Euler systems for higher K-theory of number fields, by Grzegorz Banaszak and Wojciech Gajda

We apply the Euler system of Gauss sums (introduced by K.Rubin in his proof of the Main Conjecture in Iwasawa theory) to Algebraic K-theory of number fields. Using the Euler system we construct elements in K-groups (with finite coefficients) of cyclotomic extensions of Q. The elements have many nice properties, e.g., we are able to determine explicitly their behaviour under the boundary map in the Quillen localization sequence.

The properties enabled us to perform at the level of higher K-groups a descent argument of Kolyvagin. As the result we obtain a good bound for the number of divisible elements in the p-torsion part of K_{2n}(Q), for n odd. Recall that for an abelian, p-torsion group A its group of divisible elements is by definition the intersection of the subgroups p^kA, for k>0. The group may be finite and nontrivial.

In the case of n even positive the group of divisible elements of the p-torsion part of K_{2n}(Q) should vanish, what is equivalent to the classical conjecture of Kummer and Vandiver on the class numbers of cyclotomic fields cf. Proposition 8. In the paper we also prove an index formula for the number of divisible elements when n is evan. The formula states that the number of divisible elements in the p-torsion part of the group K_{2n}(F), for n even positive and F an abelian, totally real number field equals the index of the group of cyclotomic elements of Deligne and Soule in the etale K-group K^{et}_{2n+1}(Z[{1/p}]). Our proof of the formula generalizes an argument of Bloch and Kato who proved it for Q.

ADDRESS: Department of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, POLAND

Grzegorz Banaszak <>
Wojciech Gajda <>