Congruences on the class numbers of $\mathbb{Q}(\sqrt{\pm 2p})$ for $p\equiv 3$ $(\text{mod }4)$ 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 $h(-8p)\equiv h(8p)(\Psi (2\sqrt{2p})/3-\Psi ((1+\sqrt{2p})/2)/3)$ $(\text{mod }16)$. 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 $h(-8p)\equiv (2h(8p)\Psi (2\sqrt{2p})/3)+4$ $(\text{mod }8)$ if $p\equiv 7$ $(\text{mod }8)$.

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