On K 2 and Some Classical Conjectures in Algebraic Number Theory

Abstract

Let k be any field, and let kx be the multiplicative group of k. One of the several equivalent definitions of K2k is that K2k = (kx ?z kx)/J, where J is the subgroup of the tensor product generated by all elements a 0 b with a + b = 1. Assume now that k is a finite algebraic extension of the rational field Q. An important theorem of Garland [2] implies that, in this case, K2k is a torsion group. Garland's proof, however, does not give precise information about the structure of K2k, nor does it show how this group is related to more classical invariants of k. The aim of the present paper has been to establish more precise results. Motivated by recent work of Tate [13] on the function field analogue, we show that, for each prime number 1, the i-primary subgroup of K2k is isomorphic to a certain group that arises naturally in Iwasawa's [6] theory of Z,-extensions of number fields. The precise statement of our result is given later in the paper (see Theorems 4 and 7). We then discuss various consequences of this description of K2k. In particular, we show that a classical conjecture in the theory of cyclotomic fields is equivalent to a conjecture about K2k (see Theorem 7), which, unfortunately, also seems very difficult to settle. Further, it will be apparent that our results lead, in a natural way, to a module which may possibly be an analogue for k of the Jacobian variety of a curve of genus > 1 defined over a finite field. The arguments we use rely heavily on Iwasawa's [6] theory of Z,-extensions of number fields and on cohomological methods due to Tate [13], which, in turn, were inspired by conjectures of Lichtenbaum [7]. In addition, Tate's work, and therefore indirectly ours also, depends essentially on the work of Bass [1], Garland [2], Matsumoto [8], Moore [9], and others. Finally, I wish to heartily thank J. Tate for many stimulating suggestions on the questions discussed in this paper.