Congruent Numbers and Elliptic Curves

Today's workshop is about the connection between congruent numbers and elliptic curves.

1. Daily crashing of the SAGE Notebook.
2. The Simuw SAGE Notebooks: login/password, graphics, tab completion, using mathematica/maple/etc from it.
3. Why are they called ``congruent numbers''? Connection between congruent numbers and ``congruences'' modulo $n$ .
4. 10-minute break.
5. Definition of elliptic curve.
6. Connection between congruent numbers and elliptic curves.

• COMPUTATION:
  • Find the point on an elliptic curve corresponding to the $(3,4,5)$ -triangle.
  • Find the point on an elliptic curve corresponding to a rational right triangle with area $2006$ .
  • Download a table from workshop website of the congruent numbers up to $1000$ , and look for patterns, e.g., reduce the congruent numbers modulo $8$ , etc., and see if you notice anything.
  • Find a rational number $x$ such that $x-6, x, x+6$ are all perfect squares.
  • Find a rational number $x$ such that $x-2006, x, x+2006$ are all perfect squares.
  • Let $n$ be the year you were born. Is it possible to find a rational number $x$ such that $x-n, x, x+n$ are all perfect squares?
• THEORY:
  • Prove that if $n$ is a congruent number then there exists a rational number $x$ such that $x-n$ , $x$ , and $x+n$ are all perfect squares of rational numbers.

    [[Hint: Let $X<Y<Z$ be sides of a rational right triangle with area $n$ . Let $x = (Z/2)^2$ .]]

  • Suppose $n$ is an integer and there exists a rational number $x$ such that $n-x$ , $n$ , and $n+x$ are all perfect squares. Prove that $n$ is a congruent number. [[Hint: from Koblitz's book (see page 4 of scan on website). Let $X = \sqrt{x+n} - \sqrt{x-n}$ , $Y = \sqrt{x+n} + \sqrt{x-n}$ , $Z = 2\sqrt{x}$ . Then $X,Y,Z$ are the sides of a right triangle and are all rational, and the triangle has area $n$ .]]

  • Think about problems with a similar feel:
    1. Which integers are the area of a square with rational side lengths?
    2. Which integers are the perimeter of a square with rational side lengths?
    3. Which integers are the area of a rectangle with rational side lengths?
  • Let $n$ be a rational number. Prove that there is a bijection between
    $$A = \left\{(a,b,c) \in \mathbb{Q}^3 : \frac{ab}{2} = n, a^2 + b^2 = c^2\right\}$$
    and
    $$B = \left\{(x,y) \in \mathbb{Q}^2 : y^2 = x^3 - n^2 x, \text{\rm with } y \neq 0\right\}$$
    given explicitly by the maps
    $$f(a,b,c) = \left(-\frac{nb}{a+c}, \frac{2n^2}{a+c}\right)$$
    and
    $$g(x,y) = \left(\frac{n^2-x^2}{y}, -\frac{2xn}{y}, \frac{n^2+x^2}{y}\right).$$

• RESEARCH:
  • Try to say something about which integers are the perimeter of a right triangle with rational side lengths. Try to convert this question to a question about solutions to equations.
  • Be creative and come up with similar problems, e.g., which integers are the area of an equilateral triangle with rational side lengths, etc., some very hard, and convert each to a diophantine equation.
  • Go to

    http://www.msri.org/publications/ln/msri/2000/introant/
    yui/1/index.html
    

    and read about another difficult generalization of the congruent number problem (this was a talk at a serious professional research conference, so don't be annoyed if you don't understand much of it).

For fun - the above are a bunch of elliptic curves, along with their ``Cremona labels''.