Orders in finite-dimensional algebras over number fi give rise to interesting locally symmetric spaces and algebraic varieties. Hilbert modular varieties or arithmetically defined hyperbolic 3-manifolds, compact ones as well as noncompact ones, are familiar examples. In this talk we discuss various cases related to the general linear group $GL(2)$ over orders in division algebras defined over some number field. Geometry, arithmetic, and the theory of automorphic forms are interwoven in a most fruitful way in this work. Finally we indicate a construction of non-vanishing square-integrable cohomology classes for such arithmetically defined groups.[-]

In this talk, we consider the limit multiplicity question (and some variants): how many automorphic forms of fixed infinity-type and level N are there as N grows? The question is well-understood when the archimedean representation is a discrete series, and we focus on non-tempered cohomological representations on unitary groups. Using an inductive argument which relies on the stabilization of the trace formula and the endoscopic classification, we give asymptotic counts of multiplicities, and prove the Sarnak-Xue conjecture at split level for (almost!) all cohomological representations of unitary groups. Additionally, for some representations, we derive an average Sato-Tate result in which the measure is the one predicted by functoriality. This is joint work with Rahul Dalal.[-]

A pair consisting of a K3 surface and a non-symplectic automorphism of order three is called an Eisenstein K3 surface. We introduce an invariant of Eisenstein K3 surfaces, which we obtain using the equivariant analytic torsion of an Eisenstein K3 surface and the analytic torsion of its fixed locus. Then this invariant gives rise to a function on the moduli space of Eisenstein K3 surfaces, which consists of 24 connected components and each of which is a complex ball quotient depending on the topological type of the automorphism of order three. Our main result is that, for each topological type, the invariant is expressed as the product of the Petersson norms of two kinds of automorphic forms, one is an automorphic form on the complex ball and the other is a Siegel modular form. In many cases, the automorphic form on the complex ball obtained in this way is a so-called reflective modular form. In some cases, this automorphic form is obtained as the restriction of an explicit Borcherds product to the complex ball. This is a joint work with Shu Kawaguchi.[-]