Reduction Theory and Arithmetic Groups

by Joachim Schwermer

Overview

Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.

Product Details

ISBN-13:	9781108832038
Publisher:	Cambridge University Press
Publication date:	12/15/2022
Series:	New Mathematical Monographs , #45
Pages:	374
Product dimensions:	6.93(w) x 9.88(h) x 0.98(d)

About the Author

Joachim Schwermer is Emeritus Professor of Mathematics at the University of Vienna, and recently Guest Researcher at the Max-Planck-Institute for Mathematics, Bonn. He was Director of the Erwin-Schrödinger-Institute for Mathematics and Physics, Vienna from 2011 to 2016. His research focuses on questions arising in the arithmetic of algebraic groups and the theory of automorphic forms.

Table of Contents

Part I. Arithmetic Groups in the General Linear Group: 1. Modules, lattices, and orders; 2. The general linear group over rings; 3. A menagerie of examples – a historical perspective; 4. Arithmetic groups; 5. Arithmetically defined Kleinian groups and hyperbolic 3-space; Part II. Arithmetic Groups Over Global Fields: 6. Lattices – Reduction theory for GLn; 7. Reduction theory and (semi)-stable lattices; 8. Arithmetic groups in algebraic k-groups; 9. Arithmetic groups, ambient Lie groups, and related geometric objects; 10. Geometric cycles; 11. Geometric cycles via rational automorphisms; 12. Reduction theory for adelic coset spaces; Appendices: A. Linear algebraic groups – a review; B. Global fields; C. Topological groups, homogeneous spaces, and proper actions; References; Index.