ENDOSCOPY AND COHOMOLOGY IN A TOWER OF CONGRUENCE MANIFOLDS FOR

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.