Change of Coefficients for Drinfeld Modules, Shtuka, and Abelian Sheaves

Uniformizing the Stacks of Abelian Sheaves

Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon–Rapoport–Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects “of dimension 1”. Their higher dimensional generalizations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are ...

Drinfeld Modules and Elliptic Sheaves

Frobenius actions on the de Rham cohomology of Drinfeld modules

We study the action of endomorphisms of a Drinfeld A-module φ on its de Rham cohomology HDR(φ,L) and related modules, in the case where φ is defined over a field L of finite Acharacteristic p. Among others, we find that the nilspace H0 of the total Frobenius FrDR on HDR(φ,L) has dimension h = height of φ. We define and study a pairing between the p-torsion pφ of φ and HDR(φ,L), which becomes pe...

L-functions of Φ-sheaves and Drinfeld Modules

In this paper, we apply Dwork’s p-adic methods to study the meromorphic continuation and rationality of various L-functions arising from π-adic Galois representations, Drinfeld modules and φ-sheaves. As a consequence, we prove some conjectures of Goss about the rationality of the local L-function and the meromorphic continuation of the global L-function attached to a Drinfeld module. Let Fq be ...

Introduction to Drinfeld Modules

(1) Explicit class field theory for global function fields (just as torsion of Gm gives abelian extensions of Q, and torsion of CM elliptic curves gives abelian extension of imaginary quadratic fields). Here global function field means Fp(T ) or a finite extension. (2) Langlands conjectures for GLn over function fields (Drinfeld modular varieties play the role of Shimura varieties). (3) Modular...