Abstract: The set X_{n} of isometry classes of even, unimodular lattices in Euclidean space R^{n}
has been determined for n less or equal to 24, thanks to works of Mordell, Witt and Niemeier.
The vector space of complexvalued functions on X_{n} is in a natural way a space of automorphic
forms for the orthogonal group of the quadratic form
x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}
on Q^{n}, 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 ndimensional ladic 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
nontempered and build from certain Siegel modular forms of genus 1 and 2.
As an application, it allows one to determine the pneighborhood 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.
