Nonvanishing of central values of $L$-functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters

TL;DR

This paper proves the nonvanishing of central values of twisted L-functions for certain newforms in S_2(Γ_0(dp^2)), establishing conditions under which these L-values are nonzero, with implications for rank zero quotients of twisted Jacobians.

Contribution

It introduces a new nonvanishing result for L-functions of newforms twisted by quadratic characters, extending previous work with novel estimates and trace formulas.

Findings

Existence of newforms with nonzero central L-values for large primes p

Extension of nonvanishing results to twisted L-functions in specific modular forms spaces

Implications for rank zero quotients of twisted Jacobians

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 $f \in S_2(\Gamma_0(dp^2))$ 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. 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.