International Congress of Mathematicians

158 A. Huber G. KingswhereCk(Gn,t P (xï) eis a twist of a cyclotomic unit and t p (x)(kCk(Goo,t p (x)) •= ^nCk(G n ,t p (x))-H 1 (Z[l/S],O p [G n ]®T p (x)(k))— 1) is a generator of T p (x)(k — 1). LetTheorem 5.1.2 There is a canonical isomorphism of A-determinantsdetA (H 1 (Z[l/S],A®T p (x)(kj)/c k (G 00 ,t p (x))) =detA H 2 (Z[1/S], A® T p (x)(k)).Remark For p { ord(x) this is a consequence of theorem 5.1.1 and was showndirectly by Rubin [33] with Euler system methods. The restriction at the order ofX is removed in Burns-Greither [5] and Huber-Kings [20] by different methods.The Tamagawa number conjecture for V(x)(r) (and hence for h°(F)(r) withF an abelian number field) can be deduced from theorems 5.1.1 and 5.1.2, seeBurns-Greither [5] or Huber-Kings [20]. Previous partial results were proved inMazur-Wiles [29], Wiles [37], Kato [22], [23], Kolster-Nguyen Quang Do-Fleckinger[26] and Benois-Nguyen Quang Do[2].We would like to stress that the strategy 3. is used in Huber-Kings [20] toprove theorems 5.1.1, 5.1.2 and the Tamagawa number conjecture from the classnumber formula.5.2. Elliptic curvesLet F be an elliptic curve over an imaginary quadratic field K with CM by OK-The motive ft 1 (F) considered with coefficients in K decomposes into V(ip) ® V('ip),where ip is the Grössencharacter associated to F. The F-function of V(ip) is theHecke F-function of ip, which has a zero of order 1 at 2 — k, where k > 2. YetS = Np, where N is the conductor of ip and let K n := K(E[p n ]).It is not known if Hj^(OK,K[G„] ® V(ip)(k)) has FJ[G„]-rank 1 but Deninger[13] shows that r-p ® R is surjective and that the Beilinson conjecture holds. Itis a result of Kings [25] that the image in étale cohomology of the zeta elementöp(G n ,V(ip),2 — k) given by Beilinson's Eisenstein symbol is given byefe(G„,£ p (i/0) _1 ®t p (ip) ,where ek(G n ,t p (ipj) £ H 1 (Z[l/S],O p [G„]®T p (ip)(kj) is the twist of an elliptic unit.Let A := O p [[Goo]j and e k (G oo, t p (ij))) = ljm n e k (G n ,t p (ipj).Theorem 5.2.1 There is a canonical isomorphism of determinantsdetA (H 1 (Z[l/S],A®T p (