Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.[-]

We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the characteristic function, the density, and the repartition function of this distribution in terms of higher transcendental functions, namely Legendre and Meijer functions.[-]