Forum of Mathematics, Sigma (2019), vol. , e1, 46 pages 1 doi: ENDOSCOPY AND COHOMOLOGY IN A TOWER OF CONGRUENCE MANIFOLDS FOR U(n; 1) SIMON MARSHALL1 and SUG WOO SHIN2 1Department of Mathematics University of Wisconsin – Madison 480 Lincoln Drive Madison WI 53706, USA;
[email protected] 2Department of Mathematics University of California, Berkeley 901 Evans Hall, Berkeley, CA 94720, USA / Korea Institute for Advanced Study, Dongdaemun-gu, Seoul 130-722, Republic of Korea;
[email protected] Abstract By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to U(n; 1). In the case of lattices arising from Hermitian forms, we expect that the growth exponents we obtain are sharp in all degrees. 2010 Mathematics Subject Classification: 32N10 (primary); 32M15 (secondary) 1. Introduction This paper studies the limit multiplicity problem for cohomological automorphic forms on arithmetic quotients of U(N − 1; 1). Let F be a totally real number field with ring of integers OF. Write A for the ring of adeles over F. Let U(N) = UE=F(N) denote the quasi-split unitary group with respect to a totally imaginary quadratic extension E of F. Let G be a unitary group over F that is an inner form of UE=F(N). We assume that G has signature (N − 1; 1) at one real place and compact factors at all other real places. Let S be a finite set of c The Author(s) 2019.