3 problems

Kurlberg–Rudnick's conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is the distribution of

Exponential-sum Sato–Tate conjecture. As $q\to\infty$ through primes, the family $\\{F(\chi;\psi)/\sqrt N\\}_\chi$ becomes equidistributed with respect to the Sato–Tate distributio…

Let $A\in SL_2(\mathbb{Z})$ be a hyperbolic matrix, let $U_N(A)$ be its quantization, and let $\\{\psi_j\\}$ be a Hecke basis. For a nonzero Fourier mode $n\in\mathbb{Z}^2$, consid…