Consider the equation in two variables: Square of y= cubic-polynomial in x, with rational coefficients and distinct complex roots. The rational solutions of this equation (which is called an elliptic curve) form a commutative group, and a nice theory begins.
Except for some singular elliptic curves, the following group addition works. Take any two rational solutions of the above equation, connect them by a line, and the third point where the line intersects the curve is also a rational solution! Diophantus knew of this, although the first proof is due to Poincare.
The theory of elliptic curves is one of the most advanced areas in mathematics, with the Taniyama-Weil Conjecture being one of its high points. It states that all elliptic curves over Q are of the " modular form". It is no more a conjecture, as it was finally proven by Andrew Wiles, and since Taniyama-Weil implies Fermat's last theorem, this led to a resolution of the famous theorem.
Applications :
- Cryptography
- Factorization
- Coding Theory
- Fermat's Theorem
- Diophantine equations
Last updated 18 Jun 2007