[Paper Review] A new family of elliptic curves with positive ranks arising from the Heron triangles

Farzali Izadi, Foad Khoshnam+1 more|arXiv (Cornell University)|Dec 28, 2010

Analytic Number Theory Research3 references3 citations

TL;DR

This paper introduces a new family of elliptic curves over ℚ derived from Heron triangles with rational sides and areas, using parametric formulas for sides a(k), b(k), and c(k). By transforming the Heron triangle's area formula into a Weierstrass form, the authors construct curves with torsion group ℤ/2ℤ × ℤ/2ℤ and prove the rank is at least 1 for all k ∈ ℚ. The key contribution is the explicit construction of curves with positive rank, including examples with ranks up to 6, demonstrating the existence of such curves via algebraic geometry and computational verification.

ABSTRACT

The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$. It turned out that the torsion groups of this family are of the form $\frac{\Bbb{Z}}{2\Bbb{Z}} imes \frac{\Bbb{Z}}{2\Bbb{Z}}$ and also the rank is positive.

Motivation & Objective

• To construct a new infinite family of elliptic curves over ℚ with positive Mordell-Weil rank.
• To establish that the torsion subgroup of these curves is isomorphic to ℤ/2ℤ × ℤ/2ℤ.
• To demonstrate that the rank of the constructed curves is at least 1 for all rational parameters k.
• To compute and analyze the rank distribution of the family across a large sample of k values (1 ≤ k ≤ 100), identifying curves with high ranks.

Proposed method

• Parametrize rational Heron triangles using a rational parameter k, with sides a(k), b(k), and c(k) derived from a known family of triangles with rational area.
• Transform the Heron formula S² = P(P−a)(P−b)(P−c) into a Weierstrass form via coordinate changes: (u,v) → (1/ζ, η/ζ²) and (ζ,η) → (−x/abc, y/abc), yielding y² = (x+ab)(x+bc)(x+ac).
• Apply a further coordinate shift x → x − a(k)c(k) to express the curve in the form y² = x³ + Ax² + Bx, enabling analysis of torsion and rank.
• Use Kubert’s criterion to confirm the torsion subgroup is ℤ/2ℤ × ℤ/2ℤ by verifying the non-vanishing and distinctness of the roots of the cubic.
• Employ computational tools (Sage, MWRANK) to compute the Mordell-Weil rank and Selmer rank for 100,000 curves in the range 1 ≤ k ≤ 100.
• Use the sum S(N,E) = Σₚ≤N (−aₚ + 2)/(p + 1 − aₚ) log p as a heuristic to identify promising candidates for high-rank curves before full rank computation.

Experimental results

Research questions

• RQ1 Can a new infinite family of elliptic curves with positive rank be constructed from Heron triangles with rational sides and area?
• RQ2 What is the structure of the torsion subgroup for elliptic curves derived from such Heron triangles?
• RQ3 Is the Mordell-Weil rank of these curves guaranteed to be at least 1 for all rational values of the parameter k?
• RQ4 What is the distribution of ranks within this family, and can curves of high rank (e.g., rank 6) be explicitly found?
• RQ5 How effective is the heuristic sum S(N,E) in identifying curves likely to have high rank?

Key findings

• The elliptic curve family defined by y² = (x + a(k)b(k))(x + b(k)c(k))(x + a(k)c(k)) has torsion subgroup isomorphic to ℤ/2ℤ × ℤ/2ℤ for all k ∈ ℚ \ {0, 2, -2}.
• The point (0, abc) lies on the curve and has infinite order, proving that the Mordell-Weil rank is at least 1 for all k ∈ ℚ.
• For 1 ≤ k ≤ 100, among 100,000 curves, 23.4% have rank 4, 36.7% have rank 3, and 16.2% have rank 2, indicating a high prevalence of moderate to high rank curves.
• Explicit examples of curves with ranks 1 through 6 were computed, including a curve at k = 98/625 with rank 6.
• The curve at k = 11 has rank 5, at k = 19 has rank 4, at k = 3 has rank 3, at k = 4 has rank 2, and at k = 6 has rank 1.
• Among the 100,000 curves, 19.5% had undetermined rank, suggesting that the Selmer rank or MWRANK computation did not conclusively determine the actual rank in those cases.