Appearing everywhere from state-of-the-art cryptosystems to the proof of Fermat's Last Theorem, *elliptic curves* play an important role in modern society and are the subject of much research in number theory today.

An elliptic curve $E$ defined over the rational numbers (fractions) is a smooth plane curve of the form

$$y^2 = x^3 + Ax + B$$

together with a special point $\mathcal{O}$ "at infinity".

What do elliptic curves look like? We can plot elliptic curves in the real plane:

*[above] The elliptic curve $E_1: y^2 = x^3 + 1$*

*[below] The elliptic curve $E_{17}: y^2 = x^3 + 17$*

but actually these two graphs, while they look roughly the same, hide much of the interesting *arithmetic* structure of the two curves. To elaborate on this, let us consider the set of *rational points* $E(\mathbb{Q})$: the set of points $(x,y)$ with $x$ and $y$ both rational, satisfying the equation of $E$, together with the point $\mathcal{O}$.

A fundamental problem in the study of elliptic curves is the following: given an elliptic curve $E$ defined over the rationals, find its set $E(\mathbb{Q})$ of rational points. For example, the elliptic curve $E_1: y^2 = x^3 + 1$ has precisely 6 rational points, given by the set $\{\mathcal{O}, (-1,0), (0, \pm 1), (2, \pm 3) \}$. The elliptic curve $E_{17}: y^2 = x^3 + 17$ has, among its rational points, the following:

$$\mathcal{O}, (-1, \pm 4), (-2, \pm 3), (2, \pm 5), (4, \pm 9), \left(\frac{1}{4}, \pm \frac{33}{8}\right), (8, \pm 23), \left(-\frac{8}{9}, \pm \frac{109}{27}\right), \left(\frac{19}{25}, \pm \frac{522}{125}\right), \\(43, \pm 282), (52, \pm 375), \left(-\frac{64}{25}, \pm \frac{59}{125} \right), \left(\frac{94}{25}, \pm \frac{1047}{125}\right), ...$$

While $E_1$ has finitely many rational points, $E_{17}$ has *infinitely many* rational points - something that cannot be seen by looking at the real-valued graphs above.

It turns out that the set $E(\mathbb{Q})$ of rational points enjoys additional structure: it is an *abelian group*. This means that given rational points $P$ and $Q$, there is a way to produce the sum $P+Q$, and this is another rational point. Indeed, there is a geometric way to carry out this addition. Draw the line through $P$ and $Q$. This intersects the elliptic curve in a third rational point $R$, and the reflection of $R$ in the $x$-axis is the sum $P+Q$, as seen here:

In 1922, Mordell proved that the set of rational points $E(\mathbb{Q})$ is a *finitely generated* abelian group. This means that any rational point $P$ can be written as a finite combination of certain generating points. For example, the points $P_1 = (-2, 3)$ and $P_2 = (-1, 4)$ are generators of the rational points of $E_{17}$. This means that all rational points on $E_{17}$ - even very complicated points - can be written in terms of these two points. For example, we have

$$ \begin{align} S & = \left(\frac{\small{\text{170914680007433755273838144758012}}}{\small{\text{111693884047097647373766735954481}}}, -\frac{\small{\text{5355483650472998648137959899973372475907238039375}}}{\small{\text{1180440469197546185983425190055773135812433974871}}}\right)\\ & = -5 P_1 + 5 P_2. \end{align}$$

One difficult open question is the following: given an elliptic curve $E$ over the rationals, how do we find the generators of $E(\mathbb{Q})$? In fact, giving an algorithm to calculate how many (independent infinite-order) generators there are - a quantity known as the *algebraic rank* of $E(\mathbb{Q})$ - would be a breakthrough. This is the subject of the *Birch and Swinnerton-Dyer conjecture*, one of the Clay Mathematics Institute's million-dollar Millennium Prize Problems. The conjecture is that the algebraic rank of $E(\mathbb{Q})$ is equal to another invariant associated to $E$, its *analytic rank*, the order of vanishing of an object known as the *$L$-function* $L(E,s)$ attached to $E$ at the point $s = 1$.

The elliptic curve $L$-function plays a very important role in modern number theory. Perhaps the most celebrated theorem of our era is the fact that elliptic curves over the rationals are *modular*: that is, the $L$-function of an elliptic curve defined over the rationals is equal to the $L$-function of a *cusp form*. In 1994, Wiles and Taylor-Wiles proved this result for semistable elliptic curves over the rationals and, in so doing, proved Fermat's Last Theorem. This correspondence between elliptic curves and cusp forms - through the theory of Galois representations - now motivates an exciting area of research known as the *Langlands problem*.