International Congress of Mathematicians

Tamagawa Number Conjecture for zeta Values 169The left upper arrow with a question mark shows the conjecture that the map{motives} —¥ {zeta functions} factor through automorphic representations, whichis a subject of non-abelian class field theory (Langlands correspondences). As theother question marks indicate, we do not know how to construct zeta elements ingeneral, at present.2.2. p-adic zeta elements for 1 dimensional galois representations.Let A be a commutative pro-p ring, and assume we are given a continuous homomorphismp:Gal(Q/Q)^GF„(A)which is unramified outside a finite set S of prime numbers S containing p. Letj7 — j\_(Sn on which Gal(Q/Q) acts via p, regarded as a sheaf on Spec(Z[^]) for theetale topology. We consider how to construct the p-adic zeta element £(Z[^|], T, A).In the case n = 1, we can use the "universal objects" as follows. Such p comesfrom the canonical homomorphismPu„iv : Gal(Q/Q)-*-GLi(A univ ) where A univ = Z p [[Gal(Q(0v p °°)/Q)]]for some N > 1 whose set of prime divisors coincide with S and for some continuousring homomorphism A un ; v —t A. We have T — .Funiv ®A univ A. Hence £(Z[^|], T, A)should be defined to be the image of £(Z[^|],-F un j v , A un ; v ). As is explained in [Ka 2 ]Ch. I, 3.3, £(Z[^|], .Funivj A u „iv) is the pair of the p-adic Riemann zeta function anda system of cyclotomic units. Iwasawa main conjecture is regarded as the statemnetthat this pair is a A un ; v -basis of A(Z[^], -F un j v , A un ; v ).2.3. p-adic zeta elements for 2 dimensional Galois representations.Now consider the case n = 2. The works of Hida, Wiles, and other people suggestthat the universal objects A un ; v and -F un iv for 2 dimensional Galois representationsin which the determinant of the action of the complex conjugation is -1, are givenbyA U niv = ^Tû p-adic Hecke algebras of weight 2 and of level Np n ,nFumy = hm F 1 of modular curves of level Np n .nBeilinson [Be] discovered ratinai zeta elements in K 2 of modular curves, and the imagesof these elements in the etale cohomology under the Chern class maps becomep-adic zeta elements, and the inverse limit of these p-adic zeta elements should beC(Z[iy],.Funivj A u „iv) at least conjecturally. By using this plan, the author obtainedp-adic zeta elements for motives associated to eigen cusp forms of weight > 2, fromBeilinson elements. Here it is not yet proved that these p-adic zeta elements areactually basis of A, but it can be proved that they have the desired relations withvalues L(E,x, 1) and L(f,x,r) (1 < r < k — 1) for elliptic curves over Q (whichare modular by [Wi], [BCDT]) and for eigen cusp forms of weight k > 2, and forDirichlet charcaters x- Beilinson elements are related in the Archimedean world