Drinfeld modules

E884935

algebraic structure generalization of elliptic curves object in arithmetic geometry

Drinfeld modules are algebraic structures that generalize elliptic curves to the setting of function fields, playing a central role in modern arithmetic geometry and the theory of automorphic forms.

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Statements (49)

Predicate	Object
instanceOf	algebraic structure ⓘ generalization of elliptic curves ⓘ object in arithmetic geometry ⓘ
actOn	additive group via Frobenius-type operators ⓘ
centralIn	Drinfeld’s proof of the global Langlands correspondence for GL(2) over function fields ⓘ
consideredAs	function field analogues of abelian varieties ⓘ
definedOver	fields of positive characteristic ⓘ global function fields ⓘ
fieldOfStudy	arithmetic geometry ⓘ function field arithmetic ⓘ number theory ⓘ theory of automorphic forms ⓘ
generalizes	elliptic curves over number fields ⓘ
hasAnalogueOf	L-function ⓘ Mordell–Weil theorem NERFINISHED ⓘ Néron–Ogg–Shafarevich criterion NERFINISHED ⓘ Serre–Tate theory of deformations NERFINISHED ⓘ Tate module ⓘ complex multiplication theory ⓘ modular forms ⓘ
hasComponent	ring homomorphism from A to endomorphisms of the additive group ⓘ underlying additive group scheme ⓘ
hasInvariant	conductor ⓘ endomorphism ring ⓘ height ⓘ j-invariant analogue ⓘ
hasProperty	admit a theory of isogenies ⓘ admit a theory of torsion points ⓘ admit good and bad reduction at places ⓘ form moduli spaces ⓘ have associated Galois representations ⓘ have associated exponential functions ⓘ have associated periods and quasi-periods ⓘ
introducedBy	Vladimir Drinfeld NERFINISHED ⓘ
introducedIn	1970s ⓘ
namedAfter	Vladimir Drinfeld NERFINISHED ⓘ
oftenAssume	A is a ring of functions regular away from a fixed place of a global function field ⓘ
parameterizedBy	characteristic ⓘ rank ⓘ
relatedTo	Anderson motives NERFINISHED ⓘ Drinfeld modular curves NERFINISHED ⓘ Drinfeld modular forms NERFINISHED ⓘ shtukas ⓘ t-motives NERFINISHED ⓘ
studiedIn	positive characteristic Hodge theory ⓘ
usedIn	Langlands correspondence over function fields NERFINISHED ⓘ construction of Galois representations ⓘ explicit class field theory for function fields ⓘ study of special values of L-functions ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Vladimir Drinfeld → knownFor → Drinfeld modules ⓘ