Motohashi's fourth moment identity for non-archimedean test functions and applications

Abstract

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q^{1/4}. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic L-functions, which we also use to improve the best known subconvexity bounds for automorphic L-functions in the level aspect.

arXiv comment30 pages

References in corpus (6)

• The subconvexity problem for $\GL_{2}$
• The second moment of twisted modular L-functions
• Explicit calculations with Eisenstein series
• Twisted moments of L-functions and spectral reciprocity
• The angle of large values of $L$-functions
• Kloosterman sums and Fourier coefficients of Eisenstein series

Cited by in corpus (12)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Moments and hybrid subconvexity for symmetric-square L-functions
• The fourth moment of individual Dirichlet L-functions on the critical line
• Spectral reciprocity via integral representations
• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions III: The Twisted Case
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions I: The Cuspidal Case
• Local integral transforms and global spectral decomposition
• Correction to: Equidistribution of Eisenstein Series in the Level Aspect
• $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity
• Strong Hybrid Subconvexity for Twisted Selfdual ${\mathrm{GL}}_3$ $L$-Functions
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions II: The Eisenstein Case