Wild sets and 2-ranks of class groups.

*(English)*Zbl 0880.11039The notion of reciprocity equivalence of number fields was introduced by R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland [Contemp. Math. 155, 365-387 (1994; Zbl 0807.11024)] for the purpose of studying isomorphisms between the Witt rings of such fields. Necessary and sufficient conditions for reciprocity equivalence are given in a paper by K. Szymiczek [Commun. Algebra 19, 1125-1149 (1991; Zbl 0724.11020)].

In order to differentiate this notion of equivalence from more recent notions involving higher-order reciprocity laws, the authors refer here to such an equivalence as a Hilbert-symbol equivalence. At each place of the underlying field, such an equivalence is said to be either tame or wild depending upon certain parity conditions. The set of all finite places at which the equivalence is wild is called the wild set for the equivalence; when this wild set is empty, the equivalence is referred to as tame. The ideal class groups of the number fields \(K\) and \(L\) have the same 2-rank whenever \(K\) and \(L\) admit a tame Hilbert-symbol equivalence.

In this paper, the authors establish a relationship between the minimal size of a wild set, among all Hilbert-symbol equivalences between \(K\) and \(L\), and the difference between the 2-ranks of the ideal class groups of \(K\) and \(L\). Such results are also proven for narrow ideal class groups, \(S\)-class groups, and narrow \(S\)-class groups, and, as a consequence, a result for the tame kernels \(K_2 ({\mathcal O}_K)\) and \(K_2 ({\mathcal O}_L)\) is obtained.

In order to differentiate this notion of equivalence from more recent notions involving higher-order reciprocity laws, the authors refer here to such an equivalence as a Hilbert-symbol equivalence. At each place of the underlying field, such an equivalence is said to be either tame or wild depending upon certain parity conditions. The set of all finite places at which the equivalence is wild is called the wild set for the equivalence; when this wild set is empty, the equivalence is referred to as tame. The ideal class groups of the number fields \(K\) and \(L\) have the same 2-rank whenever \(K\) and \(L\) admit a tame Hilbert-symbol equivalence.

In this paper, the authors establish a relationship between the minimal size of a wild set, among all Hilbert-symbol equivalences between \(K\) and \(L\), and the difference between the 2-ranks of the ideal class groups of \(K\) and \(L\). Such results are also proven for narrow ideal class groups, \(S\)-class groups, and narrow \(S\)-class groups, and, as a consequence, a result for the tame kernels \(K_2 ({\mathcal O}_K)\) and \(K_2 ({\mathcal O}_L)\) is obtained.

Reviewer: A.G.Earnest (Carbondale)

##### MSC:

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

11R29 | Class numbers, class groups, discriminants |