Group Law on Specific Elliptic Curve

diophantine equationselliptic-curvesgroup-theory

Let $n$ be a fixed positive integer.
Consider the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=n.$$
If $n$ is odd then there are no positive integer solutions to this equation, and if $n$ is even then the positive integer solutions can be quite large. A nice reference is this mathoverflow question.

Clearing denominators gives the equivalent equation
$$a^3+b^3+c^3+abc=(n-1)(a+b)(a+c)(b+c).$$
This is an elliptic curve (provided that you pick an identity point).

Is there a nice way to explicitly write down the group law of this elliptic curve?

Certainly it's possible to write down the group law by transforming to Weierstrass form, using the (ugly) group law formula, and transforming.
The question is whether there is a nice way to write down the group law.

Best Answer

I worked this out a while ago, but I had forgotten about this question. I don't think that there is a nice algebraic formula for the group law, but at least you can avoid transforming to Weierstrass form.

Take $O=[-1:0:1]\in\mathbb{P}^2$. The key property of $O$ is that its tangent line is $y=6(x+z)$, which passes through $O$ with multiplicity 3 (rather than 2). Then there is a group operation on $E$ with identity element $O$, and the group operation satisfies $P+Q+R=O$ for any line $L$ that intersects $E$ at $P,Q,R$. Proving this requires some algebraic geometry.

You can compute $P+Q$ as follows:

  • Let $L$ be the line through $P$ and $Q$. If $P=Q$ then $L$ is the tangent line.
  • Let $R$ be the third point of intersection between $L$ and $E$.
  • Let $L^\prime$ be the line through $R$ and $O$. If $R=O$ then $L^\prime$ is the tangent line.
  • Then $P+Q$ is the third point of intersection between $L^\prime$ and $E$.
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