The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions

Abstract

In this paper, we prove the smooth cubic moments vanish for the Hecke--Maass cusp forms, which gives a new case of the random wave conjecture. In fact, we can prove a polynomial decay for the smooth cubic moments, while for the smooth second moment (i.e. QUE) no rate of decay is known unconditionally for general Hecke--Maass cusp forms. The proof bases on various estimates of moments of central $L$-values. We prove the Lindelöf on average bound for the first moment of $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in short intervals of the subconvexity strength length, and the convexity strength upper bound for the mixed moment of $\mathrm{GL}(2)$ and the triple product $L$-functions. In particular, we prove new subconvexity bounds of certain $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions.

37 pages, incorporates the referees' comments and corrections; to appear in Mathematische Annalen. Comments welcome

References in corpus (1)

• Moments and hybrid subconvexity for symmetric-square L-functions

Cited by in corpus (1)

• Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms