The Sato-Tate law for Drinfeld modules

TL;DR

This paper establishes the Sato-Tate law for Drinfeld modules by describing the image of a key Galois representation, leading to improved bounds on related Lang-Trotter type conjectures.

Contribution

It provides the first explicit description of the Sato-Tate law for Drinfeld modules and analyzes the associated Galois representations in the generic characteristic case.

Findings

Describes the image of the Weil group representation for Drinfeld modules.

Provides improved upper bounds for the Lang-Trotter conjecture analogue.

Establishes an explicit Sato-Tate law for Drinfeld modules.

Abstract

We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module \phi defined over a field L, he constructs a continuous representation \rho_\infty : W_L \to D^* of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When the Drinfeld module has generic characteristic and L is finitely generated, we shall describe the image of this representation up to commensurability. As an application, we give improved upper bounds for the Drinfeld module analogue of the Lang-Trotter conjecture.