Elliptic Curves
From IT SPb Academy
Since childhood I was interested in Last Fermat's Theorem, and in 1994 it was proven by Andrew Wiles. Unfortunately I don't know the areas of Mathematics, required to understand the proof, such as the theory of Elliptic Curves and Modular Forms. I am trying to catch it up now and will try to explain main concepts here.
So, first, what is elliptic curve? First it is algebraic curve, i.e. the set of zeros for some polynom F(x,y) The simplest polynoms are linear: F(x,y) = a*x+b*y+c whose zeros are straight lines.
The 2nd class of polynomes are quadratic forms, like ax2+by2+c, and its zeros form quadratic curves, like ellipse, parabola, hyperbola (conic sections). It is also simple thing.
And the 3d class of polynoms are of degree 3. And it is just this class that elliptic curves correspond to. Actually the cubic curve is the set of zeros of the 3d degree polynom. And if cubic curve doesn't have singularities (intersections or cusps) then it is called elliptic curve.
It is interesting that using some rational transformations the equation for elliptic curve can be transformed to the following equation:
- y2 = x3 + a x + b
