Congruences for odd class numbers of quadratic fields with odd discriminant

Abstract: 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 $\omega_{p_1p_2}:=(1+\sqrt{p_1p_2})/2$ and let $\Psi(\omega_{p_1p_2})$ be the Hirzebruch sum of $\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\equiv h(p_1p_2)\Psi(\omega_{p_1p_2})/n$ $(\text{mod }8)$, 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 $\mathbb{Q}(\sqrt{p_1p_2})$.