Nonvanishing of central values of $L$-functions of newforms in $S_2({\Gamma }_0(d{p}^2))$ twisted by quadratic characters

Abstract

We prove that for $d\in \{2,3,5,7,13\}$ 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 which 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.

arXiv comment19 pages with notable improvements from previous versions, "Author's Draft" before publication by Canadian Mathematical Bulletin