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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 22, 2017 11:40, Moscow, Steklov Mathematical Institute
 


Non-vanishing of automorphic $L$-functions of prime power level (joint papers with O.G. Balkanova)

D. Frolenkov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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MP4 182.9 Mb

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D. Frolenkov
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Abstract: Iwaniec and Sarnak showed that at the minimum 25% of $L$-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when $N$ is square-free and $\phi(N)\sim N$. We extend the given result to the case of prime power level $N=p^{\nu}$, $\nu\geqslant 2$. The proof is based on asymptotic evaluation of twisted moments
$$ M_1(l,u,v)=\sum_{f \in H_{2k}^{*}(N)}^{h}\lambda_f(l)L_{f}(\tfrac{1}{2}+u+v), $$

$$ M_2(l,u,v)=\sum_{f \in H_{2k}^{*}(N)}^{h}\lambda_f(l)L_{f}(\tfrac{1}{2}+u+v),L_{f}(\tfrac{1}{2}+u-v), $$
and the technique of mollification.

Language: English

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