Automorphic forms and even unimodular lattices : Kneser neighbors of Niemeier lattices
 Responsibility
 Gaëtan Chenevier, Jean Lannes, translated by Reinie Erné.
 Publication
 Cham : Springer Nature Switzerland AG, [2019]
 Copyright notice
 ©2019
 Physical description
 xxi, 417 pages : illustrations ; 24 cm.
 Series
 Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 69.
At the library
Science Library (Li and Ma)
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Call number  Status 

QA243 .C44 2019  In process Request 
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Description
Creators/Contributors
 Author/Creator
 Chenevier, Gaëtan, author.
 Contributor
 Lannes, Jean, author.
 Erné, Reinie, translator.
Contents/Summary
 Bibliography
 Includes bibliographic references (p. 397408) and indexes.
 Contents

 Preface
 Introduction
 Even Unimodular Lattices
 Kneser Neighbors
 Theta Series and Siegel Modular Forms
 Automorphic Forms for the Classical Groups
 Algebraic Automorphic Representations of Small Weight
 Proofs of Theorems D and E
 A Few Applications
 Bilinear and Quadratic Algebra
 Basic Concepts in the Theory of Bilinear and Quadratic Forms
 On bModules and qModules over Z
 Root Systems and Even Unimodular Lattices
 Kneser Neighbors
 Variations on the Notion of Kneser Neighbors
 Hecke Operators Associated with the Notion of Neighborhood
 Examples
 Determination of T2 for n = 16
 Determination of T3 for n = 16
 Determination of T2 for n = 24 (Following NebeVenkov [156])
 dNeighborhoods Between a Niemeier Lattice with Roots and the Leech Lattice
 Necessary Conditions for a Niemeier Lattice with Roots to Have a dNeighbor with No Roots
 On the hNeighborhoods and (h + 1)Neighborhoods Between a Niemeier Lattice with Roots and Coxeter Number h and the Leech Lattice
 On the Stabilizers for the Action of W on PLreg (Z/d), for L a Niemeier Lattice with Roots
 Complement : on the 2Neighbors of a Niemeier Lattice with Roots, Associated with a Weyl Vector
 Automorphic Forms and Hecke Operators
 Lattices and Class Sets of Zgroups
 Linear Groups
 Orthogonal and Symplectic Groups
 SOL Versus OL
 Orthogonal Groups in Odd Dimensions
 Hecke Correspondences
 General Formalism
 A Functor from Modules to H(X)oppModules
 The Hecke Ring of a Zgroup
 Some Classical Hecke Rings
 H(SOL) Versus H(OL)
 Isogenics
 Automorpic Forms of a Zgroup
 SquareIntegrable Automorphic Forms
 The Set (G)
 Automorphic Forms for On
 Automorphic Forms for the Zgroups G with G(R) Compact
 The Case of the Groups On and SOn
 An Invariant Hermitian Inner Product
 Siegel Modular Forms
 The Classical Point of View
 Fourier Series Expansions and Cusp Forms
 The Relation Between Sw (Sp2g (Z)) and A2(PGS2g)
 The Action of Hecke Operators
 Adisc (Sp2g) May Be Deduced from Adisc (PGSp2g)
 Theta Series and Even Unimodular Lattices
 Siegel Theta Series
 Theta Series of E8, E8 and E16
 Theta Series of the Niemeier Lattices
 An Alternative Construction of I4 by Triality
 Harmonic Theta Series
 Hecke Operators Corresponding to Perestroikas
 Passage from PGOn to PGSOn
 Triality for PGSO8
 One Last Theta Series and the End of the Proof
 Appendix : a Simple Example of the Eichler Relations
 Langlands Parametrization
 Basic Facts on Reductive kGroups
 The Based Root Datum of a Reductive kGroup
 Langlands Dual
 Examples
 Representations of Split Reductive Groups in Characteristic Zero
 Satake Parametrization
 The Satake Isomorphism
 The Two Natural Bases of the Hecke Ring of G
 The Classical Groups : a Collection of Formulas
 The HarishChandra Isomorphism
 The Center of the Universal Enveloping Algebra of a Reductive Cgroup
 The Infinitesimal Character of a Unitary Representation
 The ArthurLanglands Conjecture
 Langlands Parametrization of (G) for G Semisimple over Z
 A Few Formulas
 The ArthurLanglands Conjecture
 A Few Examples
 Relations with LFunctions
