algebraic cycles versus fixed l-adic classes Tate conjecture, motives and algebraic cycles: Lecture-case 02 — Algebraic cycles versus fixed l-adic classes |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Algebraic cycles versus fixed l-adic classes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algebraic cycles versus fixed l-adic classes natural, computable, functorial, and stable under passage between local and global fields? |
S26 S14 S28 |
| 203 | Galois-invariant cohomology as test space Tate conjecture, motives and algebraic cycles: Lecture-case 03 — Galois-invariant cohomology as test space |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Galois-invariant cohomology as test space” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes galois-invariant cohomology as test space natural, computable, functorial, and stable under passage between local and global fields? |
S27 S15 S25 |
| 204 | Divisors over finite fields Tate conjecture, motives and algebraic cycles: Lecture-case 04 — Divisors over finite fields |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Divisors over finite fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes divisors over finite fields natural, computable, functorial, and stable under passage between local and global fields? |
S14 S28 S26 |
| 205 | Semisimplicity clause in conjectures Tate conjecture, motives and algebraic cycles: Lecture-case 05 — Semisimplicity clause in conjectures |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Semisimplicity clause in conjectures” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes semisimplicity clause in conjectures natural, computable, functorial, and stable under passage between local and global fields? |
S15 S25 S27 |
| 206 | Compatibility with isogeny theorems Tate conjecture, motives and algebraic cycles: Lecture-case 06 — Compatibility with isogeny theorems |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Compatibility with isogeny theorems” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes compatibility with isogeny theorems natural, computable, functorial, and stable under passage between local and global fields? |
S28 S26 S14 |
| 207 | Motivic dictionary behind cohomology Tate conjecture, motives and algebraic cycles: Lecture-case 07 — Motivic dictionary behind cohomology |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Motivic dictionary behind cohomology” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes motivic dictionary behind cohomology natural, computable, functorial, and stable under passage between local and global fields? |
S25 S27 S15 |
| 208 | Cycle classes in codimension r Tate conjecture, motives and algebraic cycles: Lecture-case 08 — Cycle classes in codimension r |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Cycle classes in codimension r” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cycle classes in codimension r natural, computable, functorial, and stable under passage between local and global fields? |
S26 S14 S28 |
| 209 | Comparison with Hodge conjecture Tate conjecture, motives and algebraic cycles: Lecture-case 09 — Comparison with Hodge conjecture |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Comparison with Hodge conjecture” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes comparison with hodge conjecture natural, computable, functorial, and stable under passage between local and global fields? |
S27 S15 S25 |
| 210 | Tate twists in fixed-space formulation Tate conjecture, motives and algebraic cycles: Lecture-case 10 — Tate twists in fixed-space formulation |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Tate twists in fixed-space formulation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate twists in fixed-space formulation natural, computable, functorial, and stable under passage between local and global fields? |
S14 S28 S26 |
| 211 | Frobenius eigenvalues and cycles Tate conjecture, motives and algebraic cycles: Lecture-case 11 — Frobenius eigenvalues and cycles |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Frobenius eigenvalues and cycles” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes frobenius eigenvalues and cycles natural, computable, functorial, and stable under passage between local and global fields? |
S15 S25 S27 |
| 212 | Zeta functions and cycle ranks Tate conjecture, motives and algebraic cycles: Lecture-case 12 — Zeta functions and cycle ranks |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Zeta functions and cycle ranks” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes zeta functions and cycle ranks natural, computable, functorial, and stable under passage between local and global fields? |
S28 S26 S14 |
| 213 | Abelian varieties as test class Tate conjecture, motives and algebraic cycles: Lecture-case 13 — Abelian varieties as test class |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Abelian varieties as test class” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes abelian varieties as test class natural, computable, functorial, and stable under passage between local and global fields? |
S25 S27 S15 |
| 214 | K3 surfaces as later proving ground Tate conjecture, motives and algebraic cycles: Lecture-case 14 — K3 surfaces as later proving ground |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “K3 surfaces as later proving ground” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes k3 surfaces as later proving ground natural, computable, functorial, and stable under passage between local and global fields? |
S26 S14 S28 |
| 215 | Endomorphism cycles Tate conjecture, motives and algebraic cycles: Lecture-case 15 — Endomorphism cycles |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Endomorphism cycles” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes endomorphism cycles natural, computable, functorial, and stable under passage between local and global fields? |
S27 S15 S25 |
| 216 | Numerical and homological equivalence Tate conjecture, motives and algebraic cycles: Lecture-case 16 — Numerical and homological equivalence |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Numerical and homological equivalence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes numerical and homological equivalence natural, computable, functorial, and stable under passage between local and global fields? |
