3 problems

Let $U(z)=y^{k/2}f(z)$, where $f$ is a holomorphic Hecke cusp form of weight $k$, with Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$…

Let $U(z)$ be a normalized restricted Hecke eigenfunction on a horocycle, and let $n$ be a positive integer. Sarnak–Reznikov's Gaussian moment conjecture. The $2n$-th moment of…

Let $\phi$ be a Hecke–Maass cusp form on the modular surface $\mathbb{X}$, normalized in the setting of the paper, and let $\mathcal{C}$ be a closed horocycle. Hejhal–Rackner conje…