Arithmetic subgroups of Lie groups are a natural generalization of SL in SL 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， Cari Ludwig Siegel and Andre Weil. They reflect their deep knowledge of the subject and their perspectives. The lecture notes of Weit 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 Therefore， this book will be a very valuable introduction and historical reference for all people who are interested in arithmetic subgroups and locally symmetric spaces.

　　This book in your hand consists of papers and lecture notes of four great mathematicians： Armand Borel， Roger Godement， Carl Ludwig Siegel and Andre Weil. Though they have different personalities， they share some common interests and good tastes， and they all write well. These writings in this book all deal with arithmetic subgroups of Lie groups and reduction theories for them， and reflect their deep knowledge of the subjects and their perspectives.
　　Before we explain the contents of these papers and the lecture notes， we first answer the question of why we publish these old writings， and why the reader should open and read this book.
　　Among these four authors mentioned above， probably the one with most opinions is Weil. In his autobiography titled The apprenticeship of a mathematician， Weil wrote：
　　I had become convinced that what really counts in the history of humanity are the truly great minds， and the only way to get to know these minds was through direct contact with the works.
　　In his commentaries on the second volume of his collected works， Weilwrote：
　　To comment on the works of Siegel has always appeared to me to be one of the
　　tasks that a present-day mathematician may most usefully undertake.
　　As the reader can tell， the papers and lecture notes by Weil in this book are his expositions and expansions of some ideas and results of Siegel.
　　To understand the relations between these writings in this book better and to put them in a historical perspective， we recall the history of arithmetic groups and reduction theories.
　　As the name suggests， the reduction theory started with the reduction of binary quadratic forms. The earliest and best known binary quadratic form is X2 + y2 as in the Pythagorean theorem. Motivated by this， Fermat studied the problem of representing primes and asserted that prime numbers of the form 4m+l can be written as sums of two squares， i.e.， represented by the quadratic form X2+y2. He also claimed several related results. Euler tried to prove them without complete success. Then in 1775， Lagrange developed a general theory of binary quadratic forms to prove these assertions of Fermat and more.