Lagmental Vicfred

Discriminant and j-Invariant Separate Smoothness from Isomorphism by Vicfred

Last updated: Mon 15 July 2019

tags: mathematics, elliptic curves, isogenies, moduli

The discriminant detects singularity, while the j-invariant classifies elliptic curves over an algebraic closure. I will separate the object being defined from the consequence being claimed.

Definitions first

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.

$$ E:y^2=x^3+Ax+B $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \Delta=-16(4A^3+27B^2)\ne0,\qquad j(E)=1728\frac{4A^3}{4A^3+27B^2} $$

A small case in full

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \begin{cases}\Delta=0&\Longrightarrow E\text{ singular},\\j=0&\Longrightarrow A=0,\\j=1728&\Longrightarrow B=0.\end{cases} $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad E:y^2=x^3+Ax+B,\\[5pt] \mathsf{C}\;&:\quad \Delta=-16(4A^3+27B^2)\ne0,\qquad j(E)=1728\frac{4A^3}{4A^3+27B^2}. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \Delta=-16(4A^3+27B^2)\ne0,\qquad j(E)=1728\frac{4A^3}{4A^3+27B^2} \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Sat 19 July 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.