The size of $2$-Selmer groups for the $\fracπ{3}$-congruent number problem
Kushal Bhowmick, Aprameyo Pal
TL;DR
This paper analyzes the average $2$-Selmer rank for the elliptic family $E_{n,\frac{\pi}{3}}: y^{2}=x(x+3n)(x-n)$, encoding the $2$-Selmer rank with $s(n)$ via $\ ext{Sel}_{2} = 2^{2+s(n)}$. Adopting Heath-Brown's $2$-descent approach, it proves the asymptotic $\\sum_{n\\in S(X,h)} 2^{s(n)} = 9\\#S(X,h) + O(X(\log X)^{-5/8}(\log\log X)^{8})$ under the condition that every prime dividing $n$ is $1$ mod $4$, yielding unconditional positive densities for $s(n)$ in residue classes modulo $24$ (0 or 2 for $n\\equiv 5\\pmod{24}$ and 1 or 3 for $n\\equiv 13\\pmod{24}$). The analysis reveals a rigidity in the Selmer-rank distribution arising from full rational $2$-torsion and a $2$-isogeny, causing deviations from the generic BKLPR heuristics and providing evidence toward the $\theta$-congruent number conjectures. The work also outlines extensions to the $\frac{2\pi}{3}$-congruent case and discusses implications for Goldfeld-type phenomena in this family.
Abstract
Our main objective in this paper is to study the average rank of the $2$-Selmer group of the elliptic curve associated with the $\fracπ{3}$-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed $2$-Selmer groups, which has several consequences towards the average of $2$-Selmer ranks and $\fracπ{3}$-congruent number problem. Indeed, we could find an unconditional positive density of $2$-Selmer rank being $1$ or $3$, among the positive square-free integers $n\equiv 13\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$ and an unconditional positive density of $2$-Selmer rank being $0$ or $2$, among the positive square-free integers $n\equiv 5\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$.