Congruent Numbers and Elliptic Curves

Transcription

1 Congruen Numbers and Ellipic Curves Pan Yan Sepember 30, Problem In an Arab manuscrip of he 10h cenury, a mahemaician saed ha he principal objec of raional righ riangles is he following quesion[] Congruen number problem (Original version) Given a posiive ineger n, find a raional square a (a Q ) such ha a ± n are boh raional squares Definiion 11 An ineger n is a congruen number if here exiss a raional square a such ha a ± n are boh raional squares Example 1 (i) 5 is a congruen number: ( ) 41 5 = 1 (ii) 6 is a congruen number: ( ) 5 6 = (iii) 7 is a congruen number: ( ) = 10 ( ) 31, 1 ( ) 1, ( ) 113, 10 ( ) = 1 ( ) = ( ) = 10 ( ) 49 1 ( ) 7 ( ) Definiion 13 A righ riangle is raional if is legs and hypoenuse are all raional numbers Congruen number problem (Triangular version) Given a posiive ineger n, find a righ riangle such ha is sides are raional and is area equals n 1

2 Proof of he equivalence of he wo versions (Origional version Triangular version) Suppose α, β, γ are arihmeic progression of raional squares wih common difference n Then he righ angled riangle wih legs and hypoenuse a = γ α, b = γ + α, c = β has an area of n (Triangular version Origional version) Conversely, suppose we have a raional righ riangle [a, b, c] wih area n Then ( ) a b (, c ) (, a+b ) is an 3-erm arihmeic progression wih common difference n Example 14 (i) 5 is he area of raional righ angled riangle [ 0 3, 3, 41 6 ] (ii) 6 is he area of raional righ angled riangle [3, 4, 5] (iii) 7 is he area of raional righ angled riangle [ 4 5, 35 1, ] Remark 15 We assume n is a square free posiive ineger, because if [a, b, c] is a righ angled riangle wih area n, hen [as, bs, cs] is also a righ angled riangle wih area ns Open Problem: (i) Give a simple crierion o deermine wheher or no a number n is congruen (ii) When n is congruen, give an effecive algorihm o find a raional righ riangle whose area is n Theorem 16 (Ferma) 1,, 3 are no congruen numbers Proof We use Ferma s Infinie Decen Mehod o prove 1 is no congruen number His argumen based on Euclidean formula: Given (a, b, c) posiive inegers, pairwise coprime, and a + b = c Then here is a pair of coprime posiive ineger (p, q) wih p + q odd such ha a = pq, b = p q, c = p + q Thus we have a congruen number generaing formula: (11) n = pq(p + q)(p q)/m Sep 1: Suppose 1 is congruen number, hen here is an inegral righ angled riangle [a, b, c] wih minimum area m = pq(p + q)(p q) Sep : Since all 4 facors of m are coprime, Sep 3: We have an equaion p = x, q = y, p + q = u, p q = v (u + v) + (u v) = (x) Sep 4: (u + v, u v, x) forms a righ angled riangle wih a smaller area y This is a conradicion

3 Corollary 17 (Ferma s Righ Triangle Theorem) If n is a square, hen n is no a congruen number Remark 18 Alhough we have formula (11) o generae congruen numbers, his algorihm is far from efficien For example, n = 157 is he area of he raional righ angled riangle wih he following legs and hypoenuse (due o Zagier): a = , b = , c = Mahemaicians can no be replaced by compuers! Ellipic Curves Connecion wih Ellipic Curves Theorem 1 For n > 0, here is a one-o-one correspondence beween he following wo ses: {(a, b, c) : a + b = c, 1 ab = n}, {(x, y) : y = x 3 n x, y 0} Muually inverse correspondences beween hese wo ses are ( nb (a, b, c) c a, n ) ( x n, (x, y) c a y, nx y, x + n y Fix a real number n 0 The real soluions (a, b, c) o each of he following equaions (1) a + b = c, 1 ab = n, describe a surface in R 3 So i is naural o expec hese wo surfaces o inersec in a curve We wan o describe such a curve, which will be y = x 3 n x under he righ choice of coordinaes Le c = + a, subsiue i ino a + b = c, we ge b = + a, or equivalenly, () a = b ) Since ab = n 0, neiher a nor b is 0, so we can wrie a = n b and subsiue i ino (): 4n b = b 3

