On non-congruent numbers with 1 modulo 4 prime factors

TL;DR

This paper explores the properties of certain non-congruent numbers with specific prime factorization patterns, using advanced methods to analyze their associated elliptic curves and Selmer groups.

Contribution

It introduces the 2-decent method to identify non-congruent numbers with prime factors congruent to 1 mod 4 and analyzes their elliptic curves' Selmer groups, extending previous results.

Findings

Identified a series of non-congruent numbers with prime factors ≡ 1 mod 4

Demonstrated these numbers' elliptic curves have second lowest Selmer groups

Extended Li and Tian's results to broader classes of numbers

Abstract

In this paper, we use the 2-decent method to find a series of odd non-congruent numbers $\equiv1\pmod 8$ whose prime factors are $\equiv1\pmod4$ such that the congruent elliptic curves have second lowest Selmer groups, which includes Li and Tian's result (Li and Tian, 2000) as special cases.