Congruences on the class numbers of $\mathbb{Q}(\sqrt{\pm 2p})$ for $p\equiv3$ $(\text{mod }4)$ a prime

TL;DR

This paper establishes congruence relations between class numbers of quadratic fields related to primes congruent to 3 mod 4, involving Hirzebruch sums and modular arithmetic, revealing new congruence patterns.

Contribution

It provides new congruence formulas connecting class numbers and Hirzebruch sums for quadratic fields associated with primes congruent to 3 mod 4.

Findings

Class number $h(-8p)$ congruence modulo 16 involving $h(8p)$ and Hirzebruch sums.

Modulo 8 congruences for $h(-8p)$ depending on $p$ mod 8.

Explicit relations between class numbers and special sums for primes with specific congruences.

Abstract

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 $\Psi(\xi)$ be the Hirzebruch sum of a quadratic irrational $\xi$. We show that $h(-8p)\equiv h(8p)\Big(\Psi(2\sqrt{2p})/3-\Psi\big((1+\sqrt{2p})/2\big)/3\Big)$ $(\text{mod }16)$. Also, we show that $h(-8p)\equiv 2h(8p)\Psi(2\sqrt{2p})/3$ $(\text{mod }8)$ if $p\equiv 3$ $(\text{mod }8)$, and $h(-8p)\equiv \big(2h(8p)\Psi(2\sqrt{2p})/3\big)+4$ $(\text{mod }8)$ if $p\equiv 7$ $(\text{mod }8)$.