Note on the class number of the $p$th cyclotomic field, III

Abstract

Let $p=2\ell^f+1$ be a prime number with $f \geq 2$ and an odd prime number $\ell$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ with $[K_t : \mathbb{Q}]=2\ell^t$. Denote by $h_{p,t}^-$ the relative class number of $K_t$, and by $h_{p,t}^+$ the class number of the maximal real subfield $K_t^+$. It is known that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is odd (and hence so is $h_{p,f}^+/h_{p,f-1}^+$) whenever $2$ is a primitive root modulo $\ell^2$. We show that $h_{p,f}^+/h_{p,f-1}^+$ is odd under a somewhat milder assumption on $\ell$ and that the ratio $h_{p,f-1}^-/h_{p,f-2}^-$ is always odd when $\ell=3$.