V.29 Rational Points on Curves and the Mordell Conjecture

Suppose that we wish to study a Diophantine equation such as x³ + y³ = z³. A simple observation we can make is that studying integer solutions to this equation is more or less equivalent to studying rational solutions to the equation a³ + b³ = 1: indeed, if we had integers x, y, and z such that x³ + y³ = z³, then we could set a = x/z and b = y/z and obtain rational numbers with a³ + b³ = 1. Conversely, given rational numbers a and b with a³ + b³ = 1, we could multiply a and b by the lowest common multiple z of their denominators and set x = az and y = bz, obtaining integers x, y, and z such that x³ + y³ = z³.

The advantage of doing this is that it reduces the number of variables by 1 and focuses our attention on the plane curve u³ + v³ = 1, which is a simpler object than the surface x³ + y³ = z³. A curve of this kind, defined by one or more polynomial equations, is called an algebraic curve.

Even though we are interested in rational points on the curve, it can be helpful to regard the curve as an abstract object that has many manifestations. (See ARITHMETIC GEOMETRY[IV. 5] for a fuller discussion of this point.) For instance, if we think of u and v as complex numbers, then the “curve” u³ + v³ = 1 becomes a two-dimensional object, which means that it starts to have a genuinely interesting geometry. To be precise, it can be regarded as a two-dimensional MANIFOLD [I.3 §6.9] living in . From a complex perspective it is a one-dimensional subset of , but from either perspective it has a potentially interesting topology. For instance, if we COMPACTIFY [III.9] the curve by considering it as a subset not of but of the complex PROJECTIVE PLANE [I.3 §6.7], then we turn it into a compact surface. As such, it must have a GENUS [III.33], which, roughly speaking, tells us how many holes it has.

Surprisingly, it turns out that this geometrical definition of the genus of a curve is intimately related to the algebraic question of how many rational points the curve contains. Consider, for instance, the curve u² + v² = 1, which corresponds to the Diophantine equation x² + y² = z². Since there are infinitely many Pythagorean triples that are not multiples of each other, there are infinitely many rational points on the curve u² + v² = 1. In order to calculate the genus of the curve, we first rewrite it as (u + iv) (u — iv) = 1. This shows that the function (u, v) u + iv is a homeomorphism from the curve to the set {0} of all nonzero complex numbers, which itself is homeomorphic to a sphere with two points removed. The compactification adds in these points, giving us a surface of genus 0, so we say that the curve u² + v² = 1 has genus 0. It turns out that a curve of genus 0 always has either no rational points or infinitely many.

In general, the larger the genus, the harder it is to find rational solutions. A curve of genus 1 is called an ELLIPTIC CURVE [III.21]. It is possible for an elliptic curve to contain infinitely many rational points as well, but the set of such points turns out to have a very restricted structure. To explain this, let us consider an elliptic curve E of the form y² = ax³ + bx² + cx + d (a form into which any elliptic curve can be put). If we think of it as a curve in , then we can define a binary operation on it as follows: given any two points P and Q on E, let L be the line through P and Q (where we define this to be the tangent to the curve at P if P = Q). In general, L intersects E in three points, of which P and Q are two; let R′ be the third. Finally, let R be the reflection of R′ in the x-axis (which also belongs to E because E has the form y² = f(x)). This construction of R from P and Q, which is illustrated in figure 1, defines a binary operation on the points of E. Remarkably, this binary operation turns E into an Abelian group, at least when we also include a point at infinity and adopt the convention that the point at infinity is the intersection of E with any vertical line. The point at infinity is the identity of the group, since a vertical line through a point P intersects E in the reflection P′ of P in the x-axis, and when we reflect P′ in the x-axis we get P again.

Figure 1 The group law for an elliptic curve.

It is laborious, but basically straightforward, to come up with a formula for the “group law” of an elliptic curve—that is, a formula for the coordinates of R in terms of the coordinates of P and Q. Once one does so, it becomes clear that if P and Q have rational coordinates, then so does R. Thus, the set of all rational points on an elliptic curve E forms a subgroup. This simple fact can be used to produce rather easily some very large solutions to the corresponding Diophantine equations. For instance, one can start with a small solution, associate with it a rational point P, and then use the formula for the binary operation to calculate 2P, then 4P, then 8P, and so on. Unless nP = 0 for some n (which can certainly happen), in no time at all one has a point on the curve with rational coordinates that have huge numerators and denominators. To give an idea of the sort of solutions that can be obtained in this way, take the elliptic curve y² = x³ - 5x and let P be the point (-1, 2) (which lies on the curve since 2² = (-1)³-5(-1)). If you calculate 5P using the group law, then you obtain the point (-5 248 681/4 020 025, 16 718 705 378/8 060 150 125). In general, the number of digits needed to express the point nP grows exponentially with n.

In the early twentieth century, POINCARÉ [VI.61] conjectured that the subgroup of rational points on an elliptic curve was finitely generated. This conjecture was proved by Louis Mordell in 1922. Thus, although a curve of genus 1 may have infinitely many rational points, there is a finite set of these points that can be used to build up all the others: this is the sense in which the structure of the set of rational solutions is restricted.

Mordell conjectured that a curve of genus at least 2 could contain only finitely many points. This was a remarkable conjecture: if true, it would apply to an extremely wide class of Diophantine equations, proving that all of them had at most finitely many solutions (up to a multiple). Just one of its many implications was that for each n 3 the Fermat equation xⁿ + yⁿ = zⁿ had at most finitely many solutions with x, y, and z coprime. However, it is one thing to make a very general conjecture and quite another to prove it, and for a long time the consensus was that the Mordell conjecture, like many other conjectures in number theory, was way beyond what anybody could prove. It therefore came as a big surprise when Gerd Faltings proved the conjecture in 1983.

As a result of Faltings′s proof, our knowledge about Diophantine equations took a huge leap forward. The theorem has subsequently been given a variety of different proofs, some of them simpler than that of Faltings. However, remarkable as these proofs are, they do have some limitations. One is that they are ineffective. That is, even though Faltings′s theorem tells us that certain curves have finitely many rational points, no known proof gives any bound on the sizes of the numerators and denominators of the coordinates of those points, so we do not have any way of knowing whether we have found all of them. This aspect of the theorem is common in number theory: another example of a famous theorem that is ineffective is ROTH′S THEOREM [V.22]. To find effective versions of these theorems would be a further remarkable breakthrough. (Variants of THE ABC CONJECTURE [V.1] would imply effective versions of these results, but the ABC conjecture seems even further out of reach now than Mordell’s conjecture seemed before Faltings proved it.)

At the beginning of this article, we simplified the equation x³ + y³ = z³ so that we were looking at a curve rather than a surface. But we obviously cannot always do that. For instance, if we apply the same procedure to the equation x⁵ + y⁵ + z⁵ = w⁵, then we obtain the two-dimensional surface t⁵ + u⁵ + v⁵ = 1. Our knowledge about rational points on varieties (that is, sets defined by polynomial equations) of dimension greater than 1 is very limited. However, there is at least a definition of a “variety of general type” that serves as an analogue of the notion of a curve of genus at least 2. One cannot expect such a variety to contain only finitely many rational points, but a higher-dimensional analogue of the Mordell conjecture, due to Serge Lang, asserts that the rational points on a variety X of general type must all be contained in a union of finitely many lower-dimensional subvarieties of X. This conjecture is considered to be well out of reach of present methods: indeed, it is not even universally believed.