Shtuka

In mathematics, a Drinfeld 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 Drinfeld 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.

Drinfeld modules were introduced by Drinfeld (1974), who used them to prove the Langlands conjectures for GL₂ of an algebraic 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 GL_n 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". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind.

We let ${\displaystyle L}$ be a field of characteristic ${\displaystyle p>0}$. The ring ${\displaystyle L\{\tau \}}$ is defined to be the ring of noncommutative (or twisted) polynomials ${\displaystyle a_{0}+a_{1}\tau +a_{2}\tau ^{2}+\cdots }$ over ${\displaystyle L}$, with the multiplication given by

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