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Arithmetic groups and reduction theory. Edited by Lizhen Ji and translated by Wolfgang Globke, Lizhen Ji, Enrico Leuzinger and Andreas Weber. (English) Zbl 1461.11001

CTM. Classical Topics in Mathematics 10. Beijing: Higher Education Press (ISBN 978-7-04-053375-0/hbk). 138 p. (2020).
Publisher’s description: Arithmetic subgroups of Lie groups are a natural generalization of \(\mathrm{SL}(n,\mathbb Z)\) in \(\mathrm{SL}(n,\mathbb R)\) and play an important role in the theory of automorphic forms and the theory of moduli spaces in algebraic geometry and number theory through locally symmetric spaces associated with arithmetic subgroups. One key component in the theory of arithmetic subgroups is the reduction theory which started with the work of Gauss on quadratic forms.
This book consists of papers and lecture notes of four great contributors of the reduction theory: Armond Borel, Roger Godement, Carl Ludwig Siegel and André Weil. They reflect their deep knowledge of the subject and their perspectives. The lecture notes of Weil are published formally for the first time, and other papers are translated into English for the first time. Therefore, this book will be a very valuable introduction and historical reference for everyone interested in arithmetic subgroups and locally symmetric spaces.
Table of contents:
1 On the reduction theory of quadratic forms by Carl Ludwig Siegel, translated by Wolfgang Globke and Andreas Weber from the original German edition [Zbl 0097.00901] (1-46)
2 Reduction of quadratic forms, according to Minkowski and Siegel by André Weil, translated by Lizhen Ji (47–54)
3 Groups of indefinite quadratic forms and alternating bilinear forms by André Weil, translated by Lizhen Ji (55–64)
4 Discontinuous subgroups of classical groups by André Weil (65-104)
5 Fundamental sets for arithmetic groups by Armand Borel, translated by Lizhen Ji from the original French paper [Zbl 0161.02603] (105–118)
6 Fundamental domains of arithmetic groups by Roger Godement, translated by Enrico Leuzinger from the original French edition [Zbl 0136.30101] (119–139)

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Exx Forms and linear algebraic groups
11F06 Structure of modular groups and generalizations; arithmetic groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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