Arithmetic subgroups of Lie groups are a natural generalization of \(SL(n, \mathbb{Z})\) in \(SL(n, \mathbb{R})\) 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: Armand 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 Therefore, this book will be a very valuable introduction and historical reference for everyone interested in arithmetic subgroups and locally symmetric spaces.