Tate’s non-archimedean uniformization of elliptic curves
E896827
Tate’s non-archimedean uniformization of elliptic curves is a foundational theory in arithmetic geometry that describes certain elliptic curves over non-archimedean fields via analytic uniformization using formal q-expansions, leading to what are now called Tate curves.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Tate’s non-archimedean uniformization of elliptic curves canonical | 1 |
| theory of Tate curves | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10973518 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Tate’s non-archimedean uniformization of elliptic curves Context triple: [John Tate, notableWork, Tate’s non-archimedean uniformization of elliptic curves]
-
A.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
B.
Introduction to Elliptic Curves and Modular Forms
Introduction to Elliptic Curves and Modular Forms is a graduate-level mathematics textbook that develops the theory of elliptic curves and their deep connections to modular forms, number theory, and arithmetic geometry.
-
C.
Lectures on Elliptic Curves
Lectures on Elliptic Curves is a classic introductory monograph by J. W. S. Cassels that systematically develops the arithmetic theory of elliptic curves for advanced undergraduates and beginning graduate students in number theory.
-
D.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
-
E.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Tate’s non-archimedean uniformization of elliptic curves Target entity description: Tate’s non-archimedean uniformization of elliptic curves is a foundational theory in arithmetic geometry that describes certain elliptic curves over non-archimedean fields via analytic uniformization using formal q-expansions, leading to what are now called Tate curves.
-
A.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
B.
Introduction to Elliptic Curves and Modular Forms
Introduction to Elliptic Curves and Modular Forms is a graduate-level mathematics textbook that develops the theory of elliptic curves and their deep connections to modular forms, number theory, and arithmetic geometry.
-
C.
Lectures on Elliptic Curves
Lectures on Elliptic Curves is a classic introductory monograph by J. W. S. Cassels that systematically develops the arithmetic theory of elliptic curves for advanced undergraduates and beginning graduate students in number theory.
-
D.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
-
E.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theory
ⓘ
theory in arithmetic geometry ⓘ uniformization theory ⓘ |
| appliesTo |
elliptic curves over non-archimedean fields
ⓘ
elliptic curves with split multiplicative reduction ⓘ |
| characterizes |
elliptic curves with non-integral j-invariant
ⓘ
elliptic curves with split multiplicative reduction over local fields ⓘ |
| context |
elliptic curves over complete non-archimedean fields
ⓘ
elliptic curves over p-adic fields ⓘ |
| describes |
elliptic curves as quotients of the multiplicative group
ⓘ
elliptic curves via analytic uniformization ⓘ |
| developedBy | John Tate ⓘ |
| field |
arithmetic geometry
ⓘ
non-archimedean analytic geometry ⓘ p-adic analysis ⓘ |
| foundationFor |
non-archimedean analytic uniformization methods
ⓘ
theory of Tate curves ⓘ |
| generalizedBy |
Mumford curves theory
ⓘ
Raynaud’s p-adic uniformization of abelian varieties ⓘ |
| hasOutcome |
classification of certain elliptic curves via q-parameters
ⓘ
explicit formulas for invariants of elliptic curves in terms of q ⓘ |
| influenced |
development of rigid analytic geometry
ⓘ
p-adic uniformization of abelian varieties ⓘ |
| involves |
parameter q with |q|<1 in a non-archimedean field
ⓘ
quotient of the multiplicative group by a discrete subgroup ⓘ |
| namedAfter | John Tate ⓘ |
| produces |
Tate curves
ⓘ
linked to:
Tate curve
|
| provides |
analytic parametrization of points on elliptic curves
ⓘ
explicit q-parameter for elliptic curves ⓘ |
| relatesTo |
Galois representations attached to elliptic curves
ⓘ
Néron models of elliptic curves ⓘ Weierstrass equation of elliptic curves ⓘ
linked to:
Weierstrass form
j-invariant of elliptic curves ⓘ |
| timePeriod | 1960s ⓘ |
| usedIn |
Iwasawa theory of elliptic curves
ⓘ
computation of conductors of elliptic curves ⓘ p-adic modular forms ⓘ study of local L-factors of elliptic curves ⓘ theory of q-expansions of modular forms ⓘ |
| usesConcept |
formal power series
ⓘ
non-archimedean absolute value ⓘ p-adic analytic functions ⓘ q-expansion ⓘ rigid analytic geometry ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Tate’s non-archimedean uniformization of elliptic curves Description of subject: Tate’s non-archimedean uniformization of elliptic curves is a foundational theory in arithmetic geometry that describes certain elliptic curves over non-archimedean fields via analytic uniformization using formal q-expansions, leading to what are now called Tate curves.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Tate’s non-archimedean uniformization of elliptic curves