4 Muliplying each side by b, we ge 4n = b 3 b Dividing by 3 ( 0, oherwise a = c and hen b = 0, bu ab = n 0) yields ( ) 4n b 3 = b Muliplying each side by n 3, we ge ( ) n = Se x = nb = nb c a and y = n ( nb ) 3 ( ) nb n = n c a 0, so y = x 3 n x Remark (i) The equaion y = x 3 n x has hree rivial raional soluions wih y = 0: (0, 0), (n, 0), ( n, 0) (ii) The correspondence preserves posiiviy (iii) The equaion y = x 3 n x is an ellipic curve! Congruen number problem (Ellipic Curve version) For a posiive number n, find a raional poin wih y 0 on he ellipic curve E n : y = x 3 n x The viewpoin of he equaion y = x 3 n x allows one o do somehing sriking: produce a new raional righ angled riangle wih area n from wo known riangles (by he group law of poins on ellipic curves) Theorem 3 (Mordell, 19) E(Q) = Z r E(Q) ors Theorem 4 (Luz-Nagell Theorem, 1937, 1935) For an ellipic curve E : y = x 3 + Ax + B over Q wih A, B Z and le D = (4A 3 + 7B ) 0 If (x, y) is a orsion poin, hen x, y Z and eiher y = 0 or y D In he case of E n : y = x 3 n x, D = 4n 6 So he orsion poins are eiher y = 0 or y 4n 6 Bu y = x 3 n x has no soluion wih y 0, x, y Z, and y 4n 6 Hence, we have he following heorem Theorem 5 E n (Q) ors = {O, (0, 0), (n, 0), ( n, 0)} Remark 6 If here is one nonrivial raional poin on he ellipic curve E n : y = x 3 n x, hen here are infiniely many raional poins on he ellipic curve E n : y = x 3 n x The argumen is as following Suppose P = (x, y) wih y 0 is a raional poin on he ellipic curve Then P can no be a orsion, so np O if n Z and n 0 This means ha P, P, 3P, are all disinc If no, hen np = mp for some n < m and hen O = mp np = (m n)p, conradicion 4

5 Theorem 7 A posiive ineger n is a congruen number if and only if he ellipic curve E n : y = x 3 n x over Q has rank greaer han 0 Remark 8 Any poin wih y 0 gives rank > 0 Theorem 9 A posiive ineger n is a congruen number if and only if here exiss a poin of infinie order on he ellipic curve E n : y = x 3 n x Crierions for Non-Congruen Numbers and Condiions for Congruen Numbers Moreover, he viewpoin of hinking abou congruen numbers in erms of he ellipic curve y = x 3 n x goes far beyond he consrucion of new raional righ angled riangle wih area n This viewpoin leads o a enaive soluion o he whole congruen number problem! In 1983, Tunnell used arihmeic propery of he ellipic curve E n : y = x 3 n x o discover a previously unknown elemenary necessary condiion on congruen numbers and he was able o prove he condiion is also sufficien if he weak Birch and Swinneron- Dyer conjecure is rue Theorem 10 (Tunnell, 1983) Le n be an squarefree posiive ineger Se a(n) = #{(x, y, z) Z 3 : x + y + 8z = n}, b(n) = #{(x, y, z) Z 3 : x + y + 3z = n}, a (n) = #{(x, y, z) Z 3 : 8x + y + 16z = n}, b (n) = #{(x, y, z) Z 3 : 8x + y + 64z = n} For odd n, if n is a congruen number, hen a(n) = b(n); for even n, if if n is a congruen number, hen a (n) = b (n) Moreover, if he weak Birch and Swinneron-Dyer conjecure is rue for he ellipic curve E n : y = x 3 n x, hen he condiions are also sufficien Remark 11 Tunnell s heorem provides an uncondiional mehod of proving a squarefree ineger n is no congruen, and a condiional mehod of proving a squarefree ineger n is congruen (i) If n is odd, and a(n) b(n), hen n is no a congruen number If n is even, and a (n) b (n), hen n is no a congruen number (ii) Suppose he weak BSD conjecure is rue for he ellipic curve E n : y = x 3 n x If n is odd and a(n) = b(n), hen n is a congruen number If n is even and a (n) = b (n), hen n is a congruen number Example 1 (i) a(1) = b(1) =, a(3) = b(3) = 4 Hence, a(1) b(1), a(3) b(3) Hence 1,3 are no congruen numbers (ii) a () = b () = Hence is no a congruen number 5

6 Theorem 13 If he weak Birch and Swinneron-Dyer conjecure is rue, hen any posiive ineger n 5, 6, 7 (mod 8) is a congruen number Proof Suppose n 5, 6, 7 (mod 8) is a posiive ineger Wriing n = a b wih b squarefree Then a is odd, oherwise 4 would be a facor of n Therefore, n b (mod 8) Thus we may assume n is squarefree If n 5, 7 (mod 8) is odd, since here is no inegral soluion o x +y 5, 7 (mod 8), we have a(n) = b(n) = 0 Hence, a(n) = b(n) If he weak BSD conjecure is rue, hen Tunnell s Theorem implies ha n is a congruen number If n 6 (mod 8) is even, hen y 6 (mod 8) has no inegral soluion, and so a (n) = b (n) = 0 Hence, a (n) = b (n) If he weak BSD conjecure is rue, hen Tunnell s Theorem implies ha n is a congruen number References [1] Keih Conrad Lecure Noe: The Congruen Number Problem [] Leonard Eugene Dickson Hisory of he Theory of Numbers, Vol (190), p 46 [3] Neal Kobliz Inroducion o Ellipic Curves and Modular Forms (GTM 97), Springer-Verlag (1984) [4] Jerrold Baes Tunnell A classical diophanine problem and modular forms of weigh 3/ Inveniones Mahemaicae (1983) 7: