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Congruent Numbers with Many Prime Factors

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Congruent Numbers with Many Prime Factors 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 10th Remark 2: The kernel A½2 and the image 2A of the multiplication century, stated that the principal object of the theory of rational by 2 on A are characterized by Gauss’ genus theory. Note that the × 2 right triangles is to find a square that when increased or diminished multiplication by 2 induces an isomorphism A½4=A½2’A½2 ∩ 2A. by a certain number, m becomes a square [Dickson LE (1971) History By Gauss’ genus theory, Condition 1 is equivalent in that there are of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In exactly an odd number of spanning subtrees in the graph whose modern language, this object is to find a rational point of infinite ; ⋯; ≠ vertices are p0 pk and whose edges are those pipj, i j,withthe order on the elliptic curve my2 = x3 − x. Heegner constructed such p m quadratic residue symbol i = − 1. It is then clear that Theorem 1 rational points in the case that are primes congruent to 5,7 mod- pj ulo 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 with follows from Theorem 2. many prime divisors and give a sketch in this report. The full details The congruent number problem is not only to determine of all the proofs will be given in ref. 1 [Tian Y (2012) Congruent whether a given integer m is congruent, but also to construct fi ðmÞ : 2 = 3 − Numbers and Heegner Points, arXiv:1210.8231]. in nite order points on the elliptic curve E my x x for congruent m. Gross-Zagier formula | modular curve | genus theory Let X0ð32Þ be the modular curve defined over Q for the congruent subgroup Γ0ð32Þ, which is a genus 1 curve, and the ∞ ′ := ; ∞ positive integer is called a congruent number if it is the area cusp is rational so that E ðX0ð32Þ Þ is an elliptic curve 2 3 of a right-angled triangle, all of whose sides have rational over Q. It is known that E′ has Weierstrass equation y = x + 4x. A 2 = 3 − : → length. The problem of determining which positive integers are Let E be the elliptic curve y x x and let f X0ð32Þ E be congruent is buried in antiquity (ref. 2, chap. 16) with it long a modular parametrization of degree 2 mapping ∞ to 0, i.e., it is MATHEMATICS being known that the numbers 5, 6, and 7 are congruent. Fermat a degree 2 isogeny from E′ to E. proved that 1 is not a congruent number, and similar arguments Let n = p0p1⋯pk be a product of distinct odd primes with show that 2 and 3 are not congruent numbers. No algorithm has p1; ⋯; pk ≡ 1 mod 8. Let m = n or 2n such that m ≡ 5; 6or fi ever been proven for infallibly deciding whether a given integer 7 mod 8. Let H be thepffiffiffiffiffiffiffiffiffi Hilbert class eld of the imaginary qua- ≥ ≥ fi = Q − = − ðn−1Þ=2 n 1 is congruent, the reason being that an integer n 1is draticffiffiffiffiffiffiffiffi eld K ð 2nÞ. Letffiffiffiffiffiffiffiffi m* ð 1Þ m so that ; p p − congruent if and only if there exists a point ðx yÞ, with x and Kð m* Þ ⊂ H and let EðQð m* ÞÞ be the group of points ≠ (n) pffiffiffiffiffiffiffiffi y rational numbers and y 0, on the elliptic curve E : ∈ Q σ = − σ 2 = 3 − 2 x Eð ð m* ÞÞ such that x ffiffiffiffiffiffiffiffix where is the nontrivialffiffiffiffiffiffiffiffi ele- y x n x. Moreover, assuming n to be square-free, a classical p p − calculation of root numbers shows that the complex L-function ment in the Galois group GalðQð m* Þ=QÞ.ThenEðQð m* ÞÞ ≅ of this curve has zero of odd order at the center of its critical EðmÞðQÞ, whose torsion subgroup is E½2. We now construct the pffiffiffiffiffiffiffiffi − ðmÞ strip precisely when n lies in one of the residue classes of 5, 6, so-called Heegner point ym ∈ EðQð m* ÞÞ ≅ E ðQÞ, which will and 7 modulo (mod) 8. Thus, in particular, the unproven con- be shown to be of infinite order for m satisfying Condition 1 in jecture of Birch and Swinnerton-Dyer (3, 4) predicts that every Theorem 2. 