Ms Amy Batsiokis

Honours thesis

The Congruent Number Probem

The congruent number problem was first discussed in the tenth century by Arabic scholars and it is still unsolved today. At present, we have an almost complete solution to the problem (but it relies on the weak form of the Birch and Swinnerton- Dyer conjecture [in Chapter 8]) being proven true. This thesis begins with some elementary work on pythagorean triples, which is followed by two equivalent definitions of congruent numbers. We then look at elliptic curves which lead to yet another alternate condition for a number to be congruent. This is followed by a look at some objects which are related to elliptic curves: height pairing, the L-function, and the Shafarevich-Tate group. Then everything is tied together in the last chapter which is about the Birch and Swinnerton-Dyer conjecture, which has not been proved yet, and Tunnel’s Theorem which will provide a complete answer to the congruent number problem when the weak form of the Birch and Swinnerton-Dyer conjecture has been proven.