Lagmental Vicfred

The Legendre Family Displays a One-Parameter Moduli Coordinate by Vicfred

Last updated: Tue 11 June 2024

tags: mathematics, elliptic curves, isogenies, moduli

Elliptic curves with ordered nonzero two-torsion can be written in Legendre form. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Start locally

An elliptic curve \(E/K\) is a smooth projective genus-one curve with a chosen point. In short Weierstrass form \(y^2=x^3+Ax+B\), smoothness is encoded by the discriminant.

$$ E_\lambda:y^2=x(x-1)(x-\lambda),\qquad\lambda\notin\{0,1\} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ j(\lambda)=256\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$

Compute before generalising

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \lambda\sim1-\lambda\sim\frac1\lambda\sim\frac1{1-\lambda}\sim\frac{\lambda-1}{\lambda}\sim\frac{\lambda}{\lambda-1} $$

The global view

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad E_\lambda:y^2=x(x-1)(x-\lambda),\qquad\lambda\notin\{0,1\},\\[5pt] \mathsf{C}\;&:\quad j(\lambda)=256\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}. \end{aligned} $$

Edge conditions

Short Weierstrass form requires characteristic different from \(2\) and \(3\). Torsion and isogeny formulas also change in inseparable characteristic.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] j(\lambda)=256\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Tue 30 August 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.