Lagmental Vicfred

An Isogeny Is a Finite Group Morphism of Elliptic Curves by Vicfred

tags: mathematics, elliptic curves, isogenies, moduli

A nonconstant morphism preserving the origin has finite kernel and is automatically surjective. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

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.

$$ \phi:E\to E',\qquad\phi(O)=O $$

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.

$$ \deg\phi=\#\ker\phi\quad\text{when }\phi\text{ is separable} $$

A small case in full

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ [n]:E\to E,\qquad\ker[n]=E[n]\cong(\mathbf Z/n\mathbf Z)^2,\qquad\deg[n]=n^2\quad(\operatorname{char}K\nmid n) $$

The reusable statement

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \phi:E\to E',\qquad\phi(O)=O,\\[5pt] \mathsf{C}\;&:\quad \deg\phi=\#\ker\phi\quad\text{when }\phi\text{ is separable}. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \deg\phi=\#\ker\phi\quad\text{when }\phi\text{ is separable} \end{gathered}} $$

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 Fri 17 June 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.