International Congress of Mathematicians

154 A. Huber G. Kingsmore precisely an element of fimHom l /(Q p [ Gn ])(lQ p [ Gn ], • ).The map A —t Q p [G„] induces an isomorphismQ P [G n ] ®A RY(Z[l/S],A®T p (kj) -> FF(Z[1/S],Q,[G„] ® Qp M p (kj) .Conjecture 3.2.1 (Non-abelian Main Conjecture) Let M and S be as in 2.,GQO as in 3., p ^ 2, TR C MR a lattice such that T p := TR ® Z p is Galois stableand k big enough (cf. section 2.). Then ö p (G 00 ,M,k) is induced by a generatorì p (G 00 ,M,k)£ [det A FF(Z[l/S],A®r p (fc)) ®det A (A®F B (fc - 1))+] .The conjecture translates into the Iwasawa Main Conjecture in the case of Dirichletcharacters or CM-elliptic curves. See section 5. for more details.Remark a) The conjecture is independent of the choice of lattice TR. The correctionfactor (A ®T B (k — 1)) + compensates different choices of lattice.b) Perrin-Riou [31] has defined a p-adic F-function and stated a Main Conjecturefor motives in the abelian case. She starts at the other side of the functional equation,where the exponential map of Bloch-Kato comes into play. Her main tool isthe "logarithme élargi", which maps Galois cohomology over K^ to a module ofp-adic analytic nature. It would be interesting to compare her approach with theabove.c) A Main Conjecture for motives and the cyclotomic tower was formulated byGreenberg [16], [17]. Ritter and Weiss consider the case of the cyclotomic towerover a finite non-abelian extension [32].Proposition 3.2.2 (see section 6.) The equivariant Bloch-Kato conjecture forM, k and all G n is equivalent to the Main Conjecture for M, k and G^.3.3. TwistingAssume that T p becomes trivial over K^,, for example let GQO be the imageof Gal(Q/Q) in Aut(F p ). Let T nvp be the Z p -module underlying T p with trivialoperation of the Galois group. The map g ® t >-¥ g ® g^1tinduces an equivariantisomorphism A ®z p T p = A ®z p T nv p . Hence there is an isomorphismdet A FF(Z[1/S], A® T p (kj) ® det A (A ®T B (k^ 1))+ ~det A FF(Z[1/S], A®T* riv (k)) ® det A (A ® T B riv (k - 1))+.Note that T B " V can be viewed as a lattice in the Betti-realization of the trivialmotive h°(Q) ® M triv = Q(0) ® M triv where M triv is M B considered as Q-vectorspace.Corollary 3.3.1 // the Main Conjecture is true for M and Q(0) ® M tnv and k,then _ _(5 p (G 0O ,M,fc) = ^(G 0O ,M triv ,fc)up to an element in FJi(A) under the above isomorphism.