International Congress of Mathematicians

152 A. Huber G. Kingsall integer values [14], [15]. The generalization to non-abelian coefficients is due toBurns and Flach [4].For simplicity of exposition, we restrict to values at very negative integers.In the absolute case this coincides with the formulation given by Kato in [23].We consider a motive M and values at 1 — k where k is big enough. In the caseM = h r (X), k big enough means that• k > inf{r,dim(X)}, (r,k) ^ (1,0); (2dim(X), dim(X) + 1) and 2k ^ r + 1.• for all £ £ S the local Euler factor Li(M p , s)^1at £ does not vanish at 1 — k.Consider the (injective) reduced norm map rn : Ki(R[G]) —¥ Z(R[G])* andrecall that L$(G, M v , 1 — fc)* £ Z(M[G])*. By strong approximation (see [4] Lemma8) there is A G Z(Q[G])* such that AF S (G, M v (l -k))* is in the image of FJi(R[G])under rn. LetAF s (G,M v (l^fc))*el R[Gn]be the corresponding generator. For k big enough, we define the fundamental linein V(Q[G]) asA f (G, M v (l - kj) = det^j F^(Z,QG] ® M (kj) ® det Q[G] (Q[G] ® M B (k - 1))+ .Here + denotes the fixed part under complex conjugation.Conjecture 2.4.1 Let M be as in 2., p ^ 2 a prime and k big enough.1. The Beilinson regulator r-p induces an isomorphismA / (G,M v (l^fc))®R-l R[G] .2. Under this isomorphism the generator (XLg(G,M v (I — k))*)^1a (unique) generatoris induced by(\- 1 ö(G,M,k)) £ A f (G,M v (l^k)).The reduced norm is an isomorphism Ki(Q p [G]) — Z(Q P [G])*. Using the operationof Ki(Q p [G]) on generators in A f (G, M v (l - kj) ® Q p , we putö p (G,M,k) := (\- 1 ó(G,M,k,))\£ Af(G,M v (l^k))®Q p .Note that this generator is independent of the choice of A.3. The p-adic regulator r p induces an isomorphismA / (G,M v (l^fc))®Q p -det^[G] H^Zll/Sl^lG] ® M p (k)) ® det Qp[G] (Q p [G] ® M B (k - 1))+.4- Let TR C MR be a lattice such that T p = TR ® Z p c M p is Galois stable.Under the last isomorphism ö p (G,M,k) is induced by a generatorì p (G,M,k) £ det Zp[G] RY^l/S],Z p [G]®T p (k))®det Zp[G] (Z p [G]®T B )(k^l))+.