2022/12/23 by Rahul Dalal, Dalal, Rahul, Mathilde Gerbelli-Gauthier +1
Mathematics · #Advanced Algebra and Geometry #Random Matrices and Applications #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2212.12138
Consider the family of automorphic representations on a unitary group with cohomological factor π0 at infinity and given split level. We compute statistics of this family as the level goes to infinity. For unramified unitary groups and a large class of π0, we use the endoscopic classification of representations to compute the exact leading term for counts of representations and averages of Satake parameters. The bounds on our error terms are similar to previous work by Shin-Templier who studied the case of discrete series at infinity. We also prove new upper bounds for all cohomological representations. This has many corollaries: new exact asymptotics on the growth of cohomology in certain towers of locally symmetric spaces, an averaged Sato-Tate equidistribution law for spectral families with specific non-tempered cohomological components at infinity, and the Sarnak-Xue density hypothesis for cohomological representations at infinity on all unitary groups of rank ≥ 5.