Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$

Abstract

Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) \in \mathbb{Q} \). We construct a class of modular polynomials $F_N(x,j)$ that relate the modular function $x$ on $X_0(N)$ to the $j$-invariant $j$, where $x$ is obtained by composing the first coordinate function of $E$ with the modular parametrization $\varphi: X_0(N) \rightarrow E$. Using $F_N(x,j)$, we can precisely determine the poles of $\varphi$, compute exact values of $\varphi$ at cusps, and develop an algorithm for calculating ramification points of $\varphi$. Moreover, $F_N(x,j)$ yields an efficient algorithm for computing the fibres of $\varphi$ over arbitrary points on $E$. In some sense, $F_N(x,j)$ also provides a ``total" formula for computing the minimal polynomial of the images of Heegner points on $X_0(N)$ under $\varphi$. Especially, we compute the semi-trace of the image $\varphi ([\frac{{ - 1 + \sqrt { - 3} }}{2}])$ of the CM-point $[\frac{-1 + \sqrt{-3}}{2}]$ on $X_{0}(389)$, under the action of a 65-element subgroup of the 260-element Galois group of $\mathbb{Q}(\sqrt{-3}, j(389 \cdot \frac{-1 + \sqrt{-3}}{2}))$. Finally, we associate a point of infinite order in~\( E(\mathbb{Q}) \) with an infinite sequence~$\{ (j(\tau_n), j(N\tau_n)) \}_{n \in \mathbb{Z}^+} $ of algebraic numbers whose degrees are bounded by the degree of~$\varphi$. This provides one seemingly practicable approach to addressing the BSD conjecture.