Nonvanishing of Central Values of L-functions of Newforms in S2(Γ0(dp2)) Twisted by Quadratic Characters

Fourn, Samuel Le

Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 329-349

Résumé

We prove that for $d\in \left\{ 2,3,5,7,13 \right\}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi $ , if a prime $p$ is large enough compared to $D$ , there is a newform $f\in {{S}_{2}}({{\Gamma }_{0}}(d{{p}^{2}}))$ with sign $(+1)$ with respect to the Atkin–Lehner involution ${{w}_{{{p}^{2}}}}$ such that $L(f\otimes \chi ,1)\ne 0$ . This result is obtained through an estimate of a weighted sum of twists of $L$ -functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$ -functions $L(f\otimes \chi ,\cdot )$ and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Détail

DOI : 10.4153/CMB-2016-085-6

Mots-clés : 14J15, 11F67, nonvanishing of L-functions of modular forms, Petersson trace formula, rank zero quotients of Jacobians