Adeles

Abstract

Throughout this Chapter, k will denote an A-field; if v is a place of k, k _v will denote the completion of k at v; if v is a finite place of k, we write r _v for the maximal compact subring of k _v and p _v for the maximal ideal of r _v , these being the subsets of k _v respectively defined by |x| _v < 1 and by |x| _v < 1. We write P _∞ for the set of the infinite places of k, and P for any finite set of places of k, containing P _∞ . For any such set P, put

$$ {k_A}(P) = \prod\limits_{v \in P} {{k_v}} \times \prod\limits_{v \notin P} {{r_v}} , $$

((1))

where the second product is taken over all the places of k, not in P. With the usual product topology, this is locally compact, since the k _v are so and the r _v are compact. On k _A (P), we put a ring structure by defining addition and multiplication componentwise; clearly this makes k _A (P) into a topological ring. Set-theoretically, k _A (P) could be defined as the subset of the product πk _v consisting of the elements x = (x _v ) of that product such that |x _v |v < 1 for all v not in P. If P′ is also a finite set of places of k, and P′ P, then k _A (P) is contained in k _A (P′); moreover, its topology and its ring structure are those induced by those of k _A (P′) and k _A (P) is an open subset of k _A (P′).