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

The element ${\displaystyle \tau }$ can be thought of as a Frobenius element: in fact, ${\displaystyle L}$ is a left module over ${\displaystyle L\{\tau \}}$, with elements of ${\displaystyle L}$ acting as multiplication and ${\displaystyle \tau }$ acting as the Frobenius endomorphism of ${\displaystyle L}$. The ring ${\displaystyle L\{\tau \}}$ can also be thought of as the ring of all (absolutely) additive polynomials

in ${\displaystyle L[x]}$, where a polynomial ${\displaystyle f}$ is called additive if ${\displaystyle f(x+y)=f(x)+f(y)}$ (as elements of ${\displaystyle L[x,y]}$). The ring of additive polynomials is generated as an algebra over ${\displaystyle L}$ by the polynomial ${\displaystyle \tau =x^{p}}$. The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.

Let F be an algebraic function field with a finite field of constants and fix a place ${\displaystyle \infty }$ of F. Define A to be the ring of elements in F that are regular at every place except possibly ${\displaystyle \infty }$. In particular, A is a Dedekind domain and it is discrete in F (with the topology induced by ${\displaystyle \infty }$). For example, we may take A to be the polynomial ring ${\displaystyle F_{q}[t]}$. Let L be a field equipped with a ring homomorphism ${\displaystyle \iota :A\to L}$.

The condition that the image of A is not in L is a non-degeneracy condition, put in to eliminate trivial cases, while the condition that ${\displaystyle d\circ \phi =\iota }$ gives the impression that a Drinfeld module is simply a deformation of the map ${\displaystyle \iota }$.