The Sato-Tate law for Drinfeld modules

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 LL, he constructs a continuous repre...

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Published in:Transactions of the American Mathematical Society 2016-03, Vol.368 (3), p.2185-2222
Main Author: Zywina, David
Format: Article
Language:English
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Research article
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Summary: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 LL, he constructs a continuous representation ρ∞:WL→D×\rho _\infty \colon W_L \to D^\times of the Weil group of LL into a certain division algebra, which encodes the Sato-Tate law. When ϕ\phi has generic characteristic and LL 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/6577