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Shtukas and the Taylor expansion of \(L\)-functions. (English) Zbl 1385.11032
Let \(X'\rightarrow X\) be an √©tale cover of degree 2 where \(X\) and \(X'\) are geometrically connected smooth complete curves over a finite field of characteristic \(> 2\) and \(F\subset F'\) function fields. Let \(G = \text{PGL}_2\). Consider an everywhere unramified cuspidal automorphic representation \(\pi\) of \(G(F_{\mathbb A})\) and its base change \(\pi_{F'}\) to \(F'\). The main result of the article is a geometric expression for the central derivatives of the \(L\)-function of \(\pi_{F'}\). The result is obtained by the comparison of two relative trace formulas. One of these is an adaptation of Jacquet’s relative trace formula, the other one is of geometric nature.
The authors define for each even number \(r\) a cycle (called Heegner-Drinfeld cycle) on a moduli stack of shtukas for \(G\) with \(r\)-modifications. The intersection number of this cycle with its transform by a Hecke function is defined. It is studied in two ways. Using intersection theory it is written as a trace. The decomposition of the cohomology of the stack under the Hecke action gives a decomposition of the Heegner-Drinfeld cycle and the spectral decomposition of the intersection number.
In order to compare the two relative trace formulas, the orbital integrals of the analytic formula are interpreted as traces of Frobenius. The orbital sides being identified (up to a factor \({(\text{log} q)}^r\)), the spectral sides are also equal. That gives the expression of the \(r\)-th derivative of the \(L\)-function as a self-intersection number of a component of the Heegner-Drinfeld cycle.

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G35 Modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14H60 Vector bundles on curves and their moduli
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[1] Ahlfors, Lars V., Complex Analysis: {A}n Introduction of the Theory of Analytic Functions of One Complex Variable, Second edition, xiii+317 pp., (1966) · Zbl 0395.30001
[2] Behrend, K.; Fantechi, B., The intrinsic normal cone, Invent. Math.. Inventiones Mathematicae, 128, 45-88, (1997) · Zbl 0909.14006
[3] Csordas, George; NSorfolk, Timothy S.; Varga, Richard S., The {R}iemann hypothesis and the {T}ur\'an inequalities, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 296, 521-541, (1986) · Zbl 0602.30030
[4] Drinfel{\cprime}d, V. G., Moduli varieties of {\(F\)}-sheaves, Funktsional. Anal. i Prilozhen.. Akademiya Nauk SSSR. Funktsional\cprime ny\u\i Analiz i ego Prilozheniya, 21, 23-41, (1987)
[5] Fulton, William, Intersection Theory, Ergeb. Math. Grenzgeb., 2, xiv+470 pp., (1998) · Zbl 0885.14002
[6] Gillet, Henri, Intersection theory on algebraic stacks and {\(Q\)}-varieties. Proceedings of the {L}uminy Conference on Algebraic {\(K\)}-Theory, J. Pure Appl. Algebra, 34, 193-240, (1984) · Zbl 0607.14004
[7] Godement, Roger; Jacquet, Herv\'e, Zeta Functions of Simple Algebras, Lecture Notes in Math., 260, ix+188 pp., (1972) · Zbl 0244.12011
[8] Goldfeld, D.; Huang, B., Super-positivity of a family of {\(L\)}-functions, (2016)
[9] Goresky, Mark; MacPherson, Robert, Intersection homology. {II}, Invent. Math.. Inventiones Mathematicae, 72, 77-129, (1983) · Zbl 0529.55007
[10] Gross, Benedict H.; Zagier, Don B., Heegner points and derivatives of {\(L\)}-series, Invent. Math.. Inventiones Mathematicae, 84, 225-320, (1986) · Zbl 0608.14019
