International Congress of Mathematicians

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 157f,g £ A such that the induced distribution (m(f n g n 1 j) n £ limZ(Q p [G„])* is thezeta distribution £s(Goo, M v , 1 — k) and the characteristic idealcoincides with the image of fg^1[RY(Z[l/S],A®T p (k))[l]]£K 0 (T)£ (F*) ab .Remark a) The conjecture is isogeny invariant, i.e., independent of the choice oflattice T p . The correction term (A®T B (kj) + vanishes.b) In the abelian case this means that the zeta distribution is a pseudo measureand generates the characteristic ideal.c) In the case of the cyclotomic tower, a similar conjecture is formulated by Greenberg,[16], [17].d) If GQO is abelian, the above conjecture is easily seen to be implied by conjecture3.2.1. The argument also works in the non-abelian case if the set of all elements ofA which, for all n, are units in Q p [G„] is an Ore set.5. Examples5.1. Dirichlet charactersLet x be a Dirichlet character, V(x) its associated motive with coefficients inF. Let Qoo = U» Q» be the cyclotomic Z p -extension of Q and GQO = Gal(Qoo/Q) =lim G n . In this case the equivariant F-function is Ls(G n ,V(x),s) = (I j s(px, s ))p,where p runs through all characters of G n and Ls(px, s) is the Dirichlet F-functionassociated to px- Yet k be big enough, i.e., k > 1.Critical case x( — 1) = ( — !)*•Here Hj i4 (Z,E[G„] ® V(x)(kj) = 0 for all n. As in section 4., the equivariantF-values give rise to the zeta distribution £s(Goo,V(x) v , l—k)€ Um E[G„]. It isa classical calculation (Stickelberger elements) that this is in fact a pseudo measure,which gives rise to the Kubota-Leopoldt p-adic F-function. Let Ö C F be the ringof integers, A = ö p [[Goo]] the Iwasawa algebra and T p (x) C V p (x) a Galois stablelattice. The Iwasawa Main Conjecture 4.2.1 amounts to the following theorem:Theorem 5.1.1 The zeta distribution £s(G 0O ,I / (x) v , 1 — k) generatesdet^1 H^Z^/S], A® T p (x)(k)) ® det A H 2 (Z[1/S], A® T p ( X )(k)).Remark This is a reformulation of the main theorem of Mazur and Wiles in [29].There is an extension to the case of totally real fields by Wiles [37] and an equivariantversion by Burns and Greither [6].Non-critical case x( — 1) = (^l)* -1 -Here Hj i4 (Z,E[G„] ® V(x)(kj) has F[G„]-rank 1. It is a theorem of Borei(resp. Soulé) that r-p ® R (resp. r p ® Q p ) is an isomorphism. By a theorem ofBeilinson-Deligne (see [21] or [19]), the image of ö p (G n ,V(x),k) under r p is givenbyc^Gn-Mx))' 1 ®t p (x)(k ^ I),