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Published online by Cambridge University Press: 17 August 2023
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.
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