🔢 mathematics

Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

This paper explicitly determines the 2-Selmer rank of elliptic curves associated with Heron triangles of area $2mn$ by analyzing quadratic twists induced by powers of 2, thereby constructing explicit families of such curves with arbitrarily large 2-Selmer ranks for any squarefree odd integer $n$ .

Original authors: Vinodkumar Ghale, Md Imdadul Islam

Published 2026-08-04

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Vinodkumar Ghale, Md Imdadul Islam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where numbers aren't just cold, hard digits on a calculator, but characters in a grand, ancient mystery. This story lives in the kingdom of Number Theory, a branch of mathematics that treats whole numbers like a secret society with hidden rules and relationships. For centuries, mathematicians have been obsessed with a specific puzzle called the "congruent number problem." It asks a simple question: Can a specific whole number be the area of a right-angled triangle where all the sides are also rational numbers (fractions)?

To solve this, mathematicians don't just draw triangles; they build bridges to a different world called Elliptic Curves. Think of these curves not as smooth lines you draw on paper, but as complex, twisting rollercoasters made of invisible mathematical tracks. The "shape" of the track tells you if the triangle puzzle has a solution. But here's the tricky part: sometimes the track looks like it has a solution, but it's actually a dead end. To figure this out, mathematicians use a special tool called the 2-Selmer group. You can think of this group as a "security checkpoint" or a "filter." It checks every possible path on the rollercoaster to see if it's a real, working route or just a fake illusion. The size of this group (its "rank") tells us how many potential solutions exist. The bigger the rank, the more complex the puzzle, and the harder it is to find the actual answer.

Now, enter the Heronian Elliptic Curves. These are a special family of these mathematical rollercoasters that are directly linked to Heron triangles—triangles with rational sides and rational areas, named after the ancient Greek mathematician Hero. While the congruent number problem is about right-angled triangles, Heron triangles can have any angle, making them a much wilder and more diverse group.

In this paper, two mathematicians, Vinodkumar Ghale and Md Imdadul Islam, decided to take a deep dive into this specific family of curves. They wanted to see what happens when you tweak a specific "knob" on the curve. This knob is a number called $m$ , which is part of the triangle's area formula ($2mn$). By changing $m$ , they aren't just making a tiny adjustment; they are performing a "quadratic twist." Imagine taking a rubber band (the curve) and twisting it around a pole. The shape changes, the tension shifts, and the way it snaps back is completely different, even though it's made of the same material.

The authors focused on what happens when they twist these curves using powers of 2 (changing $m$ from 1 to 2, to 4, to 8, and so on). They wanted to know: Does this twisting make the security checkpoint (the 2-Selmer group) get bigger or smaller? And can they make it as huge as they want?

Here is what they found. They proved that by carefully choosing the other number in the formula, $n$ , they could control the size of this security checkpoint with incredible precision. Specifically, they showed that if you pick a number $n$ made up of a certain type of prime numbers (those that leave a remainder of 1 when divided by 8), the size of the 2-Selmer group grows in a straight line. Every time you add one of these special prime factors to $n$ , the rank of the group goes up by one.

The paper provides a clear, step-by-step recipe for this. It tells you exactly how to calculate the size of the group based on how many prime factors $n$ has and whether the twisting number $m$ is odd or even. They didn't just guess; they used a rigorous mathematical method called "2-descent" to prove that their formulas are correct. They even ran computer simulations with massive numbers (some with over a dozen prime factors) to confirm their theory, and the results matched their predictions perfectly.

The most exciting part of their discovery is that there is no limit. Because you can keep adding more and more of these special prime factors to $n$ , you can construct a family of these curves where the 2-Selmer rank gets arbitrarily large. In other words, they showed you how to build a mathematical structure with a security checkpoint that is as massive as you can possibly imagine. This doesn't just solve a small puzzle; it gives mathematicians a powerful new tool to create examples of curves with huge complexity, helping them understand the deep, hidden architecture of numbers.

Technical Summary: Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

Problem Statement
This paper addresses the arithmetic of Heronian elliptic curves, specifically those arising from Heron triangles (triangles with rational sides and rational area) of area $2mn$. While the congruent number problem (right-angled triangles) is well-studied regarding the distribution of 2-Selmer ranks in quadratic twist families, the specific context of Heronian elliptic curves with a natural parameter $m$ (the exponent of 2 in the area) had not been explicitly analyzed. The authors investigate how the 2-Selmer group of the family of curves $E_{m,n}$ transforms under quadratic twisting by powers of 2 (varying $m$ ), aiming to construct explicit families with prescribed and arbitrarily large 2-Selmer ranks.

