Paper detail

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

For arithmetic applications, we extend and refine our results in \cite{YZ} 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 $Σ$ 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 $Σ$. For a cuspidal automorphic representation $π$ of $\mathrm{PGL}_{2}(\mathbb{A}_{F})$ with square-free level $Σ$, and $r\in\mathbb{Z}_{\ge0}$ whose parity matches the root number of $π_{F'}$, we prove a series of identities between: (1) The product of the central derivatives of the normalized $L$-functions $\mathcal{L}^{(a)}(π, 1/2)\mathcal{L}^{(r-a)}(π\otimesη, 1/2)$, where $η$ 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 formulas of Waldspurger and Gross--Zagier for higher derivatives of $L$-functions.