Special Lecture Series: Modular forms of classical groups and even unimodular lattices

January 14-22, 2015

Gaetan Chenevier

Abstract: The set Xn of isometry classes of even, unimodular lattices in Euclidean space Rn has been determined for n less or equal to 24, thanks to works of Mordell, Witt and Niemeier. The vector space of complex-valued functions on Xn is in a natural way a space of automorphic forms for the orthogonal group of the quadratic form
x12 + x22 + ... + xn2
on Qn, and as such, it is equipped with a collection of Hecke operators. In this series of lectures, I will explain how to determine the collection of automorphic representations, or if one prefers the n-dimensional l-adic Galois representations, associated with the Hecke eigenforms occurring in those spaces for n less or equal to 24. This provides a rather curious list of automorphic representations, which turn out to be all non-tempered and build from certain Siegel modular forms of genus 1 and 2. As an application, it allows one to determine the p-neighborhood graph of the Niemeier lattices for each prime p. (The Wikipedia page on Niemeier lattices gives a nice representation of this graph for p = 2, which is due to Borcherds.) In dimension n = 16 which is the first interesting case, I will give an "elementary" proof based on Siegel theta series and triviality. In dimension n = 24, the proof that I will present relies in particular on Arthur's recent results on the automorphic spectrum of classical groups. I will try to introduce Arthur's theory during the lectures. This is joint work with Jean Lannes.

Part Time Date Place Notes
I 3 - 4:30 PM 01/14/15 Science Center 507 PDF
II 3 - 4:30 PM 01/20/15 Science Center 507 PDF
III 3 - 4:30 PM 01/22/15 Science Center 507 PDF