S14 S28 S26 |
| 217 | Motivic weights and l-adic realizations Tate conjecture, motives and algebraic cycles: Lecture-case 17 — Motivic weights and l-adic realizations |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Motivic weights and l-adic realizations” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes motivic weights and l-adic realizations natural, computable, functorial, and stable under passage between local and global fields? |
S15 S25 S27 |
| 218 | Conjectures as field architecture Tate conjecture, motives and algebraic cycles: Lecture-case 18 — Conjectures as field architecture |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Conjectures as field architecture” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes conjectures as field architecture natural, computable, functorial, and stable under passage between local and global fields? |
S28 S26 S14 |
| 219 | The role of finite fields Tate conjecture, motives and algebraic cycles: Lecture-case 19 — The role of finite fields |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “The role of finite fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes the role of finite fields natural, computable, functorial, and stable under passage between local and global fields? |
S25 S27 S15 |
| 220 | Galois representations of geometric origin Tate conjecture, motives and algebraic cycles: Lecture-case 20 — Galois representations of geometric origin |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Galois representations of geometric origin” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes galois representations of geometric origin natural, computable, functorial, and stable under passage between local and global fields? |
S26 S14 S28 |
| 221 | Algebraic classes from cohomology Tate conjecture, motives and algebraic cycles: Lecture-case 21 — Algebraic classes from cohomology |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Algebraic classes from cohomology” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algebraic classes from cohomology natural, computable, functorial, and stable under passage between local and global fields? |
S27 S15 S25 |
| 222 | Tate motives and pure pieces Tate conjecture, motives and algebraic cycles: Lecture-case 22 — Tate motives and pure pieces |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Tate motives and pure pieces” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate motives and pure pieces natural, computable, functorial, and stable under passage between local and global fields? |
S14 S28 S26 |
| 223 | Cycle map surjectivity problem Tate conjecture, motives and algebraic cycles: Lecture-case 23 — Cycle map surjectivity problem |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Cycle map surjectivity problem” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cycle map surjectivity problem natural, computable, functorial, and stable under passage between local and global fields? |
S15 S25 S27 |
| 224 | Rational equivalence and fixed classes Tate conjecture, motives and algebraic cycles: Lecture-case 24 — Rational equivalence and fixed classes |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Rational equivalence and fixed classes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes rational equivalence and fixed classes natural, computable, functorial, and stable under passage between local and global fields? |
S28 S26 S14 |
| 225 | Naming a conjecture that organizes decades Tate conjecture, motives and algebraic cycles: Lecture-case 25 — Naming a conjecture that organizes decades |
Tate conjecture, motives and algebraic cycles 1960s–2000s |
Read “Naming a conjecture that organizes decades” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes naming a conjecture that organizes decades natural, computable, functorial, and stable under passage between local and global fields? |
S25 S27 S15 |
| 226 | Residues on algebraic curves Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 01 — Residues on algebraic curves |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Residues on algebraic curves” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes residues on algebraic curves natural, computable, functorial, and stable under passage between local and global fields? |
S09 S16 S26 |
| 227 | Duality on arithmetic surfaces Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 02 — Duality on arithmetic surfaces |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Duality on arithmetic surfaces” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes duality on arithmetic surfaces natural, computable, functorial, and stable under passage between local and global fields? |
S12 S18 S32 |
| 228 | Explicit local terms in geometry Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 03 — Explicit local terms in geometry |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Explicit local terms in geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes explicit local terms in geometry natural, computable, functorial, and stable under passage between local and global fields? |
S16 S26 S09 |
| 229 | Divisors and principal parts Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 04 — Divisors and principal parts |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Divisors and principal parts” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes divisors and principal parts natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S12 |
| 230 | Trace and residue maps Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 05 — Trace and residue maps |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Trace and residue maps” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes trace and residue maps natural, computable, functorial, and stable under passage between local and global fields? |
S26 S09 S16 |
| 231 | Weil differentials and local computation Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 06 — Weil differentials and local computation |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Weil differentials and local computation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes weil differentials and local computation natural, computable, functorial, and stable under passage between local and global fields? |
