Iwasawa invariants and class numbers of quadratic fields for the prime 3

Abstract

Let d d be a square-free integer with d ≡ 1 ( mod 3 ) d \equiv 1 \pmod {3} and d > 0 d > 0 . Put k + = Q ( d ) k^{+}=\Bbb Q(\sqrt {d}) and k − = Q ( − 3 d ) k^{-}=\Bbb Q(\sqrt {-3d}) . For the cyclotomic Z 3 \Bbb Z_3 -extension k ∞ + k^{+}_\infty of k + k^{+} , we denote by k n + k^{+}_n the n n -th layer of k ∞ + k^{+}_\infty over k + k^{+} . We prove that the 3 3 -Sylow subgroup of the ideal class group of k n + k^{+}_n is trivial for all integers n ≥ 0 n \geq 0 if and only if the class number of k − k^{-} is not divisible by the prime 3 3 . This enables us to show that there exist infinitely many real quadratic fields in which 3 3 splits and whose Iwasawa λ 3 \lambda _3 -invariant vanishes.