Rizwanur Khan

Distribution of mass of holomorphic cusp forms

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 Watsons 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.

Simultaneous non-vanishing of GL (3) × GL (2) and GL (2) L -functions

Fix g a Hecke–Maass form for SL 3 ( ). In the family of holomorphic newforms f of fixed weight and large prime level q , we find the average value of the product . From this we derive a result on the simultaneous non-vanishing of these L -functions at the central point.

On the fourth moment of Hecke–Maass forms and the random wave conjecture

Conditionally on the Generalized Lindelof Hypothesis, we obtain an asymptotic for the fourth moment of Hecke Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the Random Wave Conjecture.