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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” - PowerPoint PPT Presentation
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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