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- W4300604806 abstract "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 $Sigma$ 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 $rinmathbb{Z}_{ge0}$ 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 $mathcal{L}^{(a)}(pi, 1/2)mathcal{L}^{(r-a)}(piotimeseta, 1/2)$, where $eta$ is the quadratic id`ele class character attached to $F'/F$, and $0le ale r$; (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." @default.
- W4300604806 created "2022-10-03" @default.
- W4300604806 creator A5076699095 @default.
- W4300604806 creator A5090943702 @default.
- W4300604806 date "2017-12-21" @default.
- W4300604806 modified "2023-09-23" @default.
- W4300604806 title "Shtukas and the Taylor expansion of $L$-functions (II)" @default.
- W4300604806 doi "https://doi.org/10.48550/arxiv.1712.08026" @default.
- W4300604806 hasPublicationYear "2017" @default.
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