Distribution of mass of holomorphic cusp forms

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 $f$ of large weight $k$. 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.