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

摘要

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 ofthe $n$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 $ell eq p$. We show that $ell mid h_{n}^{-}$for all $n geq 0$ if $ell geq q-2d_{p}$ and $ell$ is a primitive root modulo$p^{2}$.