Shtukas and the Taylor expansion of $L$-functions
Annals of mathematics, Volume 186 (2017) no. 3, pp. 767-911
We define the Heegner–Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) the $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $\pi$ of $\mathrm{PGL}_{2}$, and (2)~the self-intersection number of the $\pi$-isotypic component of the Heegner–Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross–Zagier formula for higher derivatives of $L$-functions.
@article{10_4007_annals_2017_186_3_2,
author = {Zhiwei Yun and Wei Zhang},
title = {Shtukas and the {Taylor} expansion of $L$-functions},
journal = {Annals of mathematics},
pages = {767--911},
year = {2017},
volume = {186},
number = {3},
doi = {10.4007/annals.2017.186.3.2},
mrnumber = {3702678},
zbl = {06804005},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2017.186.3.2/}
}
TY - JOUR AU - Zhiwei Yun AU - Wei Zhang TI - Shtukas and the Taylor expansion of $L$-functions JO - Annals of mathematics PY - 2017 SP - 767 EP - 911 VL - 186 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2017.186.3.2/ DO - 10.4007/annals.2017.186.3.2 LA - en ID - 10_4007_annals_2017_186_3_2 ER -
Zhiwei Yun; Wei Zhang. Shtukas and the Taylor expansion of $L$-functions. Annals of mathematics, Volume 186 (2017) no. 3, pp. 767-911. doi: 10.4007/annals.2017.186.3.2
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