Low-lying zeros of L-functions for Quaternion Algebras
[Low-lying zeros of L-functions for Quaternion Algebras]
Annales de l'Institut Fourier, Online first, 42 p.

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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DOI : https://doi.org/10.5802/aif.3428
Classification : 11F66,  11F72
Mots clés : représentation automorphe, fonction L, petits zéros, type de symétrie, conjecture de densité, opérateur de Hecke, algèbre de quaternions
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Lesesvre, Didier. Low-lying zeros of L-functions for Quaternion Algebras. Annales de l'Institut Fourier, Online first, 42 p.

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