Tate curve
E896825
The Tate curve is a type of elliptic curve defined over non-archimedean local fields, central to John Tate’s work on p-adic uniformization and the study of elliptic curves with bad reduction.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
elliptic curve ⓘ mathematical object ⓘ |
| appearsIn |
Tate’s theory of rigid analytic spaces
ⓘ
Tate’s work on p-divisible groups ⓘ |
| contextOfStudy |
arithmetic geometry
ⓘ
non-archimedean analytic geometry ⓘ number theory ⓘ |
| definedOver |
non-archimedean local fields
ⓘ
p-adic fields ⓘ |
| hasCoefficient |
a_4(q) = -5\sum_{n\ge1} n^3 q^n / (1 - q^n)
ⓘ
a_6(q) = -\frac{1}{12}\sum_{n\ge1} (7n^5 + 5n^3) q^n / (1 - q^n) ⓘ |
| hasComponentGroup | component group is isomorphic to Z ⓘ |
| hasConditionOnParameter | 0 < |q| < 1 in the non-archimedean norm GENERATED ⓘ |
| hasEndomorphismStructure | compatible with the multiplicative group structure ⓘ |
| hasInvariant |
Tate module
ⓘ
discriminant expressed as a q-product ⓘ j-invariant given by a q-expansion ⓘ |
| hasMorphism | canonical map from G_m to the elliptic curve quotient ⓘ |
| hasNeronModelProperty | special fiber is a Néron n-gon (in the split multiplicative case) ⓘ |
| hasParameter | q in the maximal ideal of the valuation ring ⓘ |
| hasProperty |
admits a canonical invariant differential coming from dT/T on G_m
ⓘ
admits q-expansion description ⓘ gives explicit classification of elliptic curves with split multiplicative reduction over local fields ⓘ j-invariant is a rigid analytic function of q NERFINISHED ⓘ non-archimedean analytic uniformization ⓘ q-parameter is uniquely determined up to multiplication by a root of unity ⓘ rigid analytic elliptic curve ⓘ |
| hasReductionType | split multiplicative reduction GENERATED ⓘ |
| hasUniformization | rigid analytic uniformization by the multiplicative group ⓘ |
| hasWeierstrassEquation | y^2 + xy = x^3 + a_4(q)x + a_6(q) ⓘ |
| isIsomorphicTo | G_m / q^Z as rigid analytic groups ⓘ |
| namedAfter | John Tate NERFINISHED ⓘ |
| relatedTo |
Galois representations attached to elliptic curves
ⓘ
Lubin–Tate formal groups (by analogy in local uniformization) NERFINISHED ⓘ Néron models of elliptic curves ⓘ Serre–Tate theory of ordinary elliptic curves NERFINISHED ⓘ elliptic curves with split multiplicative reduction ⓘ p-adic Hodge theory NERFINISHED ⓘ q-parameter on modular curves ⓘ |
| usedFor |
description of the local Galois representation on the Tate module in the multiplicative reduction case
ⓘ
explicit computation of local L-factors of elliptic curves ⓘ p-adic uniformization of elliptic curves ⓘ study of elliptic curves with bad reduction ⓘ |
| usedIn |
description of the ordinary locus of modular curves
ⓘ
local study of modular curves ⓘ proofs of the Tate conjecture for abelian varieties over finite fields (via Tate modules of elliptic curves) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.