The Sato-Tate law for Drinfeld modules
Number Theory
2011-10-19 v1
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.
Keywords
Cite
@article{arxiv.1110.4098,
title = {The Sato-Tate law for Drinfeld modules},
author = {David Zywina},
journal= {arXiv preprint arXiv:1110.4098},
year = {2011}
}