The 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 ϕ \phi defined over a field L L , he constructs a continuous representation ρ ∞ : W L → D × \rho _\infty \colon W_L \to D^\times of the Weil group of L L into a certain division algebra, which encodes the Sato-Tate law. When ϕ \phi has generic characteristic and L L is finitely generated, we shall describe the image of ρ ∞ \rho _\infty up to commensurability. As an application, we give improved upper bounds for the Drinfeld module analogue of the Lang-Trotter conjecture.