V.4 The Birch–Swinnerton-Dyer Conjecture

Given an ELLIPTIC CURVE [III.21], there is a natural way of defining a binary operation on its points, and this turns the elliptic curve into an ABELIAN GROUP [I.3 §2.1]. Moreover, the points on the curve with rational coordinates form a subgroup of this group. Mordell’s theorem tells us that this subgroup is finitely generated. (These results are described in RATIONAL POINTS ON CURVES AND THE MORDELL CONJECTURE [V.29].)

Every finitely generated Abelian group is isomorphic to a group of the form ^r × C_n1, × C_n2 × · · · × C_nk, where C_n stands for the cyclic group with n elements. The number r, which measures the maximum number of independent elements of this group that have infinite order, is called the rank of the elliptic curve. Mordell’s theorem implies that the rank of every elliptic curve is finite, but it does not tell us how to calculate it. That turns out to be an extraordinarily hard problem: in fact, so hard that it is considered a remarkable achievement of Birch and Swinnerton-Dyer even to have come up with a plausible conjecture about it.

Their conjecture relates the rank of an elliptic curve to a very different object associated with that curve: an L-FUNCTION [III.47]. This is a function with properties similar to those of the RIEMANN ZETA FUNCTION [IV.2 §3], but it is defined in terms of a series of numbers N₂ (E), N₃ (E), N₅ (E), . . ., one for each prime p; the number N_p (E) is the number of points on the elliptic curve when it is considered as a curve over the FIELD [I.3 §2.2] with p elements. One of the properties of the L-function of E is that it is HOLOMORPHIC [I.3 § 5.6]. (The fact that it can be extended to a holomorphic function everywhere on the complex plane is very far from obvious: it follows from the fact that all elliptic curves are modular. See FERMAT’S LAST THEOREM [V.10].) Birch and Swinnerton-Dyer conjectured that the rank of the group associated with the elliptic curve is equal to the order of the zero of its L-function at 1. (If the L-function does not take the value 0 at 1, then this order is defined to be 0.) This can be thought of as a sophisticated LOCAL-TO-GLOBAL PRINCIPLE [III.51], in that it relates the rational solutions to the equation for the elliptic curve to the solutions mod p for each prime p.

Another remarkable feature of the conjecture is that far less was known about elliptic curves when Birch and Swinnerton-Dyer made it. Now there are many reasons to find it plausible, but then it was much more of a leap in the dark: they based it on numerical evidence gleaned from computations of N_p (E) for several elliptic curves and many primes p. In other words, they did not calculate the orders of zeros of L-functions of various elliptic curves, since that was too hard, but guessed them based on approximations.

The Birch–Swinnerton-Dyer conjecture has now been proved for curves with L-functions that have a zero of order 0 or 1 at 1, but a proof of the general case still appears to be a long way off. It is one of the problems for which the Clay Mathematics Institute offers a prize of a million dollars. For a further discussion of the problem and much more about its mathematical context, see ARITHMETIC GEOMETRY [IV.5].