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

TL;DR

This paper establishes explicit subconvex bounds for the sup-norms of automorphic forms on higher rank symmetric spaces, uniform in eigenvalue and discriminant, advancing understanding of automorphic form growth in complex algebraic structures.

Contribution

It introduces new subconvex hybrid bounds for automorphic forms on higher rank quotients, with explicit polynomial exponents in the degree of the division algebra.

Findings

Derived explicit polynomial bounds in eigenvalue and discriminant

Extended bounds to Eichler-type orders in division algebras of odd degree

Achieved uniform bounds in higher rank symmetric spaces

Abstract

Let $A$ be a central division algebra of prime degree $p$ over $\mathbb{Q}$. 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 $\operatorname{SL}_p(\mathbb{R})/\operatorname{SO}(p)$ by unit groups of orders in $A$. The exponents in the bounds are explicit and polynomial in $p$. We also prove subconvex hybrid bounds in the case of certain Eichler-type orders in division algebras of arbitrary odd degree.