A Note on Height Pairings, Tamagawa Numbers, and the Birch and Swinnerton-Dyer Conjecture

Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak). Since different choices of lifting will change G(C,,) for only a finite number of v, G(Ak) is intrinsically defined independent of the choice of Cs-scheme structure. It may happen that G(k)cG(Ak) is discrete. This will be the case, for example, if G is affine. If so, we may try to compute the volume of G(Ak)/G(k ). Writing I F = residue field at v, q,,= ~IF,,, N,,= ,t~ GOF,,), the natural volume form gives Vol(G(C~))=N~q~_ ~ for all v6S. It can happen that [IN,,q~71 does not