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, a recursive algorithm can be constructed 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). We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects.
College and Department
Physical and Mathematical Sciences
BYU ScholarsArchive Citation
Hales, Jonathan Reid, "Divisors of Modular Parameterizations of Elliptic Curves" (2020). Theses and Dissertations. 8472.
number theory, elliptic curves, modular forms