Yun, Zhiwei; Zhang, Wei Shtukas and the Taylor expansion of \(L\)-functions. II. (English) Zbl 1442.11079 Ann. Math. (2) 189, No. 2, 393-526 (2019). 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)]. Reviewer: Qiao Zhang (Fort Worth) Cited in 2 ReviewsCited in 11 Documents 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 Keywords:\(L\)-functions; Drinfeld Shtukas; Gross-Zagier formula; Waldspurger formula Citations:Zbl 1385.11032 × Cite Format Result Cite Review PDF Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.