Methodology
The authors employ a complete 2-descent on the family of elliptic curves defined by:
$E_{m,n}: y^2 = x(x - 2mn^2)(x + 2m)$
where $n$ is a squarefree odd integer such that $n^2 + 1 = 2q$ for some prime $q$ . The methodology proceeds as follows:

Torsion Analysis: The authors first establish that the rational torsion subgroup $E_{m,n}(\mathbb{Q})_{\text{tors}}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for all $m$ , ruling out points of order 4 via 2-divisibility criteria.
Descent Map Setup: They define the 2-descent map $\beta$ into $Q(S, 2) \times Q(S, 2)$ , where $S$ includes the prime factors of $n$ , $q$ , and 2. The 2-Selmer group is identified with the set of pairs $(b_1, b_2)$ for which a system of homogeneous spaces has non-trivial solutions in all local fields $\mathbb{Q}_l$ .
Local Solvability Analysis: The core of the proof involves a detailed analysis of local obstructions at $l=2$ $l = 2$ , $l=p$ $l = p$ (for $p|n$ $p ∣ n$ ), and $l=q$ $l = q$ .
- Parity of $m$ : The analysis splits into cases where $m$ is odd and where $m$ is even.
- Prime Factorization of $n$ : The integer $n$ is decomposed into $n = n_1 n_2$ , where $n_1$ contains prime factors $p \equiv \pm 1 \pmod 8$ and $n_2$ contains prime factors $p \equiv \pm 3 \pmod 8$ .
- Congruence Conditions: The authors derive necessary and sufficient conditions for pairs $(b, b)$ , $(2b, b)$ , etc., to belong to the Selmer group based on the congruence class of $q = (n^2+1)/2 \pmod 8$ and the residue classes of the prime factors of $n$ .
Explicit Construction: By eliminating pairs that fail local solvability (e.g., showing that if $p \equiv \pm 3 \pmod 8$ divides $b$ and $m$ is odd, the pair is not in the Selmer group), the authors isolate the exact generators of the 2-Selmer group.

Key Results
The paper provides explicit formulas for the 2-Selmer rank $s_2(E_{m,n})$ and the structure of the 2-Selmer group based on the parity of $m$ and the factorization of $n$ :

Case 1: $m$ is Odd
- If $n_1 = 1$ $n_{1} = 1$ (all prime factors of $n$ $n$ are $\equiv \pm 3 \pmod 8$ $\equiv \pm 3 (mod 8)$ ):
  - If $q \equiv 1 \pmod 8$ , $s_2 = 1$ .
  - If $q \equiv 5 \pmod 8$ , $s_2 = 0$ .
- If $n_1 > 1$ $n_{1} > 1$ :
  - If $q \equiv 1 \pmod 8$ , $s_2 = r + s + 1$ .
  - If $q \equiv 5 \pmod 8$ , $s_2 = r + s$ .
- Here, $r$ and $s$ are the counts of prime factors of $n_1$ congruent to $1$ and $7 \pmod 8$ , respectively.
Case 2: $m$ is Even
- If $n_2 = 1$ $n_{2} = 1$ (all prime factors of $n$ $n$ are $\equiv \pm 1 \pmod 8$ $\equiv \pm 1 (mod 8)$ ):
  - $s_2 = r + s + 1$ .
- If $n_2 > 1$ $n_{2} > 1$ :
  - $s_2 = r + s + t - 1$ , where $t$ is the number of prime factors of $n_2$ .

Significance and Claims
The authors claim that this work represents the first explicit determination of how the 2-Selmer group transforms under quadratic twisting specifically within the context of Heronian elliptic curves.

Arbitrarily Large Ranks: The results demonstrate that by choosing $n$ with a sufficiently large number of prime factors congruent to $1 \pmod 8$ (specifically in $n_1$ ), one can construct explicit families of Heronian elliptic curves with arbitrarily large 2-Selmer ranks.
Generalization: This generalizes the family studied in previous work [1] (which was restricted to specific conditions on $n$ ) to arbitrary $m$ , providing a complete arithmetic description of the twisting behavior.
Upper Bounds: Since the 2-Selmer rank provides an upper bound for the Mordell-Weil rank ( $rank E(\mathbb{Q}) \le s_2(E)$ ), these formulas provide explicit upper bounds for the rank of these curves.

The paper supports these theoretical findings with computational verification using SageMath and MAGMA, presenting tables of 2-Selmer ranks and generators for various $n$ with increasing numbers of prime factors, confirming the linear growth of the rank with respect to the number of prime factors congruent to $1 \pmod 8$ .

Technical Summary: Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

More like this