A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by p

Abstract

Let p be a prime number with $p\equiv 5\ (\mathrm{mod}\ {8})$. We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by p. Let $k_0$ be the unique subfield of $\mathbb {Q}(\zeta _p)$ of degree $(p-1)/4$ and $u_p=(t+b\sqrt{p})/2\,(>1)$ be the fundamental unit of $k:=\mathbb {Q}(\sqrt{p})$. We put $D_{m,n}:={\mathcal {L}}_m(2{\mathcal {F}}_m-{\mathcal {F}}_n{\mathcal {L}}_m)b$ for integers m and n, where $\{ {\mathcal {F}}_n \}$ and $\{ {\mathcal {L}}_n \}$ are linear recurrence sequences of degree two associated to the characteristic polynomial $P(X)=X^2-tX-1$. We assume that there exists a pair $(m_0,n_0)$ of integers satisfying certain congruence relations. Then we show that there exists a positive integer $N_q$ which satisfies the both class numbers of $k_0(\sqrt{D_{m,n}})$ and $k_0(\sqrt{pD_{m,n}})$ are divisible by p for any pairs (m, n) with $m\equiv m_0 \ (\mathrm{mod}\ {N_q}), \ n\equiv n_0 \ (\mathrm{mod}\ {N_q})$ and $n>3$. Furthermore, we show that if we assume that ERH holds, then there exists the pair $(m_0,n_0)$.