Spectral Fourth Moments of Hecke–Maaß Cusp Forms

Source abstract

Abstract In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $\mathrm{GL}(2)$. More precisely, we consider the family of Hecke–Maaß cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function $r(n)$ as well as to pointwise Diophantine exponents.