Elliptic Curves

Elliptic Curves

The way to formulate that the number of rational points on a particular elliptic curve is finite is:

Elliptic Addition

Given two complex points we can define Elliptic Operation for which if both point satisfy \( Re(z)^2 = Im(z)^3+ a Re(z) + b \) for any a,b then the resulting point will also satisfy this relation
With \(y^2=x^3+1\). Solutions are \((x,y)=(2,±3), (0,±1) ,(-1,0) \)
(2,3) + (0,1) = (-1,0)
(2,3) + (0,-1)= (2,-3)

Number of solutions mod p

The number of solutions to an equation mod p is useful

Birch-Schwinger-Dyer Conjecture

This relates to the idea that the rank of the group of the elliptic addition is related to the order of the L-function at s=1. If the rank is bigger than zero then the elliptic equation has infinite solutions.
With L(E,p) = prod{p:prime} 1/( 1 - notDivide(N,p)( (p+1 - numSol(E,p) )^(-s) + notDivide(N,p^2) p^(1-2s) ) )
Another way of stating it is that the rank is given by the following limit. The product should be only over primes:

Examples

Rank 0 curve: \( y^2-x^3-1 =0\)
Rank 1 curves: \(y^2-x^3+x-1=0\) , \( y^2-x^3-x^2-4=0\)
Rank 2 curves: \(y^2 -x^3+4x-1=0\) $$\sqrt{1}$$