COMMENTARY Congruent numbers John H. Coates1 Emmanuel College, Cambridge CB2 3AP, England; and Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea umber theory is the part of congruent numbers are of this form; for modern arithmetic geometry. The first mathematics concerned with example, the right-angled triangle with person to prove the existence of infinitely N the mysterious and hidden side lengths [225/30, 272/30, 353/30] has many square-free congruent numbers was properties of the integers and area 34). The proof of this simple general Heegner, whose now-celebrated paper rational numbers (by a rational number, assertion about the integers of Form 1 (5) published in 1952 showed that every we mean the ratio of two integers). The still seems beyond the resources of num- prime number p of the form p =8n +5 congruent number problem, the written ber theory today. However, the paper in is a congruent number. The importance history of which can be traced back at least PNAS by Tian (2) makes dramatic prog- of Heegner’s paper was realized only in a millennium, is the oldest unsolved major ress toward it, by proving that there are the late 1960s, after the discovery of the problem in number theory, and perhaps many highly composite congruent numbers conjecture of Birch and Swinnerton-Dyer in the whole of mathematics. We say that (1). We recall, without detailed explana- a right-angled triangle is “rational” if all its Tian at last sees how to tion, that the (unproven) conjecture of sides have rational length. A positive in- Birch and Swinnerton-Dyer predicts that teger N is said to be “congruent” if it is the establish a natural there are infinitely many points with ra- area of a rational right-angled triangle. If tional coordinates on an elliptic curve E if we multiply any congruent number N by generalization of and only if its complex L-series L(E, s) the square of an integer, we again get vanishes at the point s = 1 in the complex a congruent number, and so it suffices to Heegner’s argument to plane; here L(E, s)isdefined by a making consider only those integers N that are from E a certain natural Euler product square-free (meaning not divisible by the composite N. over all prime numbers, inspired by anal- square of an integer >1). The congruent ogy with the Euler product for the more number problem is simply the question of familiar Riemann zeta function. It is in fi deciding which square-free positive in- of Form 1. Speci cally, this paper shows fact known (6, 7) that there are only fi- k ≥ tegers are, or are not, congruent numbers. that, for every positive integer 1, there nitely many rational points on E if L(E, s) fi Long ago, it was realized that an integer are in nitely many square-free congruent does not vanish at s = 1, and this result can n n N ≥ 1 is congruent if and only if there numbers of the form 8 +5,8 +6, be exploited to prove the existence of exists a point (x, y) on the elliptic curve and 8n + 7 having exactly k distinct odd many noncongruent numbers (8, 9). y2 = x3 − N2x, with rational coordinates prime factors, and tells us precisely how to However, the question of proving the exis- x, y and with y ≠ 0. Until the 17th century, construct them. tence of congruent numbers is a totally different matter. Heegner’s original paper mathematicians made numerical tables of History congruent numbers by using ingenuity to makes no use whatsoever of the complex write down the corresponding rational Fermat noted that his proof that 1 is not L-series of the congruent number elliptic right-angled triangles. For example, the a congruent number also implies that curve. However, Birch (10) realized that integers 5, 6, and 7 were all known to be there are no rational numbers x and y with a variant of Heegner’s method could be 4 4 congruent, as they are the areas of the xy ≠ 0 such that x + y = 1. This is pre- used to construct rational points (which right-angled triangles, whose sides lengths sumably what led him to his claim (often he called Heegner points) on a large class are given respectively by [40/6, 9/6, 41/6], called his Last Theorem) that, for every of elliptic curves, and that these points integer n ≥ 3, there are no rational num- seemed to be of infinite order precisely [3, 4, 5], and [288/60, 175/60, 337/60]. The n n first important theoretical result about bers x and y with xy ≠ 0 such that x + y = when the complex L-series of the curve has congruent numbers was established by 1. No evidence of his claimed proof of a simple zero at s = 1. Moreover, he and Fermat, who proved in the 17th century this second assertion has ever been found, Stephens found convincing numerical evi- that 1 is not a congruent number. As ex- and the first proof