We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.

Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1,1)$. The quotient ${ℍ}^{n+1}/\Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PS{L}_2\left(\mathbb{Z}\right)\setminus PS{L}_2\left(ℝ\right)$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PS{L}_2\left(\mathbb{Z}\right)\setminus ℍ$. The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S{L}_2\left(ℝ\right)$. In the proof the key estimates come from applying...