S32 S12 S18 |
| 232 | Algebraic surfaces over arithmetic bases Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 07 — Algebraic surfaces over arithmetic bases |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Algebraic surfaces over arithmetic bases” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algebraic surfaces over arithmetic bases natural, computable, functorial, and stable under passage between local and global fields? |
S09 S16 S26 |
| 233 | Intersections and correction terms Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 08 — Intersections and correction terms |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Intersections and correction terms” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes intersections and correction terms natural, computable, functorial, and stable under passage between local and global fields? |
S12 S18 S32 |
| 234 | Local equations as arithmetic sensors Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 09 — Local equations as arithmetic sensors |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Local equations as arithmetic sensors” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local equations as arithmetic sensors natural, computable, functorial, and stable under passage between local and global fields? |
S16 S26 S09 |
| 235 | Blowups and minimal models Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 10 — Blowups and minimal models |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Blowups and minimal models” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes blowups and minimal models natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S12 |
| 236 | Component groups in families Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 11 — Component groups in families |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Component groups in families” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes component groups in families natural, computable, functorial, and stable under passage between local and global fields? |
S26 S09 S16 |
| 237 | Degeneration and special fibers Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 12 — Degeneration and special fibers |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Degeneration and special fibers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes degeneration and special fibers natural, computable, functorial, and stable under passage between local and global fields? |
S32 S12 S18 |
| 238 | Arithmetic surface exact sequences Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 13 — Arithmetic surface exact sequences |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Arithmetic surface exact sequences” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes arithmetic surface exact sequences natural, computable, functorial, and stable under passage between local and global fields? |
S09 S16 S26 |
| 239 | Cohomology with support Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 14 — Cohomology with support |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Cohomology with support” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes cohomology with support natural, computable, functorial, and stable under passage between local and global fields? |
S12 S18 S32 |
| 240 | Finite morphisms and trace Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 15 — Finite morphisms and trace |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Finite morphisms and trace” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes finite morphisms and trace natural, computable, functorial, and stable under passage between local and global fields? |
S16 S26 S09 |
| 241 | Compatibility of residues with base change Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 16 — Compatibility of residues with base change |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Compatibility of residues with base change” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes compatibility of residues with base change natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S12 |
| 242 | Explicit examples for abstract duality Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 17 — Explicit examples for abstract duality |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Explicit examples for abstract duality” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes explicit examples for abstract duality natural, computable, functorial, and stable under passage between local and global fields? |
S26 S09 S16 |
| 243 | Local rings and completions Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 18 — Local rings and completions |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Local rings and completions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local rings and completions natural, computable, functorial, and stable under passage between local and global fields? |
S32 S12 S18 |
| 244 | Valuations as geometric data Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 19 — Valuations as geometric data |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Valuations as geometric data” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes valuations as geometric data natural, computable, functorial, and stable under passage between local and global fields? |
S09 S16 S26 |
| 245 | Curves over local fields Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 20 — Curves over local fields |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Curves over local fields” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes curves over local fields natural, computable, functorial, and stable under passage between local and global fields? |
S12 S18 S32 |
| 246 | Intersection pairing normalization Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 21 — Intersection pairing normalization |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Intersection pairing normalization” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes intersection pairing normalization natural, computable, functorial, and stable under passage between local and global fields? |
S16 S26 S09 |
