The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the universal family of automorphic forms on a totally definite quaternion algebra, we determine the associated distribution for a restricted class of test functions in the analytic conductor aspect. In particular it leads to non-trivial results on densities of non-vanishing at the central point.
Pour des familles naturelles de fonctions L, la conjecture de densité de Katz et Sarnak prédit que la répartition des zéros proches de l’axe réel est régie par un groupe de symétrie. Dans le cas de la famille universelle d’une algèbre de quaternions totalement définie, nous déterminons la distribution associée pour une classe explicite de fonctions test, uniformément lorsque le conducteur analytique croît. En particulier, cela mène à des résultats non-triviaux sur les densités de non-annulation aux valeurs centrales.
Revised:
Accepted:
Online First:
Published online:
Keywords: automorphic representation, L-function, low-lying zeros, type of symmetry, density conjecture, Hecke operators, quaternion algebras
Mot clés : représentation automorphe, fonction L, petits zéros, type de symétrie, conjecture de densité, opérateur de Hecke, algèbre de quaternions
@article{AIF_2021__71_4_1635_0, author = {Lesesvre, Didier}, title = {Low-lying zeros of {L-functions} for {Quaternion} {Algebras}}, journal = {Annales de l'Institut Fourier}, pages = {1635--1676}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {71}, number = {4}, year = {2021}, doi = {10.5802/aif.3428}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3428/} }
TY - JOUR AU - Lesesvre, Didier TI - Low-lying zeros of L-functions for Quaternion Algebras JO - Annales de l'Institut Fourier PY - 2021 SP - 1635 EP - 1676 VL - 71 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3428/ DO - 10.5802/aif.3428 LA - en ID - AIF_2021__71_4_1635_0 ER -
%0 Journal Article %A Lesesvre, Didier %T Low-lying zeros of L-functions for Quaternion Algebras %J Annales de l'Institut Fourier %D 2021 %P 1635-1676 %V 71 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3428/ %R 10.5802/aif.3428 %G en %F AIF_2021__71_4_1635_0
Lesesvre, Didier. Low-lying zeros of L-functions for Quaternion Algebras. Annales de l'Institut Fourier, Volume 71 (2021) no. 4, pp. 1635-1676. doi : 10.5802/aif.3428. http://www.numdam.org/articles/10.5802/aif.3428/
[1] An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties (Clay Mathematics Proceedings), Volume 4, American Mathematical Society, 2005, pp. 1-263 | MR | Zbl
[2] Fields of Rationality of Cusp Forms, Isr. J. Math., Volume 222 (2017) no. 2, pp. 973-1028 | DOI | MR | Zbl
[3] On the Ramanujan conjecture over number fields, Ann. Math., Volume 174 (2011) no. 2, pp. 581-605 | DOI | MR | Zbl
[4] Counting cusp forms with analytic conductor (2018) (https://arxiv.org/abs/1805.00633)
[5] Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997 no. 55 | DOI
[6] On Some Results by Atkin and Lehner, Math. Ann., Volume 201 (1973), pp. 301-314 | DOI | MR | Zbl
[7] Le théorème de Paley-Wiener invariant pour les groupes réductifs. II, Ann. Sci. Éc. Norm. Supér., Volume 23 (1990) no. 2, pp. 193-228 | DOI | Numdam | Zbl
[8] La conjecture de Weil. I, Publ. Math., Inst. Hautes Étud. Sci., Volume 43 (1974), pp. 273-307 | DOI | Numdam
[9] The low lying zeros of a GL(4) and a GL(6) family of -functions, Compos. Math., Volume 142 (2006) no. 06, pp. 1403-1425 | DOI | MR | Zbl
[10] The effect of convolving families of -functions on the underlying group symmetries, Proc. Lond. Math. Soc., Volume 99 (2009) no. 3, pp. 787-820 | DOI | MR | Zbl
[11] On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of -functions, Automorphic representations and -functions (Mumbai, 2012) (Tata Institute of Fundamental Research Studies in Mathematics), Volume 22, Tata Institute of Fundamental Research, 2013, pp. 263-310 | MR | Zbl
[12] The Maximal Order of Generalized Quaternion Division Algebras, Bull. Am. Math. Soc. (1936), pp. 1-11 | MR | Zbl
[13] Low lying zeros of families of -functions, Publ. Math., Inst. Hautes Étud. Sci., Volume 91 (2000), pp. 55-131 | DOI | Numdam | Zbl
[14] Perspectives on the Analytic Theory of -functions, Geom. Funct. Anal. (2000), pp. 705-741 (Special Volume, Part II) | MR
[15] Random Matrices, Frobenius Eigenvalues, and Monodromy, Colloquium Publications, 45, American Mathematical Society, 1999
[16] Zeroes of zeta functions and symmetry, Bull. Am. Math. Soc., Volume 36 (1999), pp. 1-26 | DOI | MR | Zbl
[17] Traces of Hecke operators, Mathematical Surveys and Monographs, 133, American Mathematical Society, 2006 | DOI
[18] Arithmetic Statistics for Quaternion Algebras, Ph. D. Thesis, Université Paris 13 (2018)
[19] Counting and Equidistribution for Quaternion algebras, Math. Z., Volume 295 (2020), pp. 129-159 | DOI | MR | Zbl
[20] Low-lying zeros for -functions associated to Hilbert modular forms of large level, Acta Arith., Volume 180 (2017) no. 3, pp. 251-266 | MR | Zbl
[21] Refined dimensions of cusp forms, and equidistribution and bias of signs, J. Number Theory, Volume 188 (2018), pp. 1-17 | DOI | MR | Zbl
[22] Bounds for the Global Coefficients in the Fine Geometric Expansion of Arthur’s Trace Formula for , Isr. J. Math., Volume 205 (2015), pp. 337-396 | DOI | MR | Zbl
[23] Weyl’s law for Hecke operators on GL(n) over imaginary quadratic number fields, Am. J. Math., Volume 139 (2017) no. 1, pp. 57-145 | DOI | MR | Zbl
[24] Sato-Tate equidistribution for families of Hacke-Maass forms on SL(n)/SO(n) (2016) (https://arxiv.org/abs/1505.07285, to appear in Algebra Number Theory)
[25] Random matrices, Pure and Applied Mathematics, Academic Press Inc., 2004 no. 142
[26] One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries, Compos. Math., Volume 140 (2004) no. 4, pp. 952-992 | DOI | MR | Zbl
[27] Small zeros of quadratic -functions, Bull. Aust. Math. Soc., Volume 47 (1993) no. 2, pp. 307-319 | DOI | MR | Zbl
[28] Low-lying zeros of -functions and random matrix theory, Duke Math. J., Volume 109 (2001) no. 1, pp. 147-181 | DOI | MR | Zbl
[29] Zeroes of principal -functions and random matrix theory, Duke Math. J., Volume 81 (1996) no. 2, pp. 269-322 | DOI | Zbl
[30] Definition of Families of -functions, 2008 (available at publications.ias.edu/sarnak)
[31] Families of -functions and their symmetries, Families of automorphic forms and the trace formula (Simons Symposia), Springer, 2016, pp. 531-578 | DOI | Zbl
[32] Sato-Tate theorem for families and low-lying zeros of automorphic -functions, Invent. Math., Volume 203 (2016) no. 1, pp. 1-177 (Appendix A by Robert Kottwitz, and Appendix B by Raf Cluckers, Julia Gordon and Immanuel Halupczok) | DOI | MR | Zbl
Cited by Sources: