Relative class numbers inside the $p$th cyclotomic field

Abstract

For a prime number $p \equiv 3 \;\mbox{mod $4$}$, we write $p=2n\ell^f+1$ for some power $\ell^f$ of an odd prime number $\ell$ and an odd integer $n$ with $\ell \nmid n$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of ${\Bbb Q}(\zeta_p)$ of degree $2\ell^t$ and let $h_t^-$ be the relative class number of $K_t$. We show that for $n=1$ (resp. $n \geq 3$), a prime number $r$ does not divide the ratio $h_t^-/h_{t-1}^-$ when $r$ is a primitive root modulo $\ell^2$ and $r \geq \ell^{f-t}-1$ (resp. $r \geq (n-2)\ell^{f-t}+1$). In particular, for $n=1$ or $3$, the ratio $h_f^-/h_{f-1}^-$ at the top is not divisible by $r$ whenever $r$ is a primitive root modulo $\ell^2$. Further, we show that the $\ell$-part of $h_t^-/h_{t-1}^-$ stabilizes for ``large" $t$ under some assumption.