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

##### Author(s)

Yun, Zhiwei; Zhang, Wei
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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 Σ of F that are unramified in F', we define a collection of Heegner-Drinfeld cycles on the moduli stack of PGL 2 -Shtukas with r-modifications and Iwahori level structures at places of Σ. For a cuspidal automorphic representation π of PGL 2 (AF) with square-free level Σ, and r∈Z≥0 whose parity matches the root number of πF', we prove a series of identities between (1) the product of the central derivatives of the normalized L-functions L (a) (π, 1/2)L (r-a) (π⊗η,1/2), where η is the quadratic idèle class character attached to F'/F, and 0≤a≤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.

##### Date issued

2020-06##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Annals of mathematics

##### Publisher

Annals of Mathematics, Princeton U

##### Citation

Yun, Zhiwei and Wei Zhang. “Shtukas and the Taylor expansion of L-functions (II).” Annals of mathematics, vol. 189, no. 2, 2019, pp. 393-526 © 2019 The Author(s)

Version: Original manuscript

##### ISSN

0003-486X