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The Second Moment of Twisted Modular L-Functions

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Abstract

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment

$$\underset{\chi \bmod {q}}{\left. \sum \right.^{\ast}} L(1/2, f_1\otimes \chi) \overline{L(1/2, f_2 \otimes \chi)}$$

of central values of L-functions of any two (possibly equal) fixed cusp forms f 1, f 2 twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9 % of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and a new large sieve type bound for Kloosterman sums where the summation lengths can be below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.

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Authors and Affiliations

  1. Mathematisches Institut, Bunsenstr. 3-5, 37073, Göttingen, Germany

    Valentin Blomer

  2. Department of Mathematics, Bryn Mawr College, 101 North Merion Avenue, Bryn Mawr, PA, 19010, USA

    Djordje Milićević

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  1. Valentin Blomer
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  2. Djordje Milićević
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Correspondence to Valentin Blomer.

Additional information

V. Blomer acknowledges the support by the Volkswagen Foundation and a Starting Grant of the European Research Council. D. Milićević acknowledges the support by the National Security Agency. Project is sponsored by the NSA under Grant Number H98230-14-1-0139. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

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Blomer, V., Milićević, D. The Second Moment of Twisted Modular L-Functions. Geom. Funct. Anal. 25, 453–516 (2015). https://doi.org/10.1007/s00039-015-0318-7

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  • DOI: https://doi.org/10.1007/s00039-015-0318-7

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