Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

Blomer, Valentin and Harcos, Gergely and Milićević, D. (2016) Bounds for eigenforms on arithmetic hyperbolic 3-manifolds. Duke Mathematical Journal, 165 (4). pp. 625-659. ISSN 0012-7094

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Abstract

On a family of arithmetic hyperbolic 3-manifolds of square-free level, we prove an upper bound for the sup-norm of Hecke–Maaß 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.

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