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Nonvanishing of Central Values of L-functions of Newforms in S2(Γ0(dp2)) Twisted by Quadratic Characters
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- Journal:
- Canadian Mathematical Bulletin / Volume 60 / Issue 2 / 01 June 2017
- Published online by Cambridge University Press:
- 20 November 2018, pp. 329-349
- Print publication:
- 01 June 2017
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- Article
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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.
