Familles de fonctions L de formes automorphes et applications
Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 275-307.

Une notion importante qui a émergé de la théorie analytique des fonctions L ces dernières années, est celle de famille. Par exemple les familles de fonctions L interviennent naturellement dans le modèle probabiliste des matrices aléatoires de Katz/Sarnak qui vise à prédire la répartition des zéros des fonctions L. L’analyse des fonctions L en famille intervient également dans la résolution (inconditionnelle) de divers problèmes ayant une signification arithmétique profonde, tel que le problème de montrer la non-annulation de valeur spéciales de fonctions L ou encore celui de borner non-trivialement ces valeurs (le problème de convexité). Dans cet article, nous passons en revue les techniques analytiques mises en jeu pour résoudre ces questions et décrivons plusieurs applications de nature arithmétique.

One of the important concept that has emerged these last years in the analytic theory of L functions, is the concept of families. For instance, families of L functions occur naturally in Katz/Sarnak’s probabilistic model of random matrices whose goal is to predict the distribution of zeros of L functions. The study of L functions within families occurs also in the (unconditional) resolution of several problems having some deep arithmetical meaning : the question of non-vanishing of special values of L functions or the problem of giving non-trivial upper bounds for these special values (the subconvexity problem). In this paper, we review the analytic method involved in solution of some of these problems and give several applications.

@article{JTNB_2003__15_1_275_0,
     author = {Michel, Philippe},
     title = {Familles de fonctions $L$ de formes automorphes et applications},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {275--307},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
     zbl = {1056.11027},
     mrnumber = {2019017},
     language = {fr},
     url = {archive.numdam.org/item/JTNB_2003__15_1_275_0/}
}
Michel, Philippe. Familles de fonctions $L$ de formes automorphes et applications. Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 275-307. http://archive.numdam.org/item/JTNB_2003__15_1_275_0/

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