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November 2017 Shtukas and the Taylor expansion of $L$-functions
Zhiwei Yun, Wei Zhang
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Ann. of Math. (2) 186(3): 767-911 (November 2017). DOI: 10.4007/annals.2017.186.3.2

Abstract

We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) the $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $\pi$ of $\mathrm{PGL}_{2}$, and (2) the self-intersection number of the $\pi$-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of $L$-functions.

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Zhiwei Yun. Wei Zhang. "Shtukas and the Taylor expansion of $L$-functions." Ann. of Math. (2) 186 (3) 767 - 911, November 2017. https://doi.org/10.4007/annals.2017.186.3.2

Information

Published: November 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.186.3.2

Subjects:
Primary: 11F67
Secondary: 11F70 , 14G35 , 14H60

Keywords: $L$-functions , Drinfeld Shtukas , Gross--Zagier formula , Waldspurger formula

Rights: Copyright © 2017 Department of Mathematics, Princeton University

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Vol.186 • No. 3 • November 2017
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