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

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 ${\mathrm{SL}}_p(\mathbb{R})/\mathrm{SO}(p)$ 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.

29 pages. Minor improvements, implemented suggestions from referee. To appear in Trans. AMS. Comments are welcome!

References in corpus (4)

• Hybrid sup-norm bounds for Maass newforms of powerful level
• Local bounds for $L^p$ norms of Maass forms in the level aspect
• On sup-norm bounds part II: $GL(2)$ Eisenstein series
• Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

Cited by in corpus (1)

• Theta functions, fourth moments of eigenforms, and the sup-norm problem II