| 247 | Algebraic geometry as arithmetic language Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 22 — Algebraic geometry as arithmetic language |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Algebraic geometry as arithmetic language” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes algebraic geometry as arithmetic language natural, computable, functorial, and stable under passage between local and global fields? |
S18 S32 S12 |
| 248 | Bad fibers as invariants Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 23 — Bad fibers as invariants |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Bad fibers as invariants” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes bad fibers as invariants natural, computable, functorial, and stable under passage between local and global fields? |
S26 S09 S16 |
| 249 | Conductor terms from geometry Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 24 — Conductor terms from geometry |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Conductor terms from geometry” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes conductor terms from geometry natural, computable, functorial, and stable under passage between local and global fields? |
S32 S12 S18 |
| 250 | Local computations that globalize Arithmetic surfaces, residues and explicit algebraic geometry: Lecture-case 25 — Local computations that globalize |
Arithmetic surfaces, residues and explicit algebraic geometry 1950s–1990s |
Read “Local computations that globalize” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local computations that globalize natural, computable, functorial, and stable under passage between local and global fields? |
S09 S16 S26 |
| 251 | Letters with Serre as method trace Correspondence, comments, lectures and source-lineage: Lecture-case 01 — Letters with Serre as method trace |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Letters with Serre as method trace” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes letters with serre as method trace natural, computable, functorial, and stable under passage between local and global fields? |
S29 S31 S33 |
| 252 | Circulating manuscripts before publication Correspondence, comments, lectures and source-lineage: Lecture-case 02 — Circulating manuscripts before publication |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Circulating manuscripts before publication” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes circulating manuscripts before publication natural, computable, functorial, and stable under passage between local and global fields? |
S30 S32 S28 |
| 253 | Author comments on collected papers Correspondence, comments, lectures and source-lineage: Lecture-case 03 — Author comments on collected papers |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Author comments on collected papers” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes author comments on collected papers natural, computable, functorial, and stable under passage between local and global fields? |
S31 S33 S29 |
| 254 | Seminar notes as idea incubator Correspondence, comments, lectures and source-lineage: Lecture-case 04 — Seminar notes as idea incubator |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Seminar notes as idea incubator” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes seminar notes as idea incubator natural, computable, functorial, and stable under passage between local and global fields? |
S32 S28 S30 |
| 255 | Concise exposition as mathematical discipline Correspondence, comments, lectures and source-lineage: Lecture-case 05 — Concise exposition as mathematical discipline |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Concise exposition as mathematical discipline” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes concise exposition as mathematical discipline natural, computable, functorial, and stable under passage between local and global fields? |
S33 S29 S31 |
| 256 | Problem choice for students Correspondence, comments, lectures and source-lineage: Lecture-case 06 — Problem choice for students |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Problem choice for students” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes problem choice for students natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S32 |
| 257 | Mentorship through calibrated exercises Correspondence, comments, lectures and source-lineage: Lecture-case 07 — Mentorship through calibrated exercises |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Mentorship through calibrated exercises” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes mentorship through calibrated exercises natural, computable, functorial, and stable under passage between local and global fields? |
S29 S31 S33 |
| 258 | Lucid explanation after reflection Correspondence, comments, lectures and source-lineage: Lecture-case 08 — Lucid explanation after reflection |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Lucid explanation after reflection” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes lucid explanation after reflection natural, computable, functorial, and stable under passage between local and global fields? |
S30 S32 S28 |
| 259 | Mathematical friendship as research engine Correspondence, comments, lectures and source-lineage: Lecture-case 09 — Mathematical friendship as research engine |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Mathematical friendship as research engine” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes mathematical friendship as research engine natural, computable, functorial, and stable under passage between local and global fields? |
S31 S33 S29 |
| 260 | Perfection demanded in writing Correspondence, comments, lectures and source-lineage: Lecture-case 10 — Perfection demanded in writing |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Perfection demanded in writing” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes perfection demanded in writing natural, computable, functorial, and stable under passage between local and global fields? |
S32 S28 S30 |
