Congruences for odd class numbers of quadratic fields with odd discriminant

TL;DR

This paper establishes congruence relations involving class numbers of quadratic fields with odd discriminant, linking them through Hirzebruch sums and providing new insights into their arithmetic properties.

Contribution

It introduces novel congruence formulas connecting class numbers and Hirzebruch sums for quadratic fields with odd discriminant, extending previous understanding of their arithmetic relationships.

Findings

Proves a congruence relation modulo 8 involving class numbers and Hirzebruch sums.

Identifies specific conditions based on primes for the congruence to hold.

Analyzes real quadratic orders with conductor 2 in related fields.

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