Rational Points on Elliptic Curves

(March 07, 2008)

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)

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)

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)

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

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)

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)

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)

This diophantine equation is birationally equivalent to the elliptic curve,

E : y2+nxy=x3
birational transformation is here.


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Hisanori Mishima