We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular -functions at the central point . As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet -functions to derive that for any pair of multiplicative characters modulo a prime , there is a positive proportion of such that the central values and are simultaneously not too small.
Nous généralisons le travail de Fouvry, Kowalski et Michel sur la corrélation entre les valeurs propres de Hecke de formes modulaires et les fonctions traces dans le but d’établir une formule asymptotique pour un moment cubique généralisé de fonctions au point central . Comme application, nous exploitons notre résultat récent sur la mollification du quatrième moment des fonctions de Dirichlet et déduisons que pour deux charactères multiplicatifs modulo un nombre premier , il existe une proportion positive de telle que les valeurs centrales et soient simultanément pas trop petites.
Revised:
Accepted:
Published online:
Keywords: Modular forms, L-functions, trace functions, bilinear forms, twisted Kloosterman sums
Mot clés : Formes modulaires, fonctions $L$, fonctions traces, formes bilinéaires, sommes de Kloosterman tordues
Zacharias, Raphaël 1
@article{AIF_2019__69_4_1459_0, author = {Zacharias, Rapha\"el}, title = {Simultaneous non-vanishing for {Dirichlet} $L$-functions}, journal = {Annales de l'Institut Fourier}, pages = {1459--1524}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {4}, year = {2019}, doi = {10.5802/aif.3275}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3275/} }
TY - JOUR AU - Zacharias, Raphaël TI - Simultaneous non-vanishing for Dirichlet $L$-functions JO - Annales de l'Institut Fourier PY - 2019 SP - 1459 EP - 1524 VL - 69 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3275/ DO - 10.5802/aif.3275 LA - en ID - AIF_2019__69_4_1459_0 ER -
%0 Journal Article %A Zacharias, Raphaël %T Simultaneous non-vanishing for Dirichlet $L$-functions %J Annales de l'Institut Fourier %D 2019 %P 1459-1524 %V 69 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3275/ %R 10.5802/aif.3275 %G en %F AIF_2019__69_4_1459_0
Zacharias, Raphaël. Simultaneous non-vanishing for Dirichlet $L$-functions. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1459-1524. doi : 10.5802/aif.3275. https://aif.centre-mersenne.org/articles/10.5802/aif.3275/
[1] On moments of twisted -functions, Am. J. Math., Volume 139 (2017) no. 3, pp. 707-768 | DOI | MR | Zbl
[2] Bounds for modular -functions in the level aspect, Ann. Sci. Éc. Norm. Supér., Volume 40 (2007) no. 5, pp. 697-740 | DOI | MR | Zbl
[3] Non-vanishing of Dirichlet -functions at the central point, Int. J. Number Theory, Volume 8 (2012) no. 8, pp. 1855-1881 | DOI | MR | Zbl
[4] Nonvanishing theorems for -functions of modular forms and their derivatives, Invent. Math., Volume 102 (1990) no. 3, pp. 543-618 | DOI | MR | Zbl
[5] Simultaneous nonvanishing of Dirichlet -functions and twists of Hecke–Maass -functions, J. Ramanujan Math. Soc., Volume 30 (2015) no. 3, pp. 237-250 | MR | Zbl
[6] La conjecture de Weil. II, Publ. Math., Inst. Hautes Étud. Sci. (1980) no. 52, pp. 137-252 | DOI | MR | Zbl
[7] Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math., Volume 70 (1982) no. 2, pp. 219-288 | DOI | MR | Zbl
[8] Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method, Proc. Lond. Math. Soc., Volume 114 (2017) no. 4, pp. 684-732 | DOI | MR | Zbl
[9] The critical order of vanishing of automorphic -functions with large level, Invent. Math., Volume 119 (1995) no. 1, pp. 165-174 | DOI | MR | Zbl
[10] Bounds for automorphic -functions. II, Invent. Math., Volume 115 (1994) no. 2, pp. 219-239 | DOI | MR | Zbl
[11] The subconvexity problem for Artin -functions, Invent. Math., Volume 149 (2002) no. 3, pp. 489-577 | DOI | MR | Zbl
