Elliptic Curves

My experience is that elliptic curves are extraordinarily fun to study. Every such curve is like a whole galaxy in itself, just like the rational numbers are. An elliptic curve over  $\mathbf{Q}$ is a curve that can be put in the form

$$y^2 = x^3 + ax+b,$$

where the cubic has distinct roots and $a, b\in \mathbf{Q}$. The amazing thing is that the set of pairs

$$E(\mathbf{Q}) = \{ (x,y)\in \mathbf{Q}\times \mathbf{Q}: y^2 = x^3 + ax +b \} \cup \{\infty\}$$

has a natural structure of ``group''. In particular, this means that given two points on $E$, there is a way to ``add'' the two solutions together to get another solution.

Many exciting problems in number theory can be translated into questions about elliptic curves. For example, Fermat's Last Theorem, which asserts that $x^n + y^n = z^n$ has no positive integer solutions when $n>2$ was proved using elliptic curves. Giving a method to decide which numbers are the area of a right triangle with rational side lengths has almost, but not quite, been solved using elliptic curves.

The central question about elliptic curves is The Birch and Swinnerton-Dyer Conjecture which gives a simple conjectural criterion to decide whether or not $E(\mathbf{Q})$ is infinite (and more). Proving the BSD conjecture is one of the Clay Math Institute's million dollar prize problems. I'll tell you what this conjecture is.