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2019 | Buch

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Automorphic Forms and Even Unimodular Lattices

Kneser Neighbors of Niemeier Lattices

verfasst von: Prof. Dr. Gaëtan Chenevier, Prof. Dr. Jean Lannes

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

An overview of the purpose and results. The connecting thread is the p-neighbor problem for even unimodular lattices in dimensions 16 and 24.

Gaëtan Chenevier, Jean Lannes

Chapter 2. Bilinear and Quadratic Algebra

Abstract

This chapter essentially recalls classical material. First we introduce the definitions and results from the theory of symmetric bilinear forms, quadratic forms, alternating forms, and their associated (classical) groups, that are used in this book. Next we explain the relation between root systems and even unimodular lattices. We recall in particular the remarkable method, introduced by Venkov, for retrieving the classification, due to Niemeier, of even unimodular lattices of dimension 24.

Gaëtan Chenevier, Jean Lannes

Chapter 3. Kneser Neighbors

Abstract

First we deal with the notion of d-neighbors (d positive integer, very often prime) for even unimodular lattices, introduced by M. Kneser, and with the associated Hecke operators; numerous examples are given. Then we analyse in depth the d-neighborhoods between a Niemeier lattice with roots and the Leech lattice; this sheds some light on the “holy constructions” of the latter by Conway and Sloane. At the very end of the chapter we describe an iterative 2-neighbor algorithm, essentially due to Borcherds, which starting from any even unimodular lattice of dimension 24 produces the Leech lattice after at most 5 steps.

Gaëtan Chenevier, Jean Lannes

Chapter 4. Automorphic Forms and Hecke Operators

Abstract

We first introduce the Hecke ring of a \(\mathbb {Z}\)-group G and discuss it basic properties (local-global structure, compatibility with isogenies, criterion for commutativity…). An elementary description of the Hecke rings of classical groups is given. Then, we recall the notion of a square integrable automorphic form for G, and that of a discrete automorphic representation of G. When G is the symplectic group Sp_2g, we explain how the theory of Siegel modular forms fits into this picture. We also show how the p-neighbor problem for even unimodular lattices in rank n may be viewed as a question about automorphic representations for the orthogonal \(\mathbb {Z}\)-group O_n.

Gaëtan Chenevier, Jean Lannes

Chapter 5. Theta Series and Even Unimodular Lattices

Abstract

Most of this chapter may be read independently. We first recall known properties of the Siegel theta series of even unimodular lattices in rank 16 (Witt, Igusa, Kneser) and 24 (Erokhin, Borcherds, Nebe-Venkov…). Then we give two proofs of Theorem A of the introduction (the p-neighbor problem in dimension 16): a short one relying on a construction of Ikeda, and a self-contained one based on a novel use of the triality principle. Along the way, we provide several elementary constructions of orthogonal modular forms, and simple instances of the Eichler commutation relations.

Gaëtan Chenevier, Jean Lannes

Chapter 6. Langlands Parametrization

Abstract

This chapter starts with a general discussion of reductive groups and their root data. Our aim is to review the Satake isomorphism for the Hecke ring of a reductive group scheme over the p-adic integers \(\mathbb {Z}_p\), as well as the Harish-Chandra isomorphism for the center of the universal enveloping algebra of a complex reductive Lie algebra. In both cases, we explain Langlands’ point of view on these isomorphisms in terms of the so-called Langlands dual group. This allows to define Langlands parameters and state the important Arthur-Langlands conjecture. As another application, we obtain a deeper understanding of the Hecke rings of classical \(\mathbb {Z}\)-groups.

Gaëtan Chenevier, Jean Lannes

Chapter 7. A Few Cases of the Arthur–Langlands Conjecture

Abstract

In this chapter, we give many examples of specific cases of the Arthur-Langlands conjecture concerning automorphic forms for SO_n or Siegel modular forms. They rely on concrete constructions of automorphic forms which are either classical (using theta series and results of Rallis and Böcherer), more recent (Ikeda liftings), or new (applications of the triality principle). As an application, we prove Theorem C of the introduction and provide a table of the Langlands parameters of the first discrete automorphic representations of SO₈.

Gaëtan Chenevier, Jean Lannes

Chapter 8. Arthur’s Classification for the Classical -groups

Abstract

In this chapter, we explain Arthur’s description of the discrete automorphic representations of classical groups in terms of selfdual cuspidal automorphic representations of GL_n. In agreement with the general philosophy of this book, we restrict our exposition to the level 1 automorphic representations, but provide a concrete form of the famous Arthur multiplicity formula for those representations whose Archimedean component is a discrete series. As a prerequisite, we discuss Shelstad’s parametrization of the individual elements of a discrete series L-packet, including several examples, and its generalization to the Adams-Johnson packets. We apply this theory to Siegel modular forms and orthogonal automorphic forms; this sheds much light on the more elementary constructions of the previous chapters. We also discuss applications to standard L-functions and Galois representations.

Gaëtan Chenevier, Jean Lannes

Chapter 9. Proofs of the Main Theorems

Abstract

This long chapter is the technical heart of the book. We first apply Arthur’s theory to PGSp₄ ≃SO_3,2 to study the standard parameter of the 4 vector-valued genus 2 Siegel modular forms of interest for Niemeier lattices. Then, we apply Arthur’s theory to give two short, but conditional, proofs of Theorem E of the introduction, as well as a full proof that Theorem F implies Theorem E. We prove Theorem F by an in depth study of the Weil explicit formula applied to the L-functions of pairs of cuspidal algebraic automorphic representations of general linear groups. We then use Theorem F to classify cuspidal Siegel modular forms of weight ≤ 12 (Theorems D and G of the introduction).

Gaëtan Chenevier, Jean Lannes

Chapter 10. Applications

Abstract

In a first part we explain how Theorem E of the introduction leads to a solution of the p-neighbor problem which involves in particular the four integers τ _j,k(p) introduced in Chap. 9, which are “genus 2 analogs” of the Ramanujan τ(p). Using the analysis made in Chap. 3 of neighborhoods of the Leech lattice, we determine τ _j,k(p) for p ≤ 113 (Theorem H), hence obtain an explicit numerical solution of the p-neighbor problem for those primes. In a second part we deduce from Theorem E numerous congruences for the τ _j,k(p) which may be thought of as “genus 2 analogs” of the famous mod 691 congruence satisfied by τ(p); one of these congruences had been conjectured by Günter Harder (Theorem I). This part notably involves the theory of Galois representations.

Gaëtan Chenevier, Jean Lannes

Backmatter

Titel: Automorphic Forms and Even Unimodular Lattices
verfasst von: Prof. Dr. Gaëtan Chenevier
Prof. Dr. Jean Lannes
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-95891-0
Print ISBN: 978-3-319-95890-3
DOI: https://doi.org/10.1007/978-3-319-95891-0