Class numbers and Iwasawa invariants of quadratic fields

Abstract

Let Q ( − d ) \mathbf {Q}(\sqrt {-d}) and Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) be quadratic fields with d ≡ d \equiv 2 (mod 3) a positive integer. Let λ − , λ + \lambda ^-, \lambda ^+ be the respective Iwasawa λ \lambda -invariants of the cyclotomic Z 3 \mathbf {Z}_3 -extension of these fields. We show that if λ − = 1 \lambda ^- =1 , then 3 does not divide the class number of Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) and λ + = 0 \lambda ^+ = 0 .