The goal of this chapter is to introduce the groups of adeles of algebraic groups and develop some of their basic properties, thereby providing techniques that are fundamental in the arithmetic theory of algebraic groups. In section one, we first discuss the adelic spaces associated to arbitrary algebraic varieties defined over number fields, and then specialize this construction to linear algebraic groups. Then the group of rational points becomes a discrete subgroup of the locally compact group of adeles, which prompts one to develop a version of reduction theory in the adelic setting. This task is accomplished in the second and third sections. In section four, these results are applied to develop reduction theory for S-arithmetic groups and obtain for these the analogs of the statements established in Chapter 4 for arithmetic groups.

This chapter reviews the major results on algebraic number theory, cohomology (both abelian and nonabelian), and central simple algebras over local and global fields that are needed in later chapters.

This chapter assembles the material from the theory of linear algebraic groups that is used routinely in the remainder of the book. The first section summarizes the basic structure theory, including the classification of semisimple groups over algebraically closed fields. The second section discusses the classification of forms in terms of Galois cohomology, which is then applied in the third section to give a description of the classical groups. The fourth section contains miscellaneous results from algebraic geometry.

This chapter focuses on various properties of arithmetic groups that play a central role in the arithmetic theory of algebraic groups. One of the key results, presented in sections two and three, is the construction due to Borel and Harish-Chandra of a fundamental set in the group of real points of an algebraic group defined over the rationals with respect to the subgroup of integral points. This result is then used in the fourth section to deduce a number of group-theoretic properties of arithmetic subgroups, in particular, their finite presentation. The fifth and sixth sections contain criteria for the quotient of the group of real points by the subgroup of integral points to be compact or to have finite Haar measure. The seventh section outlines some further points concerning reduction theory, including the construction of more refined fundamental sets and extensions of the previous statements to groups defined over arbitrary number fields. The eighth section discusses an open problem concerning finite arithmetic groups, while the ninth section, written for the second edition, presents results dealing with abstract arithmetic groups.

This chapter presents the theory of algebraic groups over locally compact fields. The first section discusses the key topological and analytic properties of the sets of rational points of algebraic varieties over local fields, and then applies these to the analysis of the groups of rational points of linear algebraic groups over such fields. The second section deals with the classical case where the field of definition is either the field of real numbers or the field of complex numbers. The third and fourth sections are devoted to algebraic groups over non-archimedean fields, and rely on techniques from the theory of profinite groups, the reduction of algebraic varieties, and Bruhat–Tits theory. The concluding section summarizes some results from measure theory that are needed for subsequent chapters.