Some questions of weak approximation for algebraic tori

Abstract

Let $K$ be a global field, $T$ a $K$-torus and $S$ a finite set of places of $K$. Let $K_{v}$ be the completion at $v \in S$. Denote by $T(O_{v}) \subset T(K_{v})$ the maximal compact subgroup of the group of $K_{v}$-points of $T$. We show that the map $ T(K) \rightarrow \prod _{v \in S} T(K_{v})/T(O_{v})$ induced by the diagonal map need not be onto. As a corollary, for suitable $v$, the group $T(O_{v})$ does not cover all $R$-equivalence classes in $T(K_{v})$. D. Bourqui has recently studied the height zeta function of toric varieties over a function field in one variable over a finite field. In the course of this study he encountered a certain constant. The same type of torus as constructed for the problem above enables us to show that this constant need not always be 1.