Congruent Numbers and Elliptic Curves: II

  1. The Bijection Revisited Conceptually: (75 minutes)
    1. (2 minute) Definition of congruent number.
    2. (2 minute) Definition of elliptic curve.
    3. (40 minutes) Bijection between points and congruent triangles: revisited. In particular, here is a conceptual way to think about it. The set of rational right triangles with area $ n$ is the same as the set of simultaneous solutions to the system of equations

      $\displaystyle a^2 + b^2 = c^2, \qquad \frac{ab}{2} = n.
$

      This is the intersection of two quadratic surfaces in $ 3$ -space, hence a curve. That curve turns out, after an appropriate change of variables, to be the elliptic curve $ y^2 = x^3 - n^2 x$ . Exercise: ...6 steps... (Solution presentation: 30 minute).

  2. Break: (10 minute).

  3. Perimeter: (15 minute) Discuss the problem of which integers are the perimeter of a right triangle with rational side lengths. Again, the set of such triangles for a given $ m$ is the set of simultaneous solutions to 2 equations:

    $\displaystyle a^2 + b^2 = c^2, \qquad a+b+c = m.
$

    This is the intersection of a quadratic surface and a plane, hence a quadratic curve. (Exercise - 15 minutes trying this further; presentation of solution.)

  4. Generating New Points on Elliptic Curves from Other Points (30 minutes): Explain how it works. Show how to use SAGE to do it. (Exercise - 30 minutes doing this (and getting new triangles); presentations.)

William Stein 2006-07-07