Special values of symmetric power $L$-functions and Hecke eigenvalues

Royer, Emmanuel; Wu, Jie

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 703-753.

On calcule les moments des fonctions $L$ de puissances symétriques de formes modulaires au bord de la bande critique en les tordant par les valeurs centrales des fonctions $L$ de formes modulaires. Dans le cas des puissances paires, on montre qu’il est équivalent de tordre par la valeur au bord des fonctions $L$ de carrés symétriques. On en déduit des informations sur la taille des valeurs au bord de la bande critique de fonctions $L$ de puissances symétriques dans certaines sous-familles. Dans une deuxième partie, on étudie la répartition des petites et grandes valeurs propres de Hecke. On en déduit des informations sur des conditions d’extrémalité simultanées des valeurs de fonctions $L$ de puissances symétriques au bord de la bande critique.

We compute the moments of $L$-functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the $L$-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square $L$-functions. We deduce information on the size of symmetric power $L$-functions at the edge of the critical strip in subfamilies. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the simultaneous extremality conditions on the values of $L$-functions of symmetric powers of modular forms at the edge of the critical strip.

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