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 $F^{\mathrm{\prime }}\mathrm{/}F$ of function fields over a finite field in odd characteristic, and a finite set of places $\mathrm{\Sigma }$ of $F$ that are unramified in $F^{\mathrm{\prime }}$, we define a collection of Heegner–Drinfeld cycles on the moduli stack of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-Shtukas with $r$-modifications and Iwahori level structures at places of $\mathrm{\Sigma }$. For a cuspidal automorphic representation $\pi$ of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_F)$ with square-free level $\mathrm{\Sigma }$, and $r\in {\mathbb{Z}}_{\ge 0}$ whose parity matches the root number of ${\pi }_{F^{\mathrm{\prime }}}$, we prove a series of identities between: (1) The product of the central derivatives of the normalized $L$-functions $$\mathcal{L}$^{(a)}(\pi, 1/2)$\mathcal{L}$^{(r-a)}(\pi\otimes\eta, 1/2)$,where$\eta$ is the quadratic id`ele class character attached to $F^{\mathrm{\prime }}\mathrm{/}F$, 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.