Congruent numbers with many prime factors
Ye Tian1
Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Edited† by S. T. Yau, Harvard University, Cambridge, MA, and approved October 30, 2012 (received for review September 28, 2012)

Mohammed Ben Alhocain, in an Arab manuscript of the 10thcentury, stated that the principal object of the theory of rationalright triangles is to ﬁnd a square that when increased or diminishedby a certain number, m becomes a square [Dickson LE (1971) Historyof the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In
modern language, this object is to ﬁnd a rational point of inﬁnite
order on the elliptic curve my 2 = x 3 − x. Heegner constructed suchrational points in the case that m are primes congruent to 5,7 modulo 8 or twice primes congruent to 3 modulo 8 [Monsky P (1990)
Math Z 204:45–68]. We extend Heegner’s result to integers m withmany prime divisors and give a sketch in this report. The full detailsof all the proofs will be given in ref. 1 [Tian Y (2012) Congruent
Numbers and Heegner Points, arXiv:1210.8231].
Gross-Zagier formula| modular curve | genus theory

Apositive integer is called a congruent number if it is the areaof a right-angled triangle, all of whose sides have rational
length. The problem of determining which positive integers arecongruent is buried in antiquity (ref. 2, chap. 16) with it long
being known that the numbers 5, 6, and 7 are congruent. Fermat
proved that 1 is not a congruent number, and similar arguments
show that 2 and 3 are not congruent numbers. No algorithm has
ever been proven for infallibly deciding whether a given integern ≥ 1 is congruent, the reason being that an integer n ≥ 1 iscongruent if and only if there exists a point ðx; yÞ, with x andy rational numbers and y ≠ 0, on the elliptic curve E(n):y2 = x3 − n2 x. Moreover, assuming n to be square-free, a classicalcalculation of root numbers shows that the complex L-functionof this curve has zero of odd order at the center of its critical
strip precisely when n lies in one of the residue classes of 5, 6,and 7 modulo (mod) 8. Thus, in particular, the unproven conjecture of Birch and Swinnerton-Dyer (3, 4) predicts that every
positive integer lying in the residue classes of 5, 6, and 7 mod 8
should be a congruent number. The aim of this paper is to prove
the following partial results in this direction.
Theorem 1. For any given integer k ≥ 0, there are inﬁnitely manysquare-free congruent numbers with exactly k + 1 odd prime divisors
in each residue class of 5, 6, 7 mod 8.
Theorem 1 follows from the following result by Remark
2 below. For any abelian group A and an integer d ≥ 1, we write
A[d] for the Kernel of the multiplication by d on A.
Theorem 2. Let k ≥ 0 be an integer and n = p0 p1 ⋯pk a product ofdistinct odd primes with pi ≡ 1 mod 8 for 1 ≤ i ≤ k. Let m = n or 2nsuch that m ≡ 5; 6, or 7 mod 8. Then m is a congruent numberpﬃﬃﬃﬃﬃﬃﬃﬃﬃ
provided that the ideal class group A of the ﬁeld K = Qð −2nÞsatisﬁes the condition
0; if n ≡ ± 3 mod 8;[1]dimF2 ðA½4=A½2Þ =1;
otherwise:Remark 1: The above result when k = 0 is due to Heegner (5),and completed by Birch (6), Stephens (7), and Monsky (8); andthat when k = 1 is due to Monsky (8) and Gross (9). Actually
Heegner is the ﬁrst mathematician who found a method to construct fairly general solutions to cubic Diophantine equations (5).
The method of this paper is based on Heegner’s construction.
21256–21258 | PNAS | December 26, 2012 | vol. 109 | no. 52

