Congruent numbers

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y² = x³ − N²x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more detail later, Fermat’s discovery—and indeed every major new result proven about congruent numbers since then—has eventually led to great advances in the study of some of the deepest questions about Diophantine equations. Interest in the congruent number problem was enhanced by the discovery in the early 1960s of the celebrated conjecture of Birch and Swinnerton-Dyer (1), and the realization that, in retrospect, the congruent number problem is the oldest and most down-to-earth example of this conjecture. In particular, the conjecture predicts that every integer of the form 8n + 5, 8n + 6, 8n + 7 (n = 0, 1, 2, …)—henceforth called Form 1—should be a congruent number (but note that not all congruent numbers are of this form; for example, the right-angled triangle with side lengths [225/30, 272/30, 353/30] has area 34). The proof of this simple general assertion about the integers of Form 1 still seems beyond the resources of number theory today. However, the paper in PNAS by Tian (2) makes dramatic progress toward it, by proving that there are many highly composite congruent numbers

Tian at last sees how to establish a natural generalization of Heegner's argument to composite N.

of Form 1. Specifically, this paper shows that, for every positive integer k ≥ 1, there are infinitely many square-free congruent numbers of the form 8n + 5, 8n + 6, and 8n + 7 having exactly k distinct odd prime factors, and tells us precisely how to construct them.

History

Fermat noted that his proof that 1 is not a congruent number also implies that there are no rational numbers x and y with xy ≠ 0 such that x⁴ + y⁴ = 1. This is presumably what led him to his claim (often called his Last Theorem) that, for every integer n ≥ 3, there are no rational numbers x and y with xy ≠ 0 such that xⁿ + yⁿ = 1. No evidence of his claimed proof of this second assertion has ever been found, and the first proof was given in 1994 by Wiles (3), at the end of a vast chain of earlier developments in algebraic number theory and the theory of automorphic forms, lasting throughout the 19th and 20th century, whose roots can at least partly be traced back to the search for a solution of Fermat’s Last Theorem and the congruent number problem. In a different direction, Mordell (4) in 1922 showed that a natural generalization of Fermat’s proof that 1 is not a congruent number implies that, for every elliptic curve given by an equation with rational coefficients (henceforth we shall simply say an “elliptic curve”), the abelian group of points on the curve with rational coordinates is always finitely generated. This beautiful result was the starting point of modern arithmetic geometry. The first person to prove the existence of infinitely many square-free congruent numbers was Heegner, whose now-celebrated paper (5) published in 1952 showed that every prime number p of the form p = 8n + 5 is a congruent number. The importance of Heegner’s paper was realized only in the late 1960s, after the discovery of the conjecture of Birch and Swinnerton-Dyer (1). We recall, without detailed explanation, that the (unproven) conjecture of Birch and Swinnerton-Dyer predicts that there are infinitely many points with rational coordinates on an elliptic curve E if and only if its complex L-series L(E, s) vanishes at the point s = 1 in the complex plane; here L(E, s) is defined by a making from E a certain natural Euler product over all prime numbers, inspired by analogy with the Euler product for the more familiar Riemann zeta function. It is in fact known (6, 7) that there are only finitely many rational points on E if L(E, s) does not vanish at s = 1, and this result can be exploited to prove the existence of many noncongruent numbers (8, 9). However, the question of proving the existence of congruent numbers is a totally different matter. Heegner’s original paper makes no use whatsoever of the complex L-series of the congruent number elliptic curve. However, Birch (10) realized that a variant of Heegner’s method could be used to construct rational points (which he called Heegner points) on a large class of elliptic curves, and that these points seemed to be of infinite order precisely when the complex L-series of the curve has a simple zero at s = 1. Moreover, he and Stephens found convincing numerical evidence that these Heegner points [specifically their canonical heights, in the sense of Neron and Tate (10, 11)] were simply related to the first derivative of L(E, s) at the point s = 1, all in accord with the conjecture of Birch and Swinnerton-Dyer (1). It is now history that Gross and Zagier (12) surprised the whole number theory community by proving this conjecture of Birch and Stephens. Soon afterward, Kolyvagin (7) discovered an equally remarkable method for using these Heegner points to prove most of the conjecture of Birch and Swinnerton-Dyer for those elliptic curves E such that L(E, s) has a zero at s = 1 of order at most 1.

Recent Developments

However, all these subsequent developments relating Heegner points to L-functions bypassed the original congruent number problem. No one could see how to extend Heegner’s existence proof to the curve y² = x³ − N²x when N is of Form 1 and has more than two odd prime factors. Nor was it known how to prove that the complex L-series of this curve has a simple zero at s = 1 for a large class of composite square-free N satisfying Form 1. It is only now that Tian (2) at last sees how to establish a natural generalization of Heegner’s argument to composite N, by ingeniously combining deep ideas involving L-functions with Heegner’s original construction of points. Interestingly, Tian’s use of L-functions hinges on two quite different partial results in support of the conjecture of Birch and Swinnerton-Dyer (1). The first is an important generalization of the Gross–Zagier formula (12), due to Yuan et al. (13). The original formula of Gross–Zagier (12) does not apply to Heegner’s points on the congruent number elliptic curve because a certain ramification condition fails in this case. The second is a result of Zhao (14, 15) about the congruent number elliptic curve, not when N is of Form 1, but rather when N is either a product of k ≥ 1 distinct prime numbers, each of which is of the form 8n + 1, or twice such a product. For the congruent number elliptic curve with such N, the L-function L(E, s) does not usually vanish at s = 1, and it is known that the quantity M(E) = L(E, 1)/Ω is a rational number, where Ω denotes the smallest positive real period of E. The conjecture of Birch and Swinnerton-Dyer (1) predicts that, for these N, the rational number M(E) must always be divisible by 4^k, and Zhao (14, 15) found an ingenious proof of this fact. This divisibility plays a central role in Tian’s work (2). Both Tian (2) and Zhao (14, 15) use induction on the number k of odd prime divisors of N in their work, and it is intriguing to note that there are some striking parallels between the averaging of Heegner points attached to divisors of N used by Tian (2), and the averaging of the values M(E) over divisors of N used by Zhao (14, 15), in their separate proofs. In conclusion, Tian’s work (2) is an important milestone in the history of this ancient problem, and, as has always happened in the past, it seems only a matter of time until its generalization to all elliptic curves is established.