Spectral fourth moments of Hecke--Maa{\ss} cusp forms

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.