Caltech Logo

Number Theory Seminar

Thursday, May 6, 2021
4:00pm to 5:00pm
Add to Cal
Online Event
Sarnak's density hypothesis in horizontal families
Mikołaj Frączyk, Department of Mathematics, The University of Chicago,

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

For more information, please contact Math Department by phone at 626-395-4335 or by email at