Djordje Milićević: Beyond the spherical sup-norm problem (NTWS 193)

Abstract: The sup-norm problem on arithmetic Riemannian manifolds occupies a prominent place at the intersection of harmonic analysis, number theory, and quantum mechanics. It asks about the sup-norm of L^2-normalized joint eigenfunctions of invariant differential operators and Hecke operators — that is, automorphic forms — most classically in terms of their Laplace eigenvalues (as in the QUE problem for high-energy eigenstates), but also in terms of the volume of the manifold and other parameters.
In this talk, we will motivate the sup-norm problem and then describe our results, joint with Blomer, Harcos, and Maga, which for the first time solve it for non-spherical Maass forms of an increasing dimension of the associated K-type, on an arithmetic quotient of G=SL(2,C), with K=SU(2). We combine representation theory, spectral analysis, and Diophantine arguments, developing new Paley-Wiener theory for G and sharp estimates on spherical trace functions of arbitrary K-type on the way to a novel counting problem of Hecke correspondences close to various special submanifolds of G.


Original air date:
Thursday, December 14, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 15, 2023 (12am CST, 3am AEDT, 5am NZDT)