Revista Matemática Iberoamericana


Full-Text PDF (402 KB) | Metadata | Table of Contents | RMI summary
Volume 21, Issue 1, 2005, pp. 263–312
DOI: 10.4171/RMI/423

Published online: 2005-04-30

Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique

Emmanuel Royer[1] and Jie Wu[2]

(1) Université de Montpellier II, France
(2) Université Henri Poincaré, Vandoeuvre lès Nancy, France

For each weight $k$ and level $N$ square free and without small prime factors, we prove the existence of primitive forms $f_+$ and $f_-$ of weight $k$ and level $N$ such that $$ L(1,\sym^2f_+)\gg_{k}[\log\log(3N)]^{3} $$ and $$ L(1,\sym^2f_-)\ll_{k}[\log\log(3N)]^{-1}. $$ The result comes from a delicate study of the moments of $L(1,\sym^2 f)$. This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.

Keywords: forme automorphe, carré symétrique, fonction $L$, valeur spéciale

Royer Emmanuel, Wu Jie: Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique. Rev. Mat. Iberoam. 21 (2005), 263-312. doi: 10.4171/RMI/423