A note on the relative class number of the cyclotomic $\mathbf{Z}_{p}$-extension of $\mathbf{Q}(\sqrt{-p})$

Abstract

Let $p$ be a prime number with $p \equiv 3 \bmod 4$ and $q=(p-1)/2$. Let $k=\mathbf{Q}(\sqrt{-p})$ and $k_{\infty}/k$ be the cyclotomic $\mathbf{Z}_{p}$-extension. Denote by $h_{n}^{-}$ the relative class number of the $n$-th layer $k_{n}$. Let $\ell$ be a prime number with $\ell \neq p$. We show that, for any $n \geq 1$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) if $\ell$ is a primitive root modulo $p^{2}$ (resp. $p$) and $\ell \geq q-2$ (resp. $\ell \geq q-6$). Further, we show with the help of computer that when $p < 10000$ and $n \leq 100$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) for any prime $\ell$ which is a primitive root modulo $p^{2}$ (resp. $p$).