# ANNALES DE L'INSTITUT FOURIER

Simultaneous non-vanishing for Dirichlet $L$-functions
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1459-1524.

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 $L$-functions at the central point $s=\frac{1}{2}$. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet $L$-functions to derive that for any pair $\left({\omega }_{1},{\omega }_{2}\right)$ of multiplicative characters modulo a prime $q$, there is a positive proportion of $\chi \phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ such that the central values $L\left(\chi ,\frac{1}{2}\right),L\left(\chi {\omega }_{1},\frac{1}{2}\right)$ and $L\left(\chi {\omega }_{2},\frac{1}{2}\right)$ 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 $L$ au point central $s=\frac{1}{2}$. Comme application, nous exploitons notre résultat récent sur la mollification du quatrième moment des fonctions $L$ de Dirichlet et déduisons que pour ${\omega }_{1},{\omega }_{2}$ deux charactères multiplicatifs modulo un nombre premier $q$, il existe une proportion positive de $\chi \phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ telle que les valeurs centrales $L\left(\chi ,\frac{1}{2}\right),L\left(\chi {\omega }_{1},\frac{1}{2}\right)$ et $L\left(\chi {\omega }_{2},\frac{1}{2}\right)$ soient simultanément pas trop petites.

Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3275
Classification: 11L05, 11L07, 11M06
Keywords: Modular forms, L-functions, trace functions, bilinear forms, twisted Kloosterman sums
Zacharias, Raphaël 1

1 EPFL Mathgeom-TAN Station 8 C1015 Lausanne (Switzerland)
License: CC-BY-ND 4.0
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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/

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