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 idèle 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.