论文标题
一致性的奇数类数量的二次字段数具有奇数判别
Congruences for odd class numbers of quadratic fields with odd discriminant
论文作者
论文摘要
对于任何不同的两个普通$ p_1 \ equiv p_2 \ equiv 3 $（\ text {mod} 4）$，让$ h（-p_1）$，$ h（-p_2）$和$ h（p_1p_2）$是Quadratic Fields $ \ \ \ m nathbb {q} $ {q} $ {Q}（\ sqrt）的类别$ \ mathbb {q}（\ sqrt {-p_2}）$和$ \ mathbb {q}（\ sqrt {p_1p_2}）$，分别为$。令$ω_{p_1p_2}：=（1+ \ sqrt {p_1p_2}）/2 $，让$ψ（ω__{p_1p_2}）$为$ω___________________________________________________________{我们表明$ h（-p_1）h（-p_2）\ equiv h（p_1p_2）ψ（ω__{p_1p_2}）/n $ $（\ text {mod} 8）$，其中$ n = 6 $（分别为$ n = 2 $，如果$ \ \ \ \ \ \ \ \ \ \ {p_1，p_1，p_2 $} $ n = 6 $（分别为$ n = 2 $）。我们还考虑了带有导体$ 2 $ in $ \ mathbb {q}（\ sqrt {p_1p_2}）$的真实二次订单。
For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and $\mathbb{Q}(\sqrt{p_1p_2})$, respectively. Let $ω_{p_1p_2}:=(1+\sqrt{p_1p_2})/2$ and let $Ψ(ω_{p_1p_2})$ be the Hirzebruch sum of $ω_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\equiv h(p_1p_2)Ψ(ω_{p_1p_2})/n$ $(\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\min\{p_1,p_2\}>3$ (respectively, otherwise). We also consider the real quadratic order with conductor $2$ in $\mathbb{Q}(\sqrt{p_1p_2})$.
