Drinfel'd Module

High Quality Content by WIKIPEDIA articles! n mathematics, a Drinfel'd module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfel'd module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. Drinfel'd modules were invented in about 1973 by Vladimir Drinfel'd, who used them to prove the Langlands conjectures for GL2 of a function field in some special cases. He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL2. Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n.