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.
Publié le : 2005-03-15
Classification:
forme automorphe,
carré symétrique,
fonction $L$,
valeur spéciale,
11F12,
11F25,
11F67,
11M41,
11N36,
11N37
@article{1114176235,
author = {Royer, Emmanuel and Wu, Jie},
title = {Taille des valeurs de fonctions $L$ de carr\'es sym\'etriques au bord de la bande critique},
journal = {Rev. Mat. Iberoamericana},
volume = {21},
number = {2},
year = {2005},
pages = { 263-312},
language = {en},
url = {http://dml.mathdoc.fr/item/1114176235}
}
Royer, Emmanuel; Wu, Jie. Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique. Rev. Mat. Iberoamericana, Tome 21 (2005) no. 2, pp. 263-312. http://gdmltest.u-ga.fr/item/1114176235/