## Rational Points on Elliptic Curves

### Table of curves

 (Short) Weierstrass forms [C1]  y2 = x3 + b [C2]  y2 = x3 + ax [C3]  y2 = x3 + ax + b Related to Diophantine equations [C4]  X3 + Y3 = A [C5]  y2 = Dx4+1 [C6]  Congruent number D : y2 = x3 - D2x [C7]  Concordant number n : y2 = x(x+1)(x+n) [C8]  y2 = (x+p)(x2+p2)    (by Stroeker) [C9]  (x+y+z)(1/x+1/y+1/z)=n    (by Bremner) [C10]  x/y+y/z+z/x=n

### Results (rational point, rank, torsion)

 [C1]  y2 = x3 + b   (b∈Z) integer point (x≦109) -10000≦b≦-1  (Aug. 07, 2000)  1≦b≦10000  (Aug. 07, 2000) rank / generator -1000≦b≦-1  (Sep. 18, 2000)  1≦b≦1000  (Sep. 18, 2000) This curve is the special case of the short Weierstrass form, y2 = x3 + ax + b, a=0 If b=0, the curve becomes singular (cusp). By Hall's conjecture, if x, y are positive integers such that b=x3-y2 is nonzero, then |b| ≫ε x1/2 - ε Structure of torsion group (Aug. 16, 2000) [C2]  y2 = x3 + ax   (a∈Z) integer point (x≦109) -10000≦a≦-1  (Aug. 07, 2000)  1≦a≦10000  (Aug. 07, 2000) rank / generator -1000≦a≦-1  (Sep. 18, 2000)  1≦a≦1000  (Sep. 18, 2000) This curve is the special case of the short Weierstrass form, y2 = x3 + ax + b, b=0 If a=0, the curve becomes singular (node). Structure of torsion group (Aug. 16, 2000) [C3]  y2 = x3 + ax + b   (a,b∈Z) integer point (x≦109)  1≦a≦100, 1≦b≦100 (Aug. 16, 2000)  1≦a≦100, -100≦b≦-1 (Aug. 16, 2000) -100≦a≦-1, 1≦b≦100 (Aug. 16, 2000) -100≦a≦-1, -100≦b≦-1 (Aug. 16, 2000) rank / generator  1≦a≦20, 1≦b≦20 (Feb. 01, 2008)  1≦a≦20, -20≦b≦-1 (Feb. 01, 2008) -20≦a≦-1, 1≦b≦20 (Feb. 01, 2008) -20≦a≦-1, -20≦b≦-1 (Feb. 01, 2008) This curve is called as short Weierstrass form. If the discriminant D=a2-4b=0, i.e. (a, b) = (0, 0), (±2, 1), (±4, 4), (±6, 9), (±8, 16), (±10, 25), (±12, 36), (±14, 49), (±16, 64), (±18, 81), (±20, 100), ... the curve becomes singular. [C4]  X3 + Y3 = A   (A∈Z) rank / generator of y2=x3-423.A2 1≦A≦1000 (January 31, 2008) rational solutions of X3 + Y3 = A 1≦A≦1000 (January 31, 2008) The cubic equation is birationally equivalent to the elliptic curve, E : y2=x3-423.A2   (A ∈Z; X, Y ∈Q) under the following birational transformation. [C5]  Y2=DX4+1   (D∈Z) rank / generator of y2=x3-4Dx 1≦D≦1000 (January 31, 2008) rational solutions of Y2=DX4+1 1≦D≦1000 (February 05, 2008) This curve is birationally equivalent to, E : y2=x3-4Dx under the following birational transformation. [C6]  Congruent Numbers D rank / generator of y2=x(x2-D2) 1≦D≦999 (Jul. 30, 2002) : 361 1001≦D≦1999 (Jul. 30, 2002) : 358 2001≦D≦2999 (Jul. 30, 2002) : 354 3001≦D≦3999 (Jul. 30, 2002) : 354 4001≦D≦4999 (Jul. 30, 2002) : 349 5001≦D≦5999 (Jul. 30, 2002) : 345 6001≦D≦6999 (Jul. 30, 2002) : 351 7001≦D≦7999 (Jul. 30, 2002) : 340 8001≦D≦8999 (Jul. 30, 2002) : 347 9001≦D≦9999 (Jul. 30, 2002) : 352 The definition of congruent number and the birational transformation between the defined curve and the elliptic curve E : y2=x(x2-D2) is described here. The criteria for the congruent number was given in Tunell's paper, J. B. Tunell, "A Classical Diophantine Problem and Modular Forms of Weight 3/2", Inventiones mathematicae, 72(1983)323-334. [C7]  Concordant Numbers n rank / generator of y2=x(x+1)(x+n) 1≦n≦1000 (February 01, 2008) representations of Concordant numbers 1≦n≦1000 (February 05, 2008) Definition of concordant number If there exists the solution of following two quadratic forms simultaneously, x2 + y2 = z2 x2 + ny2 = w2 n is called as concordant. Historically, it is studied by Euler. If n is concordant, the following elliptic curve, E : y2=x(x+1)(x+n) has a rank larger than 0 (i.e. it has non-trivial rational points). Birational transformation is here. [C8]  y2=(x+p)(x2+p2)   (p∈Z, prime) rank / generator 1≦n≦1000 (January 31, 2008) This curve is studied by, R. J. Stroeker and J. Top, "On the equation Y2=(X+p)(X2+p2)", Rocky Mountain J. Math, 24:3 (1994), 1135-1161 Benjamin M. M. de Weger, "Solving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms", Experimental Mathematics, Vol. 7 (1998), No. 3, page 243-256. In the former paper, they showed that when p=2 or p=±3 (mod 8), the rank is 0. In the latter paper, they determined all integer solutions of cases p=167, 223, 337, 1201, and determined the ranks of these elliptic curves are 1, 1, 3, and 3 respectively. [C9]  (x+y+z)(1/x+1/y+1/z)=n   (n, x, y, z∈Z) rank / generator of E : y2=x3+(n2-6n-3)x2+16nx -500≦n≦-1 (February 05, 2008) 1≦n≦500 (February 05, 2008) integer solution of (x+y+z)(1/x+1/y+1/z)=n -500≦n≦-1 (February 05, 2008) 1≦n≦500 (February 05, 2008) This diophantine equation is birationally equivalent to the elliptic curve, E : y2=x3+(n2-6n-3)x2+16nx birational transformation is here. [C10]  x/y+y/z+z/x=n   (n, x, y, z∈Z) rank / generator of E : y2+nxy=x3 -100≦n≦-1 (March 07, 2008) 1≦n≦100 (March 07, 2008) integer solution of x/y+y/z+z/x=n -100≦n≦-1 (March 07, 2008) 1≦n≦100 (March 07, 2008) This diophantine equation is birationally equivalent to the elliptic curve, E : y2+nxy=x3 birational transformation is here.

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima