Description
Mathematics Colloquium
Speaker: Antareep Mandal, IIT Madras
Venue: Ramanujan Hall
Date: 4 pm, 23 November 2022
Title: Uniform sup-norm bounds for Siegel cusp forms
Abstract: Obtaining sup-norm bounds for automorphic forms is a classical problem related to fundamental problems in analytic number theory as well as in arithmetic geometry, as, for example, the Lindelöf hypothesis for automorphic L-functions, quantum ergodicity and entropy bounds, or the study of Arakelov invariants of modular curves. This talk is about obtaining uniform sup-norm bounds of Siegel cusp forms on average over an orthonormal basis using the heat kernel on Siegel upper half-space. We employ various techniques from the theory of harmonic analysis on symmetric spaces and classical as well as spectral theories of automorphic forms.
First, we relate Siegel cusp forms to the bottom of the spectrum of the respective weighted Siegel-Maass Laplacian. Then we construct and analyze the corresponding heat kernel to arrive at the conjectured optimal bounds for both cocompact and cofinite arithmetic subgroups of the symplectic group. Further, we show that these bounds are uniform over a tower of arithmetic subgroups of finite index.