Wednesday, October 4, 2017 - 2:45pm
Bryn Mawr College
Eigenfunctions of the Laplacian are basic building blocks of harmonic analysis on Riemannian manifolds. The sup-norm problem asks for bounds on the pointwise values of an L2-normalized eigenfunction in terms of its Laplacian eigen-value or other increasing parameters. Exciting progress in arithmetic cases means that this question, which is particularly interesting from the point of view of quan-tum mechanics, now occupies a prominent position at the interface of automorphic forms, analytic number theory, and analysis. In this talk, we will present our recent bounds (joint with Valentin Blomer, Gergely Harcos, and P´eter Maga) solving the sup-norm problem for spherical Hecke–Maaß newforms of square-free level for the group GL(2) over a number ﬁeld, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real ﬁelds they are as strong as the best known hybrid result over the rationals. The talk will emphasize several new features and diﬃculties that the number ﬁeld setting (and speciﬁcally complex places) introduces and new techniques we developed to address them, which are also of independent interest.