This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur.

Its connecting thread is a question about lattices of rank 24: the problem of p -neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations.

This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.

Provides an accessible introduction to the Arthur-Langlands conjectures, illustrated by numerous illuminating examples and concrete number theoretic applications Presents the arithmetic theory of automorphic forms for reductive groups over the integers, with an emphasis on phenomena not seen in the traditional GL(2) case Offers a self-contained approach to the theory of Euclidean lattices through the general theory of quadratic forms over Dedekind domains

Autorentext
Gaëtan Chenevier is a number theorist and Senior CNRS Researcher at Université Paris-Sud.
Jean Lannes is a topologist and Emeritus Professor at Université Paris Diderot.

Inhalt
Preface.-1 Introduction.- 2 Bilinear and Quadratic Algebra.- 3 Kneser Neighbors.- 4 Automorphic Forms and Hecke Operators.- 5 Theta Series and Even Unimodular Lattices.- 6 Langlands Parametrization.- 7 A Few Cases of the ArthurLanglands Conjecture.- 8 Arthur's Classification for the Classical Z-groups.- 9 Proofs of the Main Theorems.- 10 Applications.- A The BarnesWall Lattice and the Siegel Theta Series.- B Quadratic Forms and Neighbors in Odd Dimension.- C Tables.- References.-.Postface- Notation Index.-Terminology Index.