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Shtukas and the Taylor expansion of \(L\)-functions. II. (English) Zbl 1442.11079

Authors’ abstract: 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 \(\Sigma\) 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 \(r \in \mathbb{Z}_{\geq 0}\) 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 \[ \mathscr{L}^{(a)}\left(\pi, \frac{1}{2}\right) \mathscr{L}^{(r-a)}\left(\pi\otimes\eta, \frac{1}{2}\right) \] where \(\eta\) is the quadratic idèle class character attached to \(F'/F\), and \(0 \leq a \leq r\);
(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.”
For Part I, see [Ann. Math. (2) 186, No. 3, 767–911 (2017; Zbl 1385.11032)].

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations

Citations:

Zbl 1385.11032

References:

[1] Blum, A.; Stuhler, U., Drinfeld modules and elliptic sheaves. Vector Bundles on Curves–New Directions, Lecture Notes in Math., 1649, 110-193, (1997) · Zbl 0958.11045 · doi:10.1007/BFb0094426
[2] Drinfel\cprime{d}, V. G., Elliptic modules, Mat. Sb. (N.S.), 94(136), 594-627, (1974) · Zbl 0321.14014 · doi:10.1070/SM1974v023n04ABEH001731
[3] Drinfel\cprime{d}, V. G., Langlands’ conjecture for {\({\rm GL}(2)\)} over functional fields. Proceedings of the {I}nternational {C}ongress of {M}athematicians, 565-574, (1980) · Zbl 0444.12004
[4] Gross, Benedict H.; Zagier, Don B., Heegner points and derivatives of {\(L\)}-series, Invent. Math.. Inventiones Mathematicae, 84, 225-320, (1986) · Zbl 0608.14019 · doi:10.1007/BF01388809
[5] Jacquet, Herv\'{e}, Sur un r\'{e}sultat de {W}aldspurger, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`“eme S\'”{e}rie, 19, 185-229, (1986) · Zbl 0605.10015 · doi:10.24033/asens.1506
[6] Lafforgue, V., Chtoucas pour les groupes r\'eductifs et param\`etrisation de {L}anglands globale, (2012) · Zbl 1395.14017
[7] Tate, John, On the conjectures of {B}irch and {S}winnerton-{D}yer and a geometric analog [see \mr{1610977}]. Dix expos\'{e}s sur la Cohomologie des Sch\'{e}mas, Adv. Stud. Pure Math., 3, 189-214, (1968)
[8] Varshavsky, Yakov, Moduli spaces of principal {\(F\)}-bundles, Selecta Math. (N.S.). Selecta Mathematica. New Series, 10, 131-166, (2004) · Zbl 1070.14026 · doi:10.1007/s00029-004-0343-0
[9] Waldspurger, J.-L., Sur les valeurs de certaines fonctions {\(L\)} automorphes en leur centre de sym\'{e}trie, Compositio Math.. Compositio Mathematica, 54, 173-242, (1985) · Zbl 0567.10021
[10] Yun, Zhiwei; Zhang, Wei, Shtukas and the {T}aylor expansion of {\(L\)}-functions, Ann. of Math. (2). Annals of Mathematics. Second Series, 186, 767-911, (2017) · Zbl 1385.11032 · doi:10.4007/annals.2017.186.3.2
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