Drinfel'd Module

In 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 introduced by Drinfel'd (1974), 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.

"Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück” meaning “piece, item, or unit", and is unrelated to the German word Stuka, meaning dive bomber. In Russian, word "shtuka" is also used as a slang describing a thing with known properties, but having no name in a speaker's mind.

Read more about Drinfel'd Module:  Shtukas, Applications