Nonvanishing of central values of $L$-functions of newforms in $S_2 (Γ_0 (dp^2))$ twisted by quadratic characters

Samuel Le Fourn

Published 2015-06-29Version 1

We prove that for $d \in \{ 2,3,5,7,13 \}$ and a quadratic real field $K$ (or $\Q$) of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is an eigenform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg et al., and relies on classical techniques and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. This has applications in the study of rational points on some families of modular curves.