# On the sup-norm problem for eigenfunctions on arithmetic manifolds

Simple harmonics, such as monochromatic light waves or heart rhythms or standing patterns of a vibrating string, are basic building blocks of analysis. In the context of Riemannian manifolds, this role is played by Laplacian eigenfunctions, objects central in contexts ranging from spectral geometry, a field whose spirit was captured by Mark Kac’s famous question “Can you hear the shape of a drum?”, to quantum mechanics, where they represent “pure quantum states” and where their concentration of mass is closely related to geometry and dynamics.

Of critical importance in analysis, geometry, and physics is the eigenfunctions’ limiting behavior, which is closely related to the geometric and (in arithmetic cases, when they arise from automorphic forms) algebro-arithmetic and functorial structure of the underlying space. For example, while high-energy eigenfunctions on negatively curved manifolds are generically expected to exhibit rather temperate intensity fluctuations, this expectation is known to fail for a wide class of arithmetic 3-manifolds which contain immersed hyperbolic surfaces and which can be classified in terms of their invariant trace fields and invariant quaternion algebras.

In this talk, we will discuss the state of the art of the sup-norm problem on arithmetic hyperbolic manifolds and in particular present our recent upper bound (joint with Blomer and Harcos) for the sup-norm of Hecke-Maass cusp forms on a family of arithmetic hyperbolic 3-manifolds, obtained by combining spectral, diophantine, and geometric arguments in a noncommutative setting. This number-theoretic talk will be aimed at a general mathematical audience and will emphasize connections with algebra and geometry.