Shtukas and the Taylor expansion of $L$-functions

Zhiwei Yun; Wei Zhang

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arXiv · 1512.02683

Shtukas and the Taylor expansion of $L$-functions

Zhiwei Yun·Wei Zhang

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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 $PGL_{2}$; (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. 2015-12-08. Shtukas and the Taylor expansion of $L$-functions. https://arxiv.org/abs/1512.02683

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