Properties of Division Polynomials and Their Resultant

Presentation for Masters Talk

Background

• David Grant wrote a paper titled “Resultants of division polynomials II: Generalized Jacobi's derivative formula and singular torsion on elliptic curves”

• The goal, as the title suggests, was to generalize Jacobi’s derivative formula

Background, Cont.

• Jacobi’s derivative formula is:

• David Grants formula is:

My Job

• Write software that implements elliptic curves, theta functions, and the Weierstrass functions

• Test some well known and established formulae using my software and various values

• Test David Grant’s conclusions from his paper

Elliptic Curves Overview

• Elliptic curves via an equation, or via a lattice• Probably the most well known form is from

the equation

• For every equation of this form there is a lattice and vice-versa

A Prerequisite

• When I say “lattice”, I mean the following…• If and C, and they are R-independent, then we

define

as a lattice.

Elliptic Curve from Equation

• As already stated, this takes the form

where a and b are complex.

• To get a lattice (we call it the “period” of the elliptic curve), we have to use the arithmetic-geometric mean.

Arithmetic-Geometric Mean

• Defined by the recursion relation

• A subsequent theorem states that this converges, and they converge to the same limit. We denote this by

Calculating Period of Elliptic Curve

• Assume are roots of the equation , with .

Calculating Period of Elliptic Curve

• Now we can calculate the period:

• and forms the basis for the lattice, and we define the lattice by

Elliptic Curve from Lattice

• Again, let be a lattice.• For even , define the Eisenstein series as

• We let and .

Elliptic Curve From Lattice

• A theorem states that .• If we let and , then we have • Calculating the Eisenstein series is expensive.• We’ll show later a quicker way using theta

functions.

Group Properties

• We can define point-wise addition on an elliptic curve.

• Assume P and Q are two points on the elliptic curve. Let R’ be the point on the elliptic curve that intersects the line through P and Q and the curve. Let R be the reflection of R’ across the x-axis.

• R = P + Q

Division Polynomials

• Define nP = P + P + … + P (adding P to itself n times).

• Division polynomials allow us to calculate nP easily, without using traditional group addition.

Group Properties

• We can also explicitly calculate R.• Let P = , Q = , and R = ).• Then

Division Polynomials

• Given an elliptic curve of the form , define as follows:

Division Polynomials

• Finally, we define the division polynomial, as follows:

How It Can Be Used

• As stated previously, the division polynomial can be used for calculating nP:

Discriminants

• Let be a polynomial of degree n with roots .• We define the discriminant as

Resultants

• Let and be two polynomials of degrees m and n, respectfully.

• We define the resultant of and , denoted res(f,g), as