Weak approximation for algebraic tori quite fails. (Q2372795)

Let \(K\) be a global field, \(T\) a \(K\)-torus, and \(S\) a finite set of places of \(K\). Let \(T(O_v)\subset T(K_v)\) be the maximal compact subgroup, where \(v\in S\). It is shown that the diagonal homomorphism \(T(K)\to\prod_{v\in S}T(K_v)/T(O_v)\) need not be onto. As a corollary, for a suitable \(v\), the group \(T(O_v)\) does not cover all \(R\)-equivalence classes in \(T(K_v)\). There is a connection with recent work by Bourqui on the height zeta-function of toric varieties.