The Sato-Tate law for Drinfeld modules

Zywina, David

doi:10.1090/tran/6577

The Sato-Tate law for Drinfeld modules
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Trans. Amer. Math. Soc. 368 (2016), 2185-2222 Request permission

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 $\rho _\infty \colon W_L \to D^\times$ of the Weil group of $L$ into a certain division algebra, which encodes the Sato-Tate law. When $\phi$ has generic characteristic and $L$ is finitely generated, we shall describe the image of $\rho _\infty$ up to commensurability. As an application, we give improved upper bounds for the Drinfeld module analogue of the Lang-Trotter conjecture.

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