On the Relative Theory of Tamagawa Numbers

This note is an outline of some of the author's recent work on the relative theory of Tamagawa numbers of semisimple algebraic groups. The details and applications will be published elsewhere. Let k be an algebraic number field of finite degree over Q, let G be a connected semisimple algebraic group defined over k. G admits one and only one simply connected covering (G, ir) defined over k (except for isomorphisms over k). Denote by g the Galois group of k/ky where k means the algebraic closure of k. Then the finite commutative group Ker T obtains a structure of a g-module. Our purpose is to express the Tamagawa number of the isogeny w, i.e. the number r(7r)=r(G)/r(G), in terms of some invariants of the module Ker w. For the notion of the Tamagawa number, see [ó], [2]. We denote by kv the completion of k at a place v of k and use z> = p if v is nonarchimedean. We also use the standard notation in the Galois cohomology of algebraic groups [ l ] . We say that an algebraic group A defined over k is of type (K) if H(k^y ^4) = 0 for all p and the map H (k, A) —>Hv H(kv, A) is injective. Kneser has conjectured that every simply connected semisimple group defined over k is of type (K) and has verified it for many classical groups [4].