was given in 1994 by dence that these Heegner points [specifi- plained in more detail later, Fermat’s Wiles (3), at the end of a vast chain of cally their canonical heights, in the sense of discovery—and indeed every major new earlier developments in algebraic number Neron and Tate (10, 11)] were simply re- fi L E s result proven about congruent numbers theory and the theory of automorphic lated to the rst derivative of ( , )atthe s since then—has eventually led to great forms, lasting throughout the 19th and point = 1, all in accord with the conjec- advances in the study of some of the 20th century, whose roots can at least ture of Birch and Swinnerton-Dyer (1). It deepest questions about Diophantine partly be traced back to the search for is now history that Gross and Zagier (12) equations. Interest in the congruent num- a solution of Fermat’s Last Theorem and surprised the whole number theory com- ber problem was enhanced by the discov- the congruent number problem. In a dif- munity by proving this conjecture of Birch ery in the early 1960s of the celebrated ferent direction, Mordell (4) in 1922 and Stephens. Soon afterward, Kolyvagin conjecture of Birch and Swinnerton-Dyer showed that a natural generalization of (7) discovered an equally remarkable (1), and the realization that, in retrospect, Fermat’s proof that 1 is not a congruent method for using these Heegner points to the congruent number problem is the number implies that, for every elliptic prove most of the conjecture of Birch and oldest and most down-to-earth example of curve given by an equation with rational this conjecture. In particular, the conjec- coefficients (henceforth we shall simply ture predicts that every integer of the form say an “elliptic curve”), the abelian group Author contributions: J.H.C. wrote the paper. 8n +5,8n +6,8n +7(n =0,1,2,...)— of points on the curve with rational coor- The author declares no conflict of interest. henceforth called Form 1—should be a dinates is always finitely generated. This See companion article 10.1073/pnas.1216991109. congruent number (but note that not all beautiful result was the starting point of 1E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1219394110 PNAS Early Edition | 1of2 Downloaded by guest on September 30, 2021 Swinnerton-Dyer for those elliptic curves E use of L-functions hinges on two quite of Birch and Swinnerton-Dyer (1) pre- such that L(E, s)hasazeroats =1of different partial results in support of the dicts that, for these N, the rational num- k order at most 1. conjecture of Birch and Swinnerton-Dyer ber M(E) must always be divisible by 4 , (1). The first is an important generaliza- Recent Developments and Zhao (14, 15) found an ingenious tion of the Gross–Zagier formula (12), due proofofthisfact.Thisdivisibilityplays However, all these subsequent develop- to Yuan et al. (13). The original formula a central role in Tian’s work (2). Both L – ments relating Heegner points to -func- of Gross Zagier (12) does not apply to Tian (2) and Zhao (14, 15) use induction ’ tions bypassed the original congruent Heegner s points on the congruent num- on the number k of odd prime divisors number problem. No one could see how to ber elliptic curve because a certain rami- of N in their work, and it is intriguing extend Heegner’s existence proof to the fication condition fails in this case. The to note that there are some striking par- curve y2 = x3 − N2x when N is of Form 1 second is a result of Zhao (14, 15) about allels between the averaging of Heegner and has more than two odd prime factors. the congruent number elliptic curve, not N Nor was it known how to prove that the when N is of Form 1, but rather when N is points attached to divisors of used by L k ≥ Tian (2), and the averaging of the values complex -series of this curve has a simple either a product of 1 distinct prime M E N zero at s = 1 for a large class of composite numbers, each of which is of the form ( )overdivisorsof used by Zhao square-free N satisfying Form 1. It is 8n + 1, or twice such a product. For the (14, 15), in their separate proofs. In ’ only now that Tian (2) at last sees how congruent number elliptic curve with such conclusion, Tian s work (2) is an impor- to establish a natural generalization of N,theL-function L(E, s) does not usually tant milestone in the history of this Heegner’s argument to composite N,by vanish at s = 1, and it is known that the ancient problem, and, as has always hap- ingeniously combining deep ideas involv- quantity M(E)=L(E,1)/Ω is a rational pened in the past, it seems only a matter ing L-functions with Heegner’soriginal number, where Ω denotes the smallest of time until its generalization to all el- construction of points.