modularity conjecture
E921625
The modularity conjecture is a central statement in number theory asserting that every elliptic curve over the rational numbers corresponds to a modular form, a result whose proof underpins the modern proof of Fermat’s Last Theorem.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
statement in number theory ⓘ |
| alsoKnownAs |
Taniyama–Shimura conjecture
NERFINISHED
ⓘ
Taniyama–Shimura–Weil conjecture NERFINISHED ⓘ modularity theorem for elliptic curves over Q ⓘ |
| appliesTo | elliptic curves defined over Q ⓘ |
| asserts |
every elliptic curve over Q corresponds to a modular form
ⓘ
every elliptic curve over the rational numbers is modular ⓘ |
| concerns |
elliptic curves over the rational numbers
ⓘ
modular forms ⓘ |
| connectedTo |
Hasse–Weil L-function of an elliptic curve
NERFINISHED
ⓘ
cusp forms of weight 2 and level N ⓘ |
| equivalentFormulationInvolves | equality of L-functions of elliptic curves and modular forms ⓘ |
| field | number theory ⓘ |
| hasConsequence |
classification of elliptic curves over Q via modular forms
ⓘ
connections between arithmetic geometry and automorphic forms ⓘ |
| historicallyFormulatedBy |
André Weil
NERFINISHED
ⓘ
Goro Shimura NERFINISHED ⓘ Yutaka Taniyama NERFINISHED ⓘ |
| implies | Fermat’s Last Theorem NERFINISHED ⓘ |
| influenced |
development of the Langlands correspondence for GL(2)
ⓘ
modern research in arithmetic geometry ⓘ |
| involvesObject |
congruence subgroups of SL(2,Z)
ⓘ
rational points on elliptic curves ⓘ weight 2 modular forms ⓘ |
| isGeneralizedBy | modularity conjectures for higher-dimensional abelian varieties ⓘ |
| isSpecialCaseOf | Langlands reciprocity conjectures NERFINISHED ⓘ |
| originallyConjecturedInDecade | 1950s GENERATED ⓘ |
| partiallyProvedBy |
Andrew Wiles
NERFINISHED
ⓘ
Richard Taylor NERFINISHED ⓘ |
| proofCompletedInYear | 2001 ⓘ |
| provedBy |
Brian Conrad
NERFINISHED
ⓘ
Christophe Breuil NERFINISHED ⓘ Fred Diamond NERFINISHED ⓘ Richard Taylor NERFINISHED ⓘ |
| provedUsing |
Galois deformation theory
ⓘ
Iwasawa theory techniques ⓘ R=T theorems ⓘ modularity lifting theorems ⓘ properties of Hecke algebras ⓘ |
| relatedTo |
Langlands program
NERFINISHED
ⓘ
Shimura–Taniyama–Weil conjecture NERFINISHED ⓘ |
| relatesConcept |
Galois representations
ⓘ
L-functions ⓘ elliptic curves ⓘ modular curves ⓘ |
| status | proved ⓘ |
| type | modularity theorem NERFINISHED ⓘ |
| usedInProofOf | Fermat’s Last Theorem NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.