The second moment of twisted modular L-functions

Abstract

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g 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 power-saving bounds for sums of products of Kloosterman sums where the length of the sum is 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.

64 pages

References in corpus (3)

• New equidistribution estimates of Zhang type
• Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli
• Lower bounds for moments of L-functions: symplectic and orthogonal examples

Cited by in corpus (33)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Explicit calculations with Eisenstein series
• Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli
• Twisted moments of L-functions and spectral reciprocity
• Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms
• On triple correlations of divisor functions
• On the fourth moment of Hecke Maass forms and the Random Wave Conjecture
• The second moment theory of families of L-functions
• Motohashi's fourth moment identity for non-archimedean test functions and applications
• High moments of the Estermann function
• The Weyl bound for triple product L-functions
• Lectures on Applied $\ell$-adic Cohomology
• Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients
• Density Theorems for Exceptional Eigenvalues for Congruence Subgroups
• Bessel periods and the non-vanishing of Yoshida lifts modulo a prime
• The divisor function in arithmetic progressions modulo prime powers
• Spectral reciprocity via integral representations
• The Fifth Moment of modular L-functions
• Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field
• The fifth moment of Hecke L-functions in the weight aspect
• Zeros of ${\mathrm{GL}}_2$ $L$-functions on the critical line
• The twisted second moment of modular half integral weight $L$--functions
• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
• On a Twisted Version of Linnik and Selberg's Conjecture on Sums of Kloosterman Sums
• Triple correlations of Fourier coefficients of cusp forms
• Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect
• Local integral transforms and global spectral decomposition
• A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions
• Uniform Titchmarsh divisor problems
• On the regularized $L^4$-norm of Eisenstein series in the level aspect
• On the $L^2$ restriction norm of large level
• Bias of Root Numbers for Modular Newforms of Cubic Level
• On the regularized $L^4$-norm for Eisenstein series in the level aspect, Part II