The size of $2$-Selmer groups for the $\fracπ{3}$-congruent number problem

Technical Summary: The Size of 2-Selmer Groups for the $\pi/3$-Congruent Number Problem

Problem Statement
This paper investigates the average rank of the 2-Selmer group associated with the elliptic curves linked to the $\pi/3$-congruent number problem. A positive integer $n$ is a $\theta$-congruent number if it represents the area of a rational triangle with an angle $\theta$. Specifically, the authors focus on the case $\theta = \pi/3$, where the associated elliptic curve is defined as:
$$E_{n, \pi/3}: y^2 = x(x + 3n)(x - n)$$
The central object of study is the 2-Selmer rank, denoted $s(n)$, where the size of the 2-Selmer group is $2^{2+s(n)}$. The authors aim to determine the asymptotic behavior of the average value of $2^{s(n)}$ over specific families of square-free integers $n$.

Methodology
The authors adopt the strategy pioneered by Heath-Brown [H93] for congruent elliptic curves ($y^2 = x^3 - n^2x$), adapting it to the specific arithmetic structure of $E_{n, \pi/3}$. The methodology proceeds through the following steps:

1. 2-Descent and Homogeneous Systems:
   Using the 2-descent method, the authors map the quotient $E_{n, \pi/3}(\mathbb{Q})/2E_{n, \pi/3}(\mathbb{Q})$ into a product of local fields. They establish a bijection between the 2-Selmer group and the set of solvable systems of homogeneous equations over $\mathbb{Q}$. Specifically, they derive a system of equations involving four pairwise coprime square-free integers $n_1, n_2, n_3, n_4$ (where $n = n_1 n_2 n_3 n_4$):
   $$n_1 X^2 + 3n_4 W^2 = n_2 Y^2$$
   $$n_1 X^2 - n_4 W^2 = n_3 Z^2$$
   The size of the 2-Selmer group, $2^{s(n)}$, corresponds to the number of such systems solvable in $\mathbb{R}$ and all $\mathbb{Q}_p$.

2. Local Conditions and Constraints:
   The authors rigorously analyze the local solvability conditions at primes dividing $6n$. A crucial technical constraint is imposed: the study is restricted to square-free integers $n$ such that $n \equiv 5, 13 \pmod{24}$ and every prime divisor $p$ of $n$ satisfies $p \equiv 1 \pmod 4$. Under these conditions, the local groups behave in a controlled manner, allowing the global Selmer group size to be expressed as a sum over factorizations of $n$ involving Jacobi symbols.

3. Averaging and Character Sums:
   To compute the average size, the authors sum the expression for $2^{s(n)}$ over the set $S(X, h) = \{1 \le n \le X : n \equiv h \pmod{24}, n \text{ square-free}, p|n \implies p \equiv 1 \pmod 4\}$.

   • They decompose the sum over the 16 variables $n_{ij}$ arising from the factorization of $n_1, \dots, n_4$.
   • They employ Heath-Brown's technique of "linked variables" to bound character sums. Variables are considered "linked" if they appear together in a Jacobi symbol.
   • Using bounds for character sums (specifically Lemma 3.1 and Lemma 3.3 from [H93] adapted to modulus 24), they show that contributions from configurations with many "large" variables are negligible.
   • The main term arises from specific configurations of indices (9 exceptional cases identified in Lemma 4.1) where the variables decouple sufficiently to allow explicit evaluation.

Key Contributions and Results

• Main Theorem (Theorem 1.2): For $h = 5$ and $h = 13$, the authors prove the following asymptotic formula for the sum of $2^{s(n)}$:
  $$\sum_{n \in S(X,h)} 2^{s(n)} = 9 \# S(X, h) + O\left(X (\log X)^{-5/8} (\log \log X)^8\right)$$
  This result establishes that the average size of the relaxed 2-Selmer group is exactly 9 times the number of integers in the set.

• Density of Selmer Ranks:
  By combining the main theorem with parity results from Wei-Guo [WG2022], the authors derive unconditional density results:

  • Case $n \equiv 5 \pmod{24}$: Since $s(n)$ is even, the density of $s(n) = 0$ or $s(n) = 2$ is at least $7/16$.
  • Case $n \equiv 13 \pmod{24}$: Since $s(n)$ is odd, the density of $s(n) = 1$ or $s(n) = 3$ is at least $1/3$.

• Implications for Mordell-Weil Rank:
  Using the inequality $r(n) \le s(n)$, the authors provide upper bounds for the average Mordell-Weil rank $r(n)$ in these families, consistent with the bounds derived for $s(n)$.

Significance and Claims
The paper claims that its results provide unconditional evidence supporting Yoshida's conjecture regarding $\pi/3$-congruent numbers. Specifically, the derived densities suggest that a positive proportion of integers in the specified families have low Selmer ranks, which correlates with the existence of rational points on the curves.

The authors highlight a significant deviation from the general heuristics of Bhargava-Kane-Lenstra-Poonen-Rains (BKLPR). While BKLPR predicts a specific distribution based on random matrix theory, the constant 9 in the main theorem reflects a "rigid structure" originating from the 2-isogeny of the curve and the specific local conditions at primes dividing 6. This rigidity causes the distribution of 2-Selmer ranks in this family to differ from the general random matrix predictions.

The authors explicitly note that their technique is currently limited to integers where all prime factors are congruent to $1 \pmod 4$. Extending these results to the full set of square-free integers or to the $2\pi/3$-congruent number problem would require new methods, as the technical difficulties in those cases are substantial. The paper does not claim to resolve the full $\pi/3$-congruent number conjecture but provides a foundational asymptotic formula for a specific, well-behaved subfamily.