$2$-Selmer groups, $2$-class groups, and congruent numbers

TL;DR

This paper investigates necessary divisibility conditions on class numbers of imaginary quadratic fields for certain square-free integers to be congruent numbers, providing bounds and congruences related to their prime factorizations.

Contribution

It introduces new divisibility and congruence conditions involving class numbers for specific forms of congruent numbers, expanding understanding of their algebraic properties.

Findings

If such n is a congruent number, then h(-n) satisfies a specific divisibility condition.

Provides quantitative lower bounds on the count of such non-congruent numbers.

Establishes a congruence modulo powers of 2 between class numbers of related quadratic fields.

Abstract

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free integers of the form $n = p_1 p_2 \cdots p_t q,$ where each prime $p_i \equiv 5 \pmod{8}$ and $q \equiv 7 \pmod{8}$. We show that if such an integer $n$ is a congruent number, then the class number $h(-n)$ of the quadratic field $\mathbb{Q}(\sqrt{-n})$ satisfies a specific divisibility condition. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of this form. Next, we study integers of the form $n = p_1 p_2 \cdots p_t q,$ with $p_i \equiv 5 \pmod{8}$ and $q \equiv 3 \pmod{8}$. Assuming that $n$ is a congruent number, we obtain a congruence modulo powers of $2$ between the class numbers of the fields $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}\!\left(\sqrt{-p_1 p_2 \cdots p_t}\right)$.