Parametrizing elliptic curves by modular units (Q2788668)

An elliptic curve \(E/\mathbb{Q}\) of conductor \(N\) is said to be parametrized by modular units if there exist two modular units \(u,v\in \mathcal{O}(Y_1(N))^*\) such that the function field \(\mathbb{Q}(E)\) is isomorphic to \(\mathbb{Q}(u,v)\). In this paper the author shows that only finitely many elliptic curves over \(\mathbb{Q}\) can be parametrized by modular units. His proof goes as follows. If \(E\) is parametrized by two modular units \(u,v\) on \(Y_1(N)\), then there exist a finite morphism \(\varphi:X_1(N)\rightarrow E\) and two rational functions \(f,g\in \mathbb{Q}(E)^*\) such that \(\varphi^*(f)=u\) and \(\varphi^*(g)=v\). Let \(E_1\) be the \(X_1(N)\)-optimal elliptic curve in the isogeny class of \(E\), and let \(\varphi_1:X_1(N)\rightarrow E_1\) be an optimal parametrization. He shows that, if \(N\) is sufficiently large, then \(\varphi^*(\mathbb{Q}(E_1))\cap \mathcal(O)(Y_1(N))=\mathbb{Q}\), employing a lower bound due to \textit{M. Watkins} [``Explicit lower bounds on the modular degree of an elliptic curve'', Preprint \url{arXiv:math/0408126}] on the modular degree of elliptic curves.