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The Sato-Tate law for Drinfeld modules


Author: David Zywina
Journal: Trans. Amer. Math. Soc. 368 (2016), 2185-2222
MSC (2010): Primary 11G09; Secondary 11F80, 11R58
DOI: https://doi.org/10.1090/tran/6577
Published electronically: May 29, 2015
MathSciNet review: 3449237
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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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Additional Information

David Zywina
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: zywina@math.cornell.edu

DOI: https://doi.org/10.1090/tran/6577
Received by editor(s): May 31, 2013
Received by editor(s) in revised form: August 5, 2014
Published electronically: May 29, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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