Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat’s Last Theorem. More concretely, an elliptic curve is the set of zeros of a cubic polynomial in two variables. Where $ax^{3}+bx^{2}y+cxy^{2}+dy^{3}+ex^{2}+fxy+gy^{2}+hx+iy+j=0$ is the equation of a general cubic polynomial. A famous example being

$\displaystyle x^{3}+y^{3}=1$

or in homogeneous form,

$\displaystyle X^{3}+Y^{3}=Z^{3}$.

This is the first non-trivial case of Fermat’s Last Theorem.

A modular elliptic curve is an elliptic curve $E$ that admits a parametrization $X_{0}(N) \rightarrow E$ by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is a modular form in disguise.

In 1985, starting with a fictitious solution to Fermat’s last theorem (the Frey curve), G. Frey showed that he could create an unusual elliptic curve which appeared not to be modular. If the curve were not modular, then this would show that if Fermat’s last theorem were false, then the Taniyama-Shimura conjecture would also be false. Furthermore, if the Taniyama-Shimura conjecture is true, then so is Fermat’s last theorem.

However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated

By proving the semistable case of the conjecture, Andrew Wiles proved Fermat’s Last Theorem.

Some Elliptic curves: