International Congress of Mathematicians

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 151We assume the usual conjectures about the F-functions of motives, like meromorphiccontinuation and functional equation etc., to be satisfied.In order to define the equivariant F-function for G (without the Euler factorsat the primes dividing S), consider a Galois extension E/Q such that E[G] =(J) End E (V(pj), where V(p) are absolutely irreducible representations of G. Thenthe center of E[G] is Z(E[G]) = ® F and the motives V(p) ® M have coefficientsin F. We defineL S (G, M, k)* := (L S (V(P) ® M, k)*) £ Z(E ® Q C[G])*to be the element with p-component the leading coefficient at s = k of the F ®Q devaluedF-functions Ls(V(p)®M, s) without the Euler factors at S. Then Lg(G, M, k)*has actually values in Z(R[G])* (see [4] Lemma 7) and is independent of the choiceof F. We will always consider Lg(G, M, k)* as an element in Z(R[G]) c R[G].Remark In [22] Kato uses a different description of this equivariant F-function.2.3. Non-commutative determinantsWe follow the point of view of Burns and Flach. Let A be a (possibly noncommutative)ring and V(A) the category of virtual objects in the sense of Deligne[12]. V(A) is a monoidal tensor category and has a unit object 1A- Moreover it isa groupoid, i.e., all morphisms are isomorphisms. There is a functordet^ : {perfect complexes of .4-modules and isomorphisms} —¥ V(A)which is multiplicative on short exact sequences. The group of isomorphism classesof objects of V(A) is K 0 (A) andAut(l^) = Ki(A) =Gl 00 (A)/E(A)(E(A) the elementary matrices). In general Honiy^^det^X, det^F) is eitherempty or a K\ (-A)-torsor. If A —t B is a ring homomorphism, we get a functorB® : V(A) —t V(B) such that tensor product and det^ commute.Convention By abuse of notation we are going to write z £ detA X for z : 1A —^det^ X and call this a generator of det^ X.If A is commutative and local, then the category of virtual objects is equivalentto the category of pairs (L,r) where F is an invertible .4-module and r £ Z. Onerecovers the theory of determinants of Knudson and Mumford. The unit objectis 1A = (A,0) and one has Aut(l^) = Ki(A) = A*. Thus K\(A) is used asgeneralization of A* to the non-commutative case. Generators of det^ X = (L, 0)in the above sense correspond to .4-generators of F.2.4. Formulation of the conjectureThe original conjecture dates back to Beilinson [1] and Bloch-Kato [3]. Theidea of an equivariant formulation is due to Kato [23] and [22]. Fontaine and Perrin-Riou gave a uniform formulation for mixed motives and all values of F-functions at