Ideles and Adeles

Abstract

In classical number theory, one embeds a number field in the Cartesian product of its completions at the archimedean absolute values, i.e. in a Euclidean space. In more recent years (more precisely since Chevalley introduced ideles in 1936, and Weil gave his adelic proof of the Riemann- Roch theorem soon afterwards), it has been found most convenient to take the product over the completions at all absolute values, including the p-adic ones, with a suitable restriction on the components, to be explained below. This chapter merely gives the most elementary facts concerning the ideles and adeles (corresponding to a multiplicative and additive construction respectively), and their topologies. In each case, we prove a certain compactness theorem, and construct a fundamental domain. Although we use the existence of fundamental domains later, we shall not need any explicit form for them.