Monsky Matrix and 2-Selmer rank

TL;DR

This paper uses Monsky matrices to demonstrate that certain families of numbers with specific prime factorizations have elliptic curves with zero 2-Selmer rank, and proves these families contain infinitely many non-congruent numbers.

Contribution

It introduces a method to construct infinite families of non-congruent numbers with prescribed prime factors using Monsky matrices and analyzes their 2-Selmer ranks.

Findings

Infinite families of non-congruent numbers with primes in specific residue classes.

The 2-Selmer rank of associated elliptic curves is zero for these families.

Each family contains infinitely many non-congruent numbers.

Abstract

In this article, we produce infinite families of non-congruent numbers in the residue class of $1,2,$ and $3$ modulo $8$ with arbitrarily many triples or quadruples prime factors. In short, we use Monsky matrix to show that the $2$-Selmer rank of the corresponding congruent number elliptic curve is zero. We also establish some quantitative results to conclude that each such family contains infinitely many non-congruent numbers.