Shtukas and the Taylor expansion of $L$-functions (II)

Abstract

For arithmetic applications, we extend and refine our previously published results to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields over a finite field in odd characteristic, and a finite set of places $\Sigma$ of $F$ that are unramified in $F'$, we define a collection of Heegner--Drinfeld cycles on the moduli stack of $\mathrm{PGL}_2$-Shtukas with $r$-modifications and Iwahori level structures at places of $\Sigma$. For a cuspidal automorphic representation $\pi$ of $\mathrm{PGL}_2(\mathbb{A}_F)$ with square-free level $\Sigma$, and $r\in \mathbb{Z}_{\ge 0}$ whose parity matches the root number of $\pi_{F'}$, we prove a series of identities between

(1) the product of the central derivatives of the normalized $L$-functions $$\mathscr{L}^{(a)}\left(\pi, \frac{1}{2}\right)\mathscr{L}^{(r-a)}\left(\pi\otimes\eta, \frac{1}{2}\right), $$where $\eta$ is the quadratic id\'{e}le class character attached to $F'/F$, and $0\le a\le r$;

(2) the self intersection number of a linear combination of Heegner--Drinfeld cycles.

In particular, we can now obtain global $L$-functions with odd vanishing orders. These identities are function-field analogues of the formulae of Waldspurger and Gross--Zagier for higher derivatives of $L$-functions.