The rank of 2-Selmer group associate to $\theta$-congruent numbers

TL;DR

This paper investigates the parity of 2-Selmer groups related to specific congruent numbers, providing density results and conditions for non-congruence, supporting aspects of Goldfeld's conjecture.

Contribution

It offers new insights into the parity of 2-Selmer groups for certain congruent numbers and establishes density bounds for non-congruent numbers based on prime factorizations.

Findings

Positive densities for non-congruent numbers support Goldfeld's conjecture.

Necessary conditions for non-congruence involve non-trivial Shafarevich-Tate groups.

At least 75% density of non-congruent numbers for certain prime products.

Abstract

We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $\pi/3$ and $2\pi/3$-congruent numbers. Our second result gives some positive densities about $\pi/3$ and $2\pi/3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $\pi/3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $\pi/3$($resp.$ $2\pi/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.