[11] Jacquet, Herv\'e, Sur un r\'esultat de {W}aldspurger, Ann. Sci. \'Ecole Norm. Sup. (4). Annales Scientifiques de l’\'Ecole Normale Sup\'erieure. Quatri\`“‘eme S\'”’erie, 19, 185-229, (1986) · Zbl 0605.10015
[12] Kresch, Andrew, Canonical rational equivalence of intersections of divisors, Invent. Math.. Inventiones Mathematicae, 136, 483-496, (1999) · Zbl 0923.14003
[13] Kresch, Andrew, Cycle groups for {A}rtin stacks, Invent. Math.. Inventiones Mathematicae, 138, 495-536, (1999) · Zbl 0938.14003
[14] Lafforgue, V., Chtoucas pour les groupes r\'eductifs et param\`etrisation de langlands globale, (2012) · Zbl 1395.14017
[15] Lapid, Erez; Rallis, Stephen, On the nonnegativity of {\(L({1\over2},\pi)\)} for {\({\rm SO}_{2n+1}\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 157, 891-917, (2003) · Zbl 1067.11026
[16] Laszlo, Yves; Olsson, Martin, The six operations for sheaves on {A}rtin stacks. {II}. {A}dic coefficients, Publ. Math. Inst. Hautes \'Etudes Sci.. Publications Math\'ematiques. Institut de Hautes \'Etudes Scientifiques, 169-210, (2008) · Zbl 1191.14003
[17] Laumon, G\'erard, Correspondance de {L}anglands g\'eom\'etrique pour les corps de fonctions, Duke Math. J.. Duke Mathematical Journal, 54, 309-359, (1987) · Zbl 0662.12013
[18] Be\u{\i}linson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers. Analysis and Topology on Singular Spaces, {I}, Ast\'erisque, 100, 5-171, (1982)
[19] Ng\^o, Bao Ch\^au, Fibration de {H}itchin et endoscopie, Invent. Math.. Inventiones Mathematicae, 164, 399-453, (2006) · Zbl 1098.14023
[20] Ng\^o, Bao Ch\^au, Le lemme fondamental pour les alg\`“ebres de {L}ie, Publ. Math. Inst. Hautes \'”Etudes Sci.. Publications Math\'ematiques. Institut de Hautes \'Etudes Scientifiques, 1-169, (2010) · Zbl 1200.22011
[21] Sarnak, Peter, Letter to {E}. {B}achmat on positive definite {L}-functions, (2011)
[22] Stark, H. M.; Zagier, D., A property of {\(L\)}-functions on the real line, J. Number Theory. Journal of Number Theory, 12, 49-52, (1980) · Zbl 0428.10021
[23] Varshavsky, Yakov, Moduli spaces of principal {\(F\)}-bundles, Selecta Math. (N.S.). Selecta Mathematica. New Series, 10, 131-166, (2004) · Zbl 1070.14026
[24] Varshavsky, Yakov, Lefschetz-{V}erdier trace formula and a generalization of a theorem of {F}ujiwara, Geom. Funct. Anal.. Geometric and Functional Analysis, 17, 271-319, (2007) · Zbl 1131.14019
[25] Vistoli, Angelo, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math.. Inventiones Mathematicae, 97, 613-670, (1989) · Zbl 0694.14001
[26] Waldspurger, J.-L., Sur les valeurs de certaines fonctions {\(L\)} automorphes en leur centre de sym\'etrie, Compositio Math.. Compositio Mathematica, 54, 173-242, (1985) · Zbl 0567.10021
[27] Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei, The {G}ross-{Z}agier Formula on {S}himura Curves, Ann. of Math. Stud., 184, x+256 pp., (2013) · Zbl 1272.11082
[28] Yun, Zhiwei, An arithmetic fundamental lemma for function fields, (2011) · Zbl 1211.14039
[29] Yun, Zhiwei, The fundamental lemma of {J}acquet and {R}allis, Duke Math. J.. Duke Mathematical Journal, 156, 167-227, (2011) · Zbl 1211.14039
[30] Zhang, Wei, On arithmetic fundamental lemmas, Invent. Math.. Inventiones Mathematicae, 188, 197-252, (2012) · Zbl 1247.14031
[31] Zhang, Wei, Automorphic period and the central value of {R}ankin-{S}elberg {\(L\)}-function, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 27, 541-612, (2014) · Zbl 1294.11069
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