Sarnak’s density hypothesis in horizontal families

Abstract: Let $G$ be a real semi simple Lie group with an irreducible unitary representation $\pi$. The non-temperedness of $\pi$ is measured by the real parameter $p(\pi)$ which is defined as the infimum of $p$ such that $\pi$ has non-zero 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 $O(V(q)^{2/p(\pi) +\varepsilon})$, where $V(q)$ is the covolume of $\Gamma(q)$. Sarnak and Xue proved this conjecture for $G=SL(2,\mathbb R),SL(2,\mathbb C)$. I will talk about the joint work with Gergely Harcos, Peter Maga and Djordje Milicevic where we prove bounds of the same quality that hold uniformly for families of pairwise non-commensurable lattices in $G=SL(2,\mathbb R)^a\times SL(2,\mathbb C)^b$. These families of lattices, which we call horizontal, are given as unit groups of maximal orders of quaternion algebras over number fields.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar