Abstract by Jonathan Hales
Congruences of Modular Parameterizations
The modularity theorem gives that for every elliptic curve E/Q, there exists a rational map from the modular curve X0(N) to E where N is the conductor of E. This map may be expressed in terms of two modular functions X(tau) and Y(tau) (derived from the Weierstrass P-function and its derivative) where X(tau) and Y(tau) satisfy the equation for E. We examine interesting congruences between the Q-algebras generated by X(tau) and Y(tau). We also calculate the divisors of the modular functions X(tau) and Y(tau) and the preimages of rational points on E. The theory of complex multiplication gives that the traces of Heegnar-points are rational points on quadratic twists on the elliptic curve. Gross and Zagier showed that when the rank of E is one, the point constructed in this way has infinite order. We are interested in studying the preimages of rational points which do not arise from compex multiplication.