On the Iwasawa λ-invariants of quaternion extensions

Abstract

For a prime number l and a number field k, denote by λl(k) the Iwasawa λ-invariant associated with the ideal class group of the cyclotomic Zl-extension k∞(l) over k. It is conjectured that this invariant is zero for any prime l and any totally real number field k (cf. [7]). Several authors have given some sufficient conditions for the conjecture when k is a real abelian field (cf. [1]–[9]). Using them, many examples of the vanishing of λ-invariants for real abelian number fields are given. However, it seems that an example of a totally real non-abelian field has not yet been given. In this paper we give quaternion extensions K over the rational number field Q with λ2(K) = 0. A Galois extension K over Q is called a quaternion extension if the Galois group G(K/Q) of K over Q is isomorphic to the quaternion group H8 of order 8. The quaternion group H8 is a group H8 = 〈σ, τ〉 of order 8 with σ4 = 1, σ2 = τ2 and τστ−1 = σ−1. The main purpose of this paper is to prove the following: Theorem. Let p be a prime number with p ≡ 3 (mod 8), k = Q(√2,√p) and k∞(2) the cyclotomic Z2-extension of k. Then there exist natural numbers x, y with x2 − y2p = 2p. Let K∞(2) be the cyclotomic Z2-extension of K = k( √ (x+ y √ p)(2 + √ 2)). Then the Galois group G(K/Q) of K over Q is isomorphic to the quaternion group H8 and the λ-invariant λ(K∞(2)/K) of K∞(2) over K vanishes. First we recall the following lemma which plays an important role in our proof of this theorem: