BYU

Abstract by Jonathan Hales

Personal Infomation


Presenter's Name

Jonathan Hales

Degree Level

Masters

Co-Authors

Michael Griffin

Abstract Infomation


Department

Mathematics

Faculty Advisor

Michael Griffin

Title

Modular Parameterizations of Elliptic Curves

Abstract

The modularity theorem implies that for every elliptic curve E /Q there exist rational maps from the modular curve X_0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y(z) that satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, we construct a recursive algorithm to calculate the q - expansions of these parameterizations at any cusp. These functions are algebraic over Q(j(z)) and satisfy modular polynomials where each of the coefficient functions are rational functions in j(z). Using these functions, we determine the divisor of the parameterization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X_0(N).