Abstract by Jonathan Hales
Preimage of rational points on elliptical curves
The modularity theorem gives that for every elliptic curve E, there exists a rational map from the modular curve X(N) to E, where N is the conductor of E. This map may be expressed in terms of two modular functions X(z) and Y(z) (derived from the Weierstrass P function and its derivative) that satisfy the equation for E, as well as a certain differential equation. Using these two relations, a recursive algorithm can be constructed to calculate the q-expansions of these parameterizations at any cusp. These functions are algebraic over the rationals adjoined with j(z). Using these functions, we determine the divisor of the parameterization and the preimage of rational points on E.