Congruences for odd class numbers of quadratic fields with odd discriminant

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 $\mathbb{Q}(\sqrt{p_1p_2})$.

No reviews yet.

No comments yet.