Shtukas and the Taylor expansion of $L$-functions

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 $\mathrm{PGL}_{2}$;

(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.

Authors

Zhiwei Yun

Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511

Wei Zhang

Department of Mathematics, Columbia University, MC 4423, 2990 Broadway, New York, NY 10027