Drinfel'd Module - Shtukas

Shtukas

Suppose that X is a curve over the finite field F_p. A (right) shtuka of rank r over a scheme (or stack) U is given by the following data:

• Locally free sheaves E, E′ of rank r over U×X together with injective morphisms

E → E′ ← (Fr×1)*E,

whose cokernels are supported on certain graphs of morphisms from U to X (called the zero and pole of the shtuka, and usually denoted by 0 and ∞), and are locally free of rank 1 on their supports. Here (Fr×1)*E is the pullback of E by the Frobenius endomorphism of U.

A left shtuka is defined in the same way except that the direction of the morphisms is reversed. If the pole and zero of the shtuka are disjoint then left shtukas and right shtukas are essentially the same.

By varying U, we get an algebraic stack Shtukar of shtukas of rank r, a "universal" shtuka over Shtukar×X and a morphism (∞,0) from Shtukar to X×X which is smooth and of relative dimension 2r − 2. The stack Shtukar is not of finite type for r > 1.

Drinfel'd modules are in some sense special kinds of shtukas. (This is not at all obvious from the definitions.) More precisely, Drinfel'd showed how to construct a shtuka from a Drinfel'd module. See Drinfel'd, V. G. Commutative subrings of certain noncommutative rings. Funkcional. Anal. i Prilovzen. 11 (1977), no. 1, 11–14, 96. for details.