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Filling a gap in the literature, this text gives a solid, rigorous foundation in the subject of the arithmetic of algebraic groups and reduction theory. It follows different developments in this area, geometric as well as number theoretical, at a level suitable for graduate students and researchers in related fields.
List 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.
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.
Summary
Filling a gap in the literature, this text gives a solid, rigorous foundation in the subject of the arithmetic of algebraic groups and reduction theory. It follows different developments in this area, geometric as well as number theoretical, at a level suitable for graduate students and researchers in related fields.