 The Generalized Ramanujan Conjecture
 A Few Cases of the ArthurLanglands Conjecture
 The Eichler Relations Revisited
 The Point of View of Rallis
 A Refinement : Passage to the Spin Groups
 Disc(O8) and Triality
 A Few Consequences of the Work of Ikeda and Böcherer
 A Table of the First Elements of disc(SO8)
 The Space Mdet(O24)
 Arthur's Classification for the Classical Zgroups
 Standard Parameters for the Classical Groups
 SelfDual Representations of PGLn
 Duality in disc(PGLn)
 Regular Algebraic Representations
 Representations of GLn(R)
 The Ramanujan Conjecture and Galois Representations
 LFunctions of Pairs of Algebraic Representations
 Arthur's Multiplicity Formula
 Arthur's SymplecticOrthogonal Alternative
 The Multiplicity Formula : General Assumptions
 The Group C and the Character E
 The Case of the Chevalley Groups
 Discrete Series
 Discrete Series, Following HarishChandra
 Shelstad's Canonical Parametrization, the Case of Split Groups
 Dual Interpretation and Link with Arthur Packets
 Example : the Holomorphic Discrete Series of Sp2g(R)
 Pure Forms of the Split Groups
 AdamsJohnson Packets
 Example : AdamsJohnson Parameters of Sp2g
 Dual Parametrization of AJ
 AdamsJohnson Packets and Arthur Packets
 Explicit Multiplicity Formulas
 Explicit Formula for Sp2g
 Explicit Formula for SOn with n = ±1 mod 8
 Explicit Formula for SOn with n = 0 mod 8
 Compatibility with the Theta Correspondence
 Compatibility with Bocherer's Lfunction
 Proofs of the Main Theorems
 Tsushima's Modular Forms of Genus 2
 Tsushima's Dimension Formula
 Standard Parameters of the First Six Forms of Genus 2
 A Few Eigenvalues of Hecke Operators
 Where We Explain the Occurrence of the j, k in Table 7.1
 Disc(SO24) and the NebeVenkov Conjecture
 A Characterization of Table 1.2
 Statements and an Overview of the Proofs
 Theorem 9.2.5 Implies Theorem 9.2.6
 First, Conditional, Proof of Theorem 9.2.5
 Second Proof of Theorem 9.2.5, Modulo Conjecture 8.4.22
 Algebraic Representations of Motivic Weight at Most 22
 A Classification Statement
 The Explicit Formula for the LFunctions of Pairs
 Odlyzko's Function
 Beginning of the Proof of Theorem 9.3.2 : the Case w 20
 Intermezzo : a Geometric Criterion
 End of the Proof of Theorem 9.3.2 : the Case of Motivic Weights 21 and 22
 Complements
 Proof of Theorem E
 A New Proof of Theorem A
 Proof of Theorem E
 Siegel Modular Forms of Weight at Most 12
 Forms of Weight 12 and a Proof of Theorem D of the Introduction
 Forms of Weight at Most 11
 Toward a New Proof of the Equality (X24) = 24
 A Few Elements of disc(SOn) for n = 15, 17 and 23
 Applications
 24 lAdic Galois Representations
 Back to pNeighbors of Niemeier Lattices
 Determination of the Tj, k (q) for Small Values of q
 Determination of the Tj, k (p) for p 113
 Determination of the Tj, k (p2) for p 29
 HarderType Congruences
 The BarnesWall Lattice and the Siegel Theta Series of Even Unimodular Lattices of Dimension 16
 Quadratic Forms and Neighbors in Odd Dimension
 Basic Concepts in the Theory of Quadratic Forms on a Projective Module of Odd Constant Rank
 On the qiModules over Z
 The Theory of pNeighbors for qiModules over Z
 The Theory of pNeighbors for Even Lattices of Determinant 2
 Examples
 Determination of T2 for n = 17
 Determination of T2 for n = 15
 On the Determination of T2 for n = 23
 Tables
 References
 Postface
 Notation Index
 Terminology Index.
 Summary

This book includes a selfcontained 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 pneighborhoods 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, Lfunctions 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 nonspecialists.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2019
 Series
 Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 69
 ISBN
 9783319958903 hardbound : EUR106.99
 3319958909 (hardcover)
 3319958917 (ebook)
 9783319958910 (ebook)