[Low-lying zeros of L-functions for Quaternion Algebras]
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.
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.
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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
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@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 = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Tome 71 (2021) no. 4, pp. 1635-1676. doi : 10.5802/aif.3428. https://aif.centre-mersenne.org/articles/10.5802/aif.3428/
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