| 261 | Delayed publication and long influence Correspondence, comments, lectures and source-lineage: Lecture-case 11 — Delayed publication and long influence |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Delayed publication and long influence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes delayed publication and long influence natural, computable, functorial, and stable under passage between local and global fields? |
S33 S29 S31 |
| 262 | Correspondence as preprint culture Correspondence, comments, lectures and source-lineage: Lecture-case 12 — Correspondence as preprint culture |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Correspondence as preprint culture” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes correspondence as preprint culture natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S32 |
| 263 | Examples in conversations Correspondence, comments, lectures and source-lineage: Lecture-case 13 — Examples in conversations |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Examples in conversations” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes examples in conversations natural, computable, functorial, and stable under passage between local and global fields? |
S29 S31 S33 |
| 264 | Sounding board for unfinished ideas Correspondence, comments, lectures and source-lineage: Lecture-case 14 — Sounding board for unfinished ideas |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Sounding board for unfinished ideas” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes sounding board for unfinished ideas natural, computable, functorial, and stable under passage between local and global fields? |
S30 S32 S28 |
| 265 | Source comments as reconstruction data Correspondence, comments, lectures and source-lineage: Lecture-case 15 — Source comments as reconstruction data |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Source comments as reconstruction data” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes source comments as reconstruction data natural, computable, functorial, and stable under passage between local and global fields? |
S31 S33 S29 |
| 266 | Advisor style and research autonomy Correspondence, comments, lectures and source-lineage: Lecture-case 16 — Advisor style and research autonomy |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Advisor style and research autonomy” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes advisor style and research autonomy natural, computable, functorial, and stable under passage between local and global fields? |
S32 S28 S30 |
| 267 | Letters as theorem infrastructure Correspondence, comments, lectures and source-lineage: Lecture-case 17 — Letters as theorem infrastructure |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Letters as theorem infrastructure” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes letters as theorem infrastructure natural, computable, functorial, and stable under passage between local and global fields? |
S33 S29 S31 |
| 268 | Seminars on class field theory Correspondence, comments, lectures and source-lineage: Lecture-case 18 — Seminars on class field theory |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Seminars on class field theory” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes seminars on class field theory natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S32 |
| 269 | Student problem ladders Correspondence, comments, lectures and source-lineage: Lecture-case 19 — Student problem ladders |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Student problem ladders” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes student problem ladders natural, computable, functorial, and stable under passage between local and global fields? |
S29 S31 S33 |
| 270 | Writing after the idea stabilizes Correspondence, comments, lectures and source-lineage: Lecture-case 20 — Writing after the idea stabilizes |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Writing after the idea stabilizes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes writing after the idea stabilizes natural, computable, functorial, and stable under passage between local and global fields? |
S30 S32 S28 |
| 271 | Generosity and mathematical exactness Correspondence, comments, lectures and source-lineage: Lecture-case 21 — Generosity and mathematical exactness |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Generosity and mathematical exactness” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes generosity and mathematical exactness natural, computable, functorial, and stable under passage between local and global fields? |
S31 S33 S29 |
| 272 | Understatement as style Correspondence, comments, lectures and source-lineage: Lecture-case 22 — Understatement as style |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Understatement as style” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes understatement as style natural, computable, functorial, and stable under passage between local and global fields? |
S32 S28 S30 |
| 273 | Collaborative maturation of conjectures Correspondence, comments, lectures and source-lineage: Lecture-case 23 — Collaborative maturation of conjectures |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Collaborative maturation of conjectures” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes collaborative maturation of conjectures natural, computable, functorial, and stable under passage between local and global fields? |
S33 S29 S31 |
| 274 | Work process visible in archives Correspondence, comments, lectures and source-lineage: Lecture-case 24 — Work process visible in archives |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Work process visible in archives” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes work process visible in archives natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S32 |
| 275 | Collected works as methodological record Correspondence, comments, lectures and source-lineage: Lecture-case 25 — Collected works as methodological record |
Correspondence, comments, lectures and source-lineage 1950s–2000s |
Read “Collected works as methodological record” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes collected works as methodological record natural, computable, functorial, and stable under passage between local and global fields? |
S29 S31 S33 |
