The density hypothesis for horizontal families of lattices

Abstract: Let G be a semisimple real Lie group without compact factors and Gamma an arithmetic lattice in G. Sarnak and Xue formulated a conjecture on the multiplicity with which a given irreducible unitary representation of G occurs in the right regular representation of G on L^2(Gamma\G). It is known for the groups SL(2,R) and SL(2,C) by the work of Sarnak-Xue (1991) and Huntley-Katznelson (1993). I will report on recent joint work with Mikołaj Frączyk, Péter Maga, and Djordje Milićević, where we prove a strong, effective version of the conjecture for products of SL(2,R)'s and SL(2,C)'s. We consider congruence lattices coming from quaternion algebras over number fields of bounded degree, and we address uniformity in all archimedean parameters.

number theory

Audience: researchers in the topic

Number theory during lockdown

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