Distribution of mass of holomorphic cusp forms

TL;DR

This paper establishes bounds on the L^4-norm and geodesic-restricted L^2-norm of large-weight holomorphic cusp forms, using Watson's formula and mean value estimates of degree 6 L-functions.

Contribution

It introduces new bounds for norms of holomorphic cusp forms and applies these to restriction problems and subconvexity bounds of higher-degree L-functions.

Findings

Upper bounds for L^4-norm of cusp forms

Bounds for L^2-norm restricted to geodesics

Applications to Siegel modular forms and degree 8 L-functions

Abstract

We prove an upper bound for the L^4-norm and for the L^2-norm restricted to the vertical geodesic of a holomorphic Hecke cusp form of large weight. The method is based on Watson's formula and estimating a mean value of certain L-functions of degree 6. Further applications to restriction problems of Siegel modular forms and subconvexity bounds of degree 8 L-functions are given.