2025/09/18 by Wang, SanMin
#11F03 #14H05 #FOS: Mathematics #Number Theory (math.NT) #Primary 11G05 #Secondary 11Y16
paper · doi:10.48550/arxiv.2509.14747
Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) ∈ ℚ \). We construct a class of modular polynomials FN(x,j) that relate the modular function x on X0(N) to the j-invariant j, where x is obtained by composing the first coordinate function of E with the modular parametrization φ: X0(N) → E. Using FN(x,j), we can precisely determine the poles of φ, compute exact values of φ at cusps, and develop an algorithm for calculating ramification points of φ. Moreover, FN(x,j) yields an efficient algorithm for computing the fibres of φ over arbitrary points on E. In some sense, FN(x,j) also provides a ``total" formula for computing the minimal polynomial of the images of Heegner points on X0(N) under φ. Especially, we compute the semi-trace of the image φ([\frac - 1 + √ - 3 2]) of the CM-point [(-1 + √(-3))/(2)] on X0(389), under the action of a 65-element subgroup of the 260-element Galois group of ℚ(√(-3), j(389 ⋅ (-1 + √(-3))/(2))). Finally, we associate a point of infinite order in~\( E(ℚ) \) with an infinite sequence~\ (j(τn), j(Nτn)) \n ∈ ℤ+ of algebraic numbers whose degrees are bounded by the degree of~φ. This provides one seemingly practicable approach to addressing the BSD conjecture.