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Washington State University
Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics and Statistics
  August 19, Monday, 4:10 - 5:00 PM


Abhishek Saha

  School of Mathematical Sciences
  Queen Mary University of London

Title: Some analytic aspects of automorphic forms and L-functions

The eigenfunctions (of the Laplacian) on various geometric spaces constitute a class of mathematical objects of fundamental importance. From the point of view of quantum mechanics, the eigenfunctions correspond to particles moving with a certain energy, which leads naturally to questions motivated by subfields of physics. For example, one also has the so-called sup-norm problem, which asks how high the peaks of an eigenfunctions can be.
       In this talk, I will give a gentle introduction to some of these problems in a setting where number theory plays a key role. In the special case when the manifold is a surface of constant negative structure, and is constructed from "quaternion algebras", a famous result of Iwaniec and Sarnak improves upon the trivial bound for the sup-norm using number-theoretic techniques. I will explain this result, and then talk about recent progress on an analogous question where the underlying surface is itself allowed to vary. There are also many interesting connections between these questions and deep open conjectures in number theory such as the Grand Riemann hypothesis, which I will attempt to explain.