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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Tate curve canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10973512 — 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 curve Context triple: [John Tate, notableWork, Tate curve]
-
A.
Twisted Edwards curve
A Twisted Edwards curve is a type of elliptic curve with a specific algebraic form that enables especially fast and secure implementations of cryptographic operations such as digital signatures and key exchange.
-
B.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
C.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
-
D.
Tate pairing
The Tate pairing is a bilinear, non-degenerate pairing on the points of an elliptic curve (or abelian variety) over a finite field, fundamental in number theory and widely used in pairing-based cryptography.
-
E.
Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
- 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 curve Target entity description: 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.
-
A.
Twisted Edwards curve
A Twisted Edwards curve is a type of elliptic curve with a specific algebraic form that enables especially fast and secure implementations of cryptographic operations such as digital signatures and key exchange.
-
B.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
-
C.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
-
D.
Tate pairing
The Tate pairing is a bilinear, non-degenerate pairing on the points of an elliptic curve (or abelian variety) over a finite field, fundamental in number theory and widely used in pairing-based cryptography.
-
E.
Mordell curve
A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\) over a field, central to number theory and Diophantine geometry.
- F. None of above. chosen
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) ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Tate curve Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.