Divisors of Modular Parametrizations of Elliptic Curves

The modularity theorem implies that for every elliptic curve \(E /\mathbb{Q}\) 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 \(Y(z)\) where \(X(z)\) and \(Y(z)\) satisfy the Weierstrass equation for \(E\) as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the \(q\) - expansions of these parametrizations at any cusp. %These functions are algebraic over \(\mathbb{Q}(j(z))\) 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 parametrization 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.