Density conjecture for horizontal families of lattices in SL(2)

Let G be a real semi-simple Lie group with an irreducible unitary representation \pi. The non-temperedness of \pi is measured by the parameter p(\pi) which is defined as the infimum of p\geq 2 such that \pi has matrix coefficients in L^p(G). Sarnak and Xue conjectured that for any arithmetic lattice \Gamma \subset G and principal congruence subgroup \Gamma(q)\subset \Gamma, the multiplicity of \pi in L^2(G/\Gamma(q)) is at most O(V(q)^{2/p(\pi)+\epsilon}) where V(q) is the covolume of \Gamma(q). In some contexts such estimate is a decent substitute for the Ramanujan conjecture. For G of real rank 1 Sarnak and Xue translate the estimate into a diophantine counting problem which they managed to solve SL(2,R) and SL(2,C). In this talk I will explain how one can get the same multiplicity bounds for families of pairwise non-commensurable lattices in G=SL(2,R),SL(2,C) given as unit groups of maximal orders of quaternion algebras over number fields (“horizontal families”). Namely: m(\pi,\Gamma) V^{2/p(\pi)+\varepsilon}, where V is the covolume of \Gamma. I will also discuss similar bounds on multiplicities of representations \pi_1\times \pi_2 of G=SL(2,R)^2 where \pi_1 is fixed non-tempered but \pi_2 is allowed to vary together with the lattice. Talk is based on a joint work with Gergely Harcos, Peter Maga and Djordje Milicevic.