Remark 2: The kernel A½2 and the image 2A of the multiplication
by 2 on A are characterized by Gauss’ genus theory. Notethat the×2
multiplication by 2 induces an isomorphism A½4=A½2 ’ A½2 ∩ 2A.
By Gauss’ genus theory, Condition 1 is equivalent in that there areexactly an odd number of spanning subtrees in the graph whose
vertices are p0 ; ⋯; pk and whose edges are those pi pj , i ≠ j, with thequadratic residue symbolpipj= − 1. It is then clear that Theorem 1

follows from Theorem 2.
The congruent number problem is not only to determine
whether a given integer m is congruent, but also to constructinﬁnite order points on the elliptic curve EðmÞ : my2 = x3 − x forcongruent m.
Let X0 ð32Þ be the modular curve deﬁned over Q for thecongruent subgroup Γ0 ð32Þ, which is a genus 1 curve, and thecusp ∞ is rational so that E′ : = ðX0 ð32Þ; ∞Þ is an elliptic curve
over Q. It is known that E′ has Weierstrass equation y2 = x3 + 4x.
Let E be the elliptic curve y2 = x3 − x and let f : X0 ð32Þ → E bea modular parametrization of degree 2 mapping ∞ to 0, i.e., it isa degree 2 isogeny from E′ to E.
Let n = p0 p1 ⋯pk be a product of distinct odd primes withp1 ; ⋯; pk ≡ 1 mod 8. Let m = n or 2n such that m ≡ 5; 6 or
7 mod 8. Let H be theHilbertclass ﬁeld of the imaginary quapﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
dratic= ð−1Þðn−1Þ=2 m so thatpﬃﬃﬃﬃﬃﬃﬃﬃﬁeld K = Qð −2nÞ. pLet
ﬃﬃﬃﬃﬃﬃﬃﬃ m*−Kð m* Þp⊂ﬃﬃﬃﬃﬃﬃﬃ
H ﬃ and let EðQð m* ÞÞ be the group of pointsx ∈ EðQð m* ÞÞ such that σx = p− ﬃﬃﬃﬃﬃﬃﬃx whereσ is the nontrivial
ﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃ element in the Galois group GalðQð m* Þ=QÞ. Then EðQð m* ÞÞ− ≅E½2.
EðmÞ ðQÞ, whose torsion subgroup ispﬃﬃﬃﬃﬃﬃﬃ
ﬃ We now construct theso-called Heegner point ym ∈ EðQð m* ÞÞ− ≅ EðmÞ ðQÞ, which will
be shown to be of inﬁnite order for m satisfying Condition 1 inTheorem 2.
1. If n ≡ 5 modpﬃﬃﬃﬃﬃ8, then m = m* = n. Let P ∈ X0 ð32Þ be theimage of i 2n=8 on the upper half plane H via the complex
1uniformization X0 ð32Þ = Γp0 ð32Þ∖ðHﬃﬃﬃ
pﬃﬃ∪ﬃ P ðQÞÞ. It turns out thatthe point z : = f ðPÞ + ð1 + 2; 2 + 2Þ on E is deﬁned overpﬃﬃﬃtheHilbert class ﬁeld H, and that yn : = TrH=KðpﬃﬃnÞ z ∈ EðQð nÞÞ.
pﬃﬃﬃ
Moreover, yn (resp. 2yn ) belongs to EðQð nÞÞ− if k ≥ 1 (resp.
k = 0).
2. If n ≡ 3; 7pmod
ﬃﬃﬃﬃﬃ 8, then m* = − m. Let P ∈ X0 ð32Þ be the image of ði 2n + p2Þ=8ﬃﬃﬃ viapthe
ﬃﬃﬃ complex uniformization and letz : = f ðPÞ + ð1 + 2; 2 + 2Þ, which turns out to be a point deﬁned over HðiÞ. Identify A with G = GalðHðiÞ=KðiÞÞ and let
σ 0 ∈ GalðHðiÞ=KðiÞÞ
beﬃ the element corresponding to thepﬃﬃﬃﬃﬃﬃﬃﬃ
ideal class of ð2; p−2n
ﬃﬃﬃﬃﬃﬃﬃﬃ Þ. Let χ be the character of G factor
through GalðKði; m* Þ=KðiÞÞ, which is nontrivial if m ≠ 2n.
Let ϕ ⊂ A bePa complete representative of G=hσ 0 i and deﬁneym : = ym;ϕ = σ∈ϕ χðσÞzσ .
The following theorem is our main result, from which Theorem
2 follows.

Author contributions: Y.T. designed research, performed research, and wrote the paper.
The author declares no conﬂict of interest.
See Commentary on page 21182.
†This Direct Submission article had a prearranged editor.

1

To whom correspondence should be addressed. E-mail: ytian@math.ac.cn.

www.pnas.org/cgi/doi/10.1073/pnas.1216991109