"Bounds For Eigenforms On Arithmetic Hyperbolic 3-Manifolds"
Duke Mathematical Journal, Vol. 165, Issue 4, Pages 625–659, Duke University Press, March 15, 2016. Written with Valentin Blomer and Gergely Harcos.
On a family of arithmetic hyperbolic 3-manifolds of square-free level, we prove an upper bound for the sup-norm of Hecke-Maa beta cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of Diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.