| 276 | Reformulate before solving Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 01 — Reformulate before solving |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Reformulate before solving” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes reformulate before solving natural, computable, functorial, and stable under passage between local and global fields? |
S25 S29 S32 |
| 277 | Make the right object unavoidable Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 02 — Make the right object unavoidable |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Make the right object unavoidable” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes make the right object unavoidable natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S33 |
| 278 | Compress a field into a map Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 03 — Compress a field into a map |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Compress a field into a map” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes compress a field into a map natural, computable, functorial, and stable under passage between local and global fields? |
S29 S32 S25 |
| 279 | Name the invariant at the right moment Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 04 — Name the invariant at the right moment |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Name the invariant at the right moment” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes name the invariant at the right moment natural, computable, functorial, and stable under passage between local and global fields? |
S30 S33 S28 |
| 280 | Test the theory on the smallest field Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 05 — Test the theory on the smallest field |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Test the theory on the smallest field” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes test the theory on the smallest field natural, computable, functorial, and stable under passage between local and global fields? |
S32 S25 S29 |
| 281 | Bad places as diagnostic instruments Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 06 — Bad places as diagnostic instruments |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Bad places as diagnostic instruments” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes bad places as diagnostic instruments natural, computable, functorial, and stable under passage between local and global fields? |
S33 S28 S30 |
| 282 | Local objects first, global theorem second Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 07 — Local objects first, global theorem second |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Local objects first, global theorem second” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local objects first, global theorem second natural, computable, functorial, and stable under passage between local and global fields? |
S25 S29 S32 |
| 283 | Examples that determine conventions Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 08 — Examples that determine conventions |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Examples that determine conventions” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes examples that determine conventions natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S33 |
| 284 | Duality as a habit of thought Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 09 — Duality as a habit of thought |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Duality as a habit of thought” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes duality as a habit of thought natural, computable, functorial, and stable under passage between local and global fields? |
S29 S32 S25 |
| 285 | Functoriality as quality control Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 10 — Functoriality as quality control |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Functoriality as quality control” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes functoriality as quality control natural, computable, functorial, and stable under passage between local and global fields? |
S30 S33 S28 |
| 286 | A theorem as reusable infrastructure Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 11 — A theorem as reusable infrastructure |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “A theorem as reusable infrastructure” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes a theorem as reusable infrastructure natural, computable, functorial, and stable under passage between local and global fields? |
S32 S25 S29 |
| 287 | Letters that preserve the birth of concepts Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 12 — Letters that preserve the birth of concepts |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Letters that preserve the birth of concepts” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes letters that preserve the birth of concepts natural, computable, functorial, and stable under passage between local and global fields? |
S33 S28 S30 |
| 288 | Mathematical concepts bearing a name Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 13 — Mathematical concepts bearing a name |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Mathematical concepts bearing a name” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes mathematical concepts bearing a name natural, computable, functorial, and stable under passage between local and global fields? |
S25 S29 S32 |
| 289 | From one proof to a field Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 14 — From one proof to a field |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “From one proof to a field” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes from one proof to a field natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S33 |
| 290 | Classification through shadows Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 15 — Classification through shadows |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Classification through shadows” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes classification through shadows natural, computable, functorial, and stable under passage between local and global fields? |
S29 S32 S25 |
