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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Sato-Tate law for Drinfeld modules
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by David Zywina PDF
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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Additional Information
  • David Zywina
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • MR Author ID: 871503
  • Email: zywina@math.cornell.edu
  • Received by editor(s): May 31, 2013
  • Received by editor(s) in revised form: August 5, 2014
  • Published electronically: May 29, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 2185-2222
  • MSC (2010): Primary 11G09; Secondary 11F80, 11R58
  • DOI: https://doi.org/10.1090/tran/6577
  • MathSciNet review: 3449237