Congruences on the class numbers of $\mathbb{Q}(\sqrt{\pm 2p})$ for $p\equiv3$ $(\text{mod }4)$ a prime

Abstract

For a prime $p\equiv 3$ $(\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\mathbb{Q}(\sqrt{-2p})$ and $\mathbb{Q}(\sqrt{2p})$, respectively. Let $\Psi(\xi)$ be the Hirzebruch sum of a quadratic irrational $\xi$. We show that $h(-8p)\equiv h(8p)\Big(\Psi(2\sqrt{2p})/3-\Psi\big((1+\sqrt{2p})/2\big)/3\Big)$ $(\text{mod }16)$. Also, we show that $h(-8p)\equiv 2h(8p)\Psi(2\sqrt{2p})/3$ $(\text{mod }8)$ if $p\equiv 3$ $(\text{mod }8)$, and $h(-8p)\equiv \big(2h(8p)\Psi(2\sqrt{2p})/3\big)+4$ $(\text{mod }8)$ if $p\equiv 7$ $(\text{mod }8)$.