[12] An inverse theorem for Gowers norms of trace functions over , Math. Proc. Camb. Philos. Soc., Volume 155 (2013) no. 2, pp. 277-295 | DOI | MR | Zbl
[13] Algebraic trace functions over the primes, Duke Math. J., Volume 163 (2014) no. 9, pp. 1683-1736 | DOI | MR | Zbl
[14] Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal., Volume 25 (2015) no. 2, pp. 580-657 | DOI | MR | Zbl
[15] Automorphic forms on adèle groups, Annals of Mathematics Studies, 83, Princeton University Press; University of Tokyo Press, 1975, x+267 pages | MR | Zbl
[16] Automorphic representations and -functions for the general linear group. Volume I, Cambridge Studies in Advanced Mathematics, 129, Cambridge University Press, 2011, xx+550 pages (With exercises and a preface by Xander Faber) | DOI | MR | Zbl
[17] Topics in classical automorphic forms, Graduate Studies in Mathematics, 17, American Mathematical Society, 1997, xii+259 pages | DOI | MR | Zbl
[18] Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, American Mathematical Society; Revista Matemática Iberoamericana, 2002, xii+220 pages | DOI | MR | Zbl
[19] Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004, xii+615 pages | DOI | MR | Zbl
[20] Dirichlet -functions at the central point, Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997), Walter de Gruyter, 1999, pp. 941-952 | MR | Zbl
[21] The non-vanishing of central values of automorphic -functions and Landau–Siegel zeros, Isr. J. Math., Volume 120 (2000) no. A, pp. 155-177 | DOI | MR | Zbl
[22] Sommes exponentielles, Astérisque, 79, Société Mathématique de France, 1980, 209 pages (Course taught at the University of Paris, Orsay, Fall 1979, With a preface by Luc Illusie, Notes written by Gérard Laumon, With an English summary) | MR | Zbl
[23] Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, 116, Princeton University Press, 1988, x+246 pages | DOI | MR | Zbl
[24] Exponential sums and differential equations, Annals of Mathematics Studies, 124, Princeton University Press, 1990, xii+430 pages | DOI | MR | Zbl
[25] Nonvanishing of Dirichlet -functions, Algebra Number Theory, Volume 10 (2016) no. 10, pp. 2081-2091 | DOI | MR | Zbl
[26] Functoriality for the exterior square of and the symmetric fourth of , J. Am. Math. Soc., Volume 16 (2003) no. 1, pp. 139-183 (With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak) | DOI | MR
[27] Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms, Mem. Am. Math. Soc., Volume 224 (2013) no. 1055, vi+132 pages | DOI | MR | Zbl
[28] The analytic rank of and zeros of automorphic -functions, Duke Math. J., Volume 100 (1999) no. 3, pp. 503-542 | DOI | MR | Zbl
[29] Bilinear forms with Kloosterman sums and applications, Ann. Math., Volume 186 (2017) no. 2, pp. 413-500 | DOI | MR | Zbl
[30] Rankin–Selberg -functions in the level aspect, Duke Math. J., Volume 114 (2002) no. 1, pp. 123-191 | DOI | MR | Zbl
[31] Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math., Inst. Hautes Étud. Sci. (1987) no. 65, pp. 131-210 | DOI | MR | Zbl
[32] On the nonvanishing of Rankin–Selberg -functions, Duke Math. J., Volume 69 (1993) no. 2, pp. 411-425 | DOI | MR | Zbl
[33] Probabilistic aspects of short sums of trace functions over finite fields, Ecole polytechnique fédérale de Lausanne (Switzerland) (2016) (Ph. D. Thesis)
[34] The subconvexity problem for Rankin–Selberg -functions and equidistribution of Heegner points, Ann. Math., Volume 160 (2004) no. 1, pp. 185-236 | DOI | MR | Zbl
[35] Analytic number theory and families of automorphic -functions, Automorphic forms and applications (IAS/Park City Mathematics Series), Volume 12, American Mathematical Society, 2007, pp. 181-295 | MR | Zbl
[36] Simultaneous nonvanishing of twists of automorphic -functions, Compos. Math., Volume 134 (2002) no. 2, pp. 135-191 | DOI | MR | Zbl
[37] A twisted fourth moment of Dirichlet -functions (2016) (https://arxiv.org/abs/1611.09582)
Cited by Sources: