Lagmental Vicfred

The Weil Pairing Detects Two-Dimensional Torsion by Vicfred

tags: mathematics, elliptic curves, isogenies, moduli

For n prime to the characteristic, the n-torsion carries a nondegenerate alternating pairing into roots of unity. I will separate the object being defined from the consequence being claimed.

The data

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.

$$ e_n:E[n]\times E[n]\to\mu_n $$

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.

$$ e_n(P,Q)=e_n(Q,P)^{-1},\qquad e_n(P,P)=1 $$

Derivation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ e_n(aP+bQ,cP+dQ)=e_n(P,Q)^{ad-bc},\qquad\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc $$

Invariant content

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad e_n:E[n]\times E[n]\to\mu_n,\\[5pt] \mathsf{C}\;&:\quad e_n(P,Q)=e_n(Q,P)^{-1},\qquad e_n(P,P)=1. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] e_n(P,Q)=e_n(Q,P)^{-1},\qquad e_n(P,P)=1 \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Wed 13 June 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.