Hybrid bounds for the sup-norm of automorphic forms in higher rank

Abstract

Let A A be a central division algebra of prime degree p p over Q \mathbb {Q} . We obtain subconvex hybrid bounds, uniform in both the eigenvalue and the discriminant, for the sup-norm of Hecke-Maaß forms on the compact quotients of SL p ⁡ ( R ) / SO ⁡ ( p ) \operatorname {SL}_p(\mathbb {R})/\operatorname {SO}(p) by unit groups of orders in A A . The exponents in the bounds are explicit and polynomial in p p . We also prove subconvex hybrid bounds in the case of certain Eichler-type orders in division algebras of arbitrary odd degree.