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Nonvanishing of Central Values of $L$-functions of Newforms in $S_2 (\Gamma_0 (dp^2))$ Twisted by Quadratic Characters

  Published:2017-01-26
 Printed: Jun 2017
  • Samuel Le Fourn,
    ENS de Lyon, Lyon, France
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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.
Keywords: nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
MSC Classifications: 14J15, 11F67 show english descriptions Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
14J15 - Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
 

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