Class numbers of real quadratic fields

Abstract

Let k - ℚ(√m) be a real quadratic field. It is well known that if 3 divides the class number of k, then 3 divides the class number of ℚ(√-3m), and thus it divides B1,χω-1, where χ and ω are characters belonging to the fields k and ℚ(√-3) respectively. In general, the main conjecture of Iwasawa theory implies that if an odd prime p divides the class number of k, then p divides B1,χω-1, where ω is the Teichmüller character for p. The aim of this paper is to examine its converse when p splits in k. Let k∞ be the ℤp-extension of k = k0 and hn be the class number of kn, the nth layer of the ℤp-extension. We shall prove that if p | B1,χω-1, then p | hn or all n ≥ 1. In terms of Iwasawa theory, this amounts to saying that if M∞/k∞ is nontrivial, then L∞/k∞ is nontrivial, where M∞ and L∞ are the maximal abelian p-extensions unramified outside p and unramified everywhere respectively.