| 291 | Canonical normalization as ethics Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 16 — Canonical normalization as ethics |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Canonical normalization as ethics” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes canonical normalization as ethics natural, computable, functorial, and stable under passage between local and global fields? |
S30 S33 S28 |
| 292 | Six-decade continuity of themes Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 17 — Six-decade continuity of themes |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Six-decade continuity of themes” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes six-decade continuity of themes natural, computable, functorial, and stable under passage between local and global fields? |
S32 S25 S29 |
| 293 | Source spine as control system Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 18 — Source spine as control system |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Source spine as control system” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes source spine as control system natural, computable, functorial, and stable under passage between local and global fields? |
S33 S28 S30 |
| 294 | Comments that reveal method Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 19 — Comments that reveal method |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Comments that reveal method” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes comments that reveal method natural, computable, functorial, and stable under passage between local and global fields? |
S25 S29 S32 |
| 295 | Conjectures that outlive their maker Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 20 — Conjectures that outlive their maker |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Conjectures that outlive their maker” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes conjectures that outlive their maker natural, computable, functorial, and stable under passage between local and global fields? |
S28 S30 S33 |
| 296 | Minimalism with hidden depth Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 21 — Minimalism with hidden depth |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Minimalism with hidden depth” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes minimalism with hidden depth natural, computable, functorial, and stable under passage between local and global fields? |
S29 S32 S25 |
| 297 | Architecture over accumulation Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 22 — Architecture over accumulation |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Architecture over accumulation” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes architecture over accumulation natural, computable, functorial, and stable under passage between local and global fields? |
S30 S33 S28 |
| 298 | Local calculation global consequence Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 23 — Local calculation global consequence |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Local calculation global consequence” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes local calculation global consequence natural, computable, functorial, and stable under passage between local and global fields? |
S32 S25 S29 |
| 299 | Research culture as mathematical force Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 24 — Research culture as mathematical force |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Research culture as mathematical force” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes research culture as mathematical force natural, computable, functorial, and stable under passage between local and global fields? |
S33 S28 S30 |
| 300 | Tate as language designer of arithmetic Synthetic Tate method: reformulation, examples and long-horizon programs: Lecture-case 25 — Tate as language designer of arithmetic |
Synthetic Tate method: reformulation, examples and long-horizon programs 1950–2006 |
Read “Tate as language designer of arithmetic” not as an isolated topic but as a controlled passage between explicit arithmetic, representation, cohomology, and canonical normalization. Which reformulation makes tate as language designer of arithmetic natural, computable, functorial, and stable under passage between local and global fields? |
S25 S29 S32 |
The compact official citation: Tate’s 1950 thesis, class field theory, Lubin-Tate formal groups, rigid analytic spaces, Hodge-Tate theory, Tate modules, duality, heights, isogeny, Honda-Tate, and named conjectures.
Biographical spine: Harvard AB, Princeton PhD under Emil Artin, Harvard career, University of Texas chair, and major honors.
Two-volume source spine containing Tate’s published mathematical papers across more than six decades, author comments, and a selection of letters.
Survey/review organizing Tate’s work into Hecke L-series, Galois cohomology, Lubin-Tate spaces, abelian varieties, Tate conjecture, rigid analytic spaces, unpublished papers, letters, and comments.
Milne’s overview of Tate’s work in the context of the “great reformulation,” useful for understanding the page’s method rather than only its chronology.
Memorial material edited by Barry Mazur and Kenneth Ribet, with recollections of Tate’s mentorship, writing style, correspondence, seminars, and influence.
Departmental memorial and links to Tate’s genealogy, collected works, interviews, obituary material, and videos.
Biographical overview and historical placement of Tate’s career and mathematical influence.
Tate’s 1950 thesis / Hecke L-functions
Hecke L-functions become zeta integrals over the ideles; Fourier analysis gives continuation and functional equation.
Artin–Tate class field theory
Reciprocity laws become consequences of class formations, fundamental classes, and cohomological duality.
Abelian varieties and isogeny theorems
Geometry of abelian varieties is read through l-adic Galois representations on torsion.
Tate algorithm and elliptic curves
A local Weierstrass equation is transformed into reduction type, conductor contribution, and arithmetic data.
Rigid analytic spaces and Tate curve
Naive p-adic topology is replaced by admissible affinoid geometry, enabling p-adic analytic uniformization.
Algebraic cycles and Galois representations
The conjectural bridge asks whether Galois-fixed l-adic cohomology classes are accounted for by algebraic cycles.