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

Abstract

Let $p$ be a prime number with $p \equiv 3\,\mathrm{mod}\,4$, and let $k=\mathbf{Q}(\sqrt{-p})$. Denote by $h_{n}^{-}$ the relative class number of the $n$th layer of the cyclotomic $\mathbf{Z}_{p}$-extension over $k$. Let $q=(p-1)/2$ and $d_{p}$ be the largest divisor of $q$ with $d_{p} < q$. Let $\ell$ be a prime number with $\ell \neq p$. We show that $\ell \nmid h_{n}^{-}$ for all $n \geq 0$ if $\ell \geq q-2d_{p}$ and $\ell$ is a primitive root modulo $p^{2}$.