Properties of Division Polynomials and Their Resultant

1. Properties of Division Polynomials and Their Resultant Presentation for Masters Talk

2. 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

3. Background, Cont. • Jacobi’s derivative formula is: • David Grants formula is:

4. 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

5. 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

6. A Prerequisite • When I say “lattice”, I mean the following… • If and C, and they are R-independent, then we define as a lattice.

7. 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.

8. 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

9. Calculating Period of Elliptic Curve • Assume are roots of the equation , with .

10. Calculating Period of Elliptic Curve • Now we can calculate the period: • and forms the basis for the lattice, and we define the lattice by

11. Elliptic Curve from Lattice • Again, let be a lattice. • For even , define the Eisenstein series as • We let and .

12. 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.

13. 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

14. 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.

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

16. Division Polynomials • Given an elliptic curve of the form , define as follows:

17. Division Polynomials • Finally, we define the division polynomial, as follows:

18. How It Can Be Used • As stated previously, the division polynomial can be used for calculating nP:

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

20. Resultants • Let and be two polynomials of degrees m and n, respectfully. • We define the resultant of and , denoted res(f,g), as