Class number parity of a quadratic twist of a cyclotomic field of prime power conductor

Abstract

Let $p$ be a fixed odd prime number and $K_{n}$ the $p^{n+1}$-st cyclotomic field. For a fixed integer $d \in \boldsymbol{Z}$ with $\sqrt{d} \notin K_{0}$, denote by $L_{n}$ the imaginary quadratic subextension of the biquadratic extension $K_{n}(\sqrt{d})/K_{n}^{+}$ with $L_{n} \neq K_{n}$. Let $h_{n}^{*}$ and $h_{n}^{-}$ be the relative class numbers of $K_{n}$ and $L_{n}$, respectively. We give an explicit constant $n_{d}$ depending on $p$ and $d$ such that (i) for any integer $n \geq n_{d}$, the ratio $h_{n}^{-}/h_{n-1}^{-}$ is odd if and only if $h_{n}^{*}/h_{n-1}^{*}$ is odd and (ii) for $1 \leq n < n_{d}$, $h_{n}^{-}/h_{n-1}^{-}$ is even.