On the connected identity component of the adèle-class group of an algebraic torus

Abstract

The structure of the connected component of the identity of the idele-class group of a number field is given in Artin-Tate [1]. It seems worthwhile to investigate whether this result is the trivial of a theorem in the arithmetic of algebraic tori, i.e. the case corresponding to the multiplicative group Gm (as is true of the unit theorem and the finiteness of the class number). It turns out that the formalism of the theory of tori defined over number fields allows one to generalize Artin's proof line-by-line, obtaining the structure of the identity component of the adele-class group of any such torus. I am grateful to Professor T. Ono for suggesting the problem to me. I give now a summary of the definitions and results that I shall need (for all this, see Ono [3]). I denote by Z, Q, R, and C the rings of respectively integers, rationals, real numbers, and complex numbers, and by T an algebraic torus defined over a number field k. T is the group of rational characters of T. If the dimension of T is d, then T is a free abelian group of rank d [3, ?1.1]. T is split by a finite extension K of k, which I choose to be normal over Q. Then the group TK of points of T rational over K is isomorphic to the product (K*)d of K* with itself d times. Every & 'T is a rational map defined over K; hence if g denotes the galois group of K over k, T has the structure of a g-module. The group Hom(Ti, K*) has the structure of a gmodule, given byfo() = (f(a)), and we have g-isomorphisms