Universal Families and Ruling out Small Primes

Bryden Cais

These are notes for a talk given in the 2003-2004 VIGRE Number Theory Working Group. The goal of that seminar was to understand Mazur's fundamental paper Modular Curves and the Eisenstein Ideal.

The purpose of this talk is to give the general flavor of the background (i.e. pre Mazur) material on the possibilities for torsion subgroups of elliptic curves over Q. To that end, there are three sections: in the first, we show how to write down a one parameter family of elliptic curves defined over Q having a rational point of of order N for N between 3 and 10 and N=12. We then give some motivation as to why it is these values of N that occur and why A Ogg. conjectured that these are the only possible values. In the second section, we will show that no elliptic curve over Q can have a rational point of order 11. We do this by showing that the modular curve X_1(11), which classifies pairs (E, P) of elliptic curves E and points P of E having order 11, has no rational points other than cusps. In the third section, we tackle the same problem for points of order 35. This is substantially more subtle than the previous case as the genus of X_1(35) is large.