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

$$ {k_{\rm A}}\left( {\rm P} \right) = \mathop \prod \limits_{v \in {\rm P}} \,{k_v}{\rm X}\,\mathop \prod \limits_{v \notin {\rm P}} \,{r_{{v^,}}}$$

((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’).