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 1. If n 5 modpffiffiffiffiffi 8, then m m* n. Let P X0ð32Þ be the the following partial results in this direction. image of i 2n=8 on the upper half plane H via the complex = Γ ∖ ∪ P1 Q uniformization X0ð32Þ p0ðffiffiffi32Þ ðHpffiffiffi ð ÞÞ. It turns out that ≥ fi := + + ; + fi Theorem 1. For any given integer k 0, there are in nitely many the point z fðPÞ ð1 2 2 2Þ on E is de ned overpffiffiffi the fi := pffiffi ∈ Q square-free congruent numbers with exactly k + 1 odd prime divisors Hilbert class eld H, and that yn TrH=Kð nÞz Eð ð nÞÞ. pffiffiffi − in each residue class of 5, 6, 7 mod 8. Moreover, yn (resp. 2yn) belongs to EðQð nÞÞ if k ≥ 1 (resp. Theorem 1 follows from the following result by Remark k = 0). 2 below. For any abelian group A and an integer d ≥ 1, we write ≡ ; = − ∈ 2. If n 3 7p modffiffiffiffiffi 8, then m* m. Let P X0ð32Þ be the im- A[d] for the Kernel of the multiplication by d on A. + = age of ði 2n p2Þffiffiffi8 viap theffiffiffi complex uniformization and let := + + ; + Theorem 2. ≥ = ⋯ z fðPÞ ð1 2 2 2Þ, which turns out to be a point de- Let k 0 be an integer and n p0p1 pk a product of fi = = ≡ ≤ ≤ = ned over HðiÞ. Identify A with G GalðHðiÞ KðiÞÞ and let distinct odd primes with pi 1 mod 8 for 1 i k. Let m nor2n σ ∈ = ≡ ; 0 GalðHðiÞ KðpiÞÞffiffiffiffiffiffiffiffiffibe the element corresponding to the such that m 5 6, or 7 mod 8. Then m is a congruent numberpffiffiffiffiffiffiffiffiffi ; − χ fi = Q − ideal class of ð2 pffiffiffiffiffiffiffiffi2nÞ. Let be the character of G factor provided that the ideal class group A of the eld K ð 2nÞ through GalðKði; m* Þ=KðiÞÞ, which is nontrivial if m ≠ 2n. satisfies the condition Let ϕ ⊂ A beP a complete representative of G=hσ0i and define σ ym := ym;ϕ = σ∈ϕχðσÞz . 0; if n ≡ ± 3 mod 8; dimF ðA½4=A½2Þ = [1] 2 1; otherwise: The following theorem is our main result, from which Theorem 2 follows. Remark 1: The above result when k = 0 is due to Heegner (5), and completed by Birch (6), Stephens (7), and Monsky (8); and that when k = 1 is due to Monsky (8) and Gross (9). Actually Author contributions: Y.T. designed research, performed research, and wrote the paper. Heegner is the first mathematician who found a method to con- The author declares no conflict of interest. † struct fairly general solutions to cubic Diophantine equations (5). This Direct Submission article had a prearranged editor. The method of this paper is based on Heegner’s construction. 1To whom correspondence should be addressed. E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1216991109 PNAS Early Edition | 1of3 Downloaded by guest on September 27, 2021 Theorem 3. = ⋯ X Let n p0p1 pk be a product of distinct odd primes k 2 y0 − yd ∈ E½2: [2] with pi ≡ 1 mod 8 for 1 ≤ i ≤ k. Let m = n or 2n such that m ≡ 5; 6 fi p0jdjn or 7 mod 8. Let ympbeffiffiffiffiffiffiffiffi the point de ned as above. Then we have k−1 − that ym ∈ 2 EðQð m* ÞÞ + E½2. Moreover, if Conditionffiffiffiffiffiffiffiffi 1 in p − We claim thatffiffiffi for each proper divisor d of n divisible by Theorem 2 is satisfied, then the point y ∉ 2kEðQð m* ÞÞ + E½2. p − m p , y ∈ 2kEðQð dÞÞ + E½2 (and we will show this key result Moreover, by Theorem 3 and a simple computation on 2-Selmer 0 d at the end using induction hypothesis).pffiffiffi Assuming this, we can group, we obtain the following result on Birch and Swinnerton- = k ′ + ′ ′ ∈ Q − ∈ write yd 2 Pyd td for some yd Eð ð dÞÞ and td E½2.Thus Dyer conjecture: k yn = 2 ðy0 − ; ≠ y′dÞ + t for some t ∈ E½2.Considerthe p0jdjn d n ffiffiffi ffiffiffi ffiffiffi p k+1 p 1 p Theorem 4. For the elliptic curve EðmÞ : my2 = x3 − x, where m is (injective) Kummer map EðQð nÞÞ=2 EðQð nÞÞ → H ðQð nÞ; + m k 1 fl an integer as in Theorem 2, we have rankZEð ÞðQÞ = 1 = E½2 Þ and the exact in ation-restriction sequence À À À ffiffiffiÁÁ Â ÃÁ ðmÞ ðmÞ + p ords=1LðE ; sÞ. Moreover, the Shafarevich–Tate group of E is → 1 =Q ; 1 H Gal H0 n E 2 finite and has odd cardinality. À ÀpffiffiffiÁ Â ÃÁ À Â ÃÁ → 1 Q ; k+1 → 1 +; k+1 : H n E 2 H H0 E 2 Sketch of Proof. We now explain our method in the case p0 ≡ P k+1 + = − ′ ∈ 5 mod 8 (other cases are similar).
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