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On mean values and non-vanishing of derivatives of L-functions in a nonlinear family

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  20 April 2010

Ritabrata Munshi
Ritabrata Munshi*
Affiliation:
Department of Mathematics, Rutgers University, Hill Center, Piscataway, NJ 08854, USA (email: rmunshi@math.ias.edu)
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Abstract

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We prove a mean-value result for derivatives of L-functions at the center of the critical strip for a family of forms obtained by twisting a fixed form by quadratic characters with modulus which can be represented as sum of two squares. Such a family of forms is related to elliptic fibrations given by the equation q(t)y2=f(x) where q(t)=t2+1 and f(x) is a cubic polynomial. The aim of the paper is to establish a prototype result for such quadratic families. Though our method can be generalized to prove similar results for any positive definite quadratic form in place of sum of two squares, we refrain from doing so to keep the presentation as clear as possible.

Keywords

derivatives of L-functionsnon-vanishing of central values

MSC classification

Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11M41: Other Dirichlet series and zeta functions 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 19 - 34
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

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