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 $\mathrm{PGL}_{2}$, 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.