III.21 Elliptic Curves

Jordan S. Ellenberg

An elliptic curve over a field k can be defined as an algebraic curve of genus 1 over K, endowed with a point defined over k. If this definition is too abstract for your tastes, then an equivalent definition is the following: an elliptic curve is a curve in the plane determined by an equation of the form

When the characteristic of k is not 2, we can transform this equation into the simpler form y² = f (x), for some cubic polynomial f. In this sense, an elliptic curve is a rather concrete object. However, this definition has given rise to a subject of seemingly inexhaustible mathematical interest, which has provided a tremendous fund of ideas, examples, and problems in number theory and algebraic geometry. This is in part because there are many values of “X” for which it is the case that “the simplest interesting example of X is an elliptic curve.”

For instance, the points of an elliptic curve E with coordinates in K naturally form an Abelian group, which we call E(K). The connected projective VARIETIES [III.95] that admit a group law of this kind are called Abelian varieties; and elliptic curves are just the Abelian varieties that are one dimensional. The Mordell-Weil theorem tells us that, when K is a number field and A is an Abelian variety, A(K) is actually a finitely generated Abelian group, called a Mordell-Weil group; these Abelian groups are much studied but have retained much of their mystery (see RATIONAL POINTS ON CURVES AND THE MORDELL CONJECTURE [V.29]). Even when A is an elliptic curve, in which case we would call it E instead, there is a great deal that we do not know, though THE BIRCH-SWINNERTON-DYER CONJECTURE [V.4] offers a conjectural formula for the rank of the group E(K). For much more on the topic of rational points on elliptic curves, see ARITHMETIC GEOMETRY [IV.5].

Since E(K) forms an Abelian group, given any prime p one can look at the subgroup of elements P such that p^p =0. This subgroup is called E(K) [p]. In particular, we can take the algebraic closure of K and look at E()[p]. It turns out that, when k is a NUMBER FIELD [III.63] (or, for that matter, any field of characteristic not equal to p), this group is isomorphic to (/p)², no matter what choice of E we started with. If the group is the same for all elliptic curves, why is it interesting? Because it turns out that the GALOIS GROUP [V.21] Gal(/K) permutes the set E() [p]. In fact, the action of Gal(/K) on the group (/p)² gives rise to a REPRESENTATION [III.77] of the Galois group. This is a foundational example in the theory of Galois representations, which has become central to contemporary number theory. Indeed, the proof of FERMAT’S LAST THEOREM [V.10] by Andrew Wiles is in the end a theorem about the Galois representations that arise from elliptic curves. And what Wiles proved about these special Galois representations is itself a small special case of the family of conjectures known as the Langlands program, which proposes a thoroughgoing correspondence between Galois representations and automorphic forms, which are generalized versions of the classical analytic functions called MODULAR FORMS [III.59].

In another direction, if E is an elliptic curve over , then the set of points of E with complex coordinates, which we denote E(), is a COMPLEX MANIFOLD [III.88 §3]. It turns out that this manifold can always be expressed as the quotient of the complex plane by a certain group Λ of transformations. What is more, these transformations are just translations: each map sends z to z + c for some complex number c. (This expression of E() as a quotient is carried out with the help of ELLIPTIC FUNCTIONS [V.31].) Each elliptic curve gives rise in this way to a subset—indeed, a subgroup—of the complex numbers; the elements of this subgroup are called periods of the elliptic curve. This construction can be regarded as the very beginning of Hodge theory, a powerful branch of algebraic geometry with a reputation for extreme difficulty. (The Hodge conjecture, a central question in the theory, is one of the Clay Institute’s million-dollar-prize problems.)

Yet another point of view is presented by the MODULI SPACE [IV.8] of elliptic curves, denoted M_1,1. This is itself a curve, but not an elliptic one. (In fact, if I am completely honest, I should say that M_{1, l} is not quite a curve at all—it is an object called, depending on whom you ask, an ORBIFOLD [IV.4 §7] or an algebraic stack— you can think of it as a curve from which someone has removed a few points, folded the points in half or into thirds, and then glued the folded-up points back in. You might find it reassuring to know that even professionals in the subject find this process rather difficult to visualize.) The curve M_{1, l} is a “simplest example” in two ways: it is the simplest modular curve, and simultaneously the simplest moduli space of curves.