论文标题
$ \ mathbb {q}的类数量的一致性（\ sqrt {\ sqrt {\ pm 2p}）$ for $ p \ equiv3 $ $（\ text {mod} 4）$ a prime
Congruences on the class numbers of $\mathbb{Q}(\sqrt{\pm 2p})$ for $p\equiv3$ $(\text{mod }4)$ a prime
论文作者
论文摘要
对于prime $ p \ equiv 3 $ $（\ text {mod} 4）$，让$ h（-8p）$和$ h（8p）$是$ \ mathbb {q}（\ sqrt {-2p}）$的班级数字，而$ \ sqrt {-2p}）$和$ \ \ \ \ \ mathbb {q}（q}（q}（q}（\ sqrt}）令$ψ（ξ）$为二次非理性$ξ$的Hirzebruch总和。我们表明$ h（-8p）\ equiv H（8p）\ big（ψ（2 \ sqrt {2p}）/3-ψ\ big（（（1+ \ sqrt {2p} {2p}）/2 \ big）/2 \ big）/3 \ big）另外，我们表明$ h（-8p）\ equiv 2H（8p）ψ（2 \ sqrt {2p}）/3 $ $（\ text {mod} 8）$如果$ p \ equiv 3 $ 3 $（\ text {mod} {mod} 8）$ \ big（2H（8p）ψ（2 \ sqrt {2p}）/3 \ big）+4 $ $（\ text {mod} 8）$如果$ p \ equiv 7 $ $（\ text {mod} 8）$。
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 $Ψ(ξ)$ be the Hirzebruch sum of a quadratic irrational $ξ$. We show that $h(-8p)\equiv h(8p)\Big(Ψ(2\sqrt{2p})/3-Ψ\big((1+\sqrt{2p})/2\big)/3\Big)$ $(\text{mod }16)$. Also, we show that $h(-8p)\equiv 2h(8p)Ψ(2\sqrt{2p})/3$ $(\text{mod }8)$ if $p\equiv 3$ $(\text{mod }8)$, and $h(-8p)\equiv \big(2h(8p)Ψ(2\sqrt{2p})/3\big)+4$ $(\text{mod }8)$ if $p\equiv 7$ $(\text{mod }8)$.
