Curve equations from expansions of 1-forms at a nonrational point

TL;DR

This paper presents an algorithm to derive algebraic curve equations from power series expansions of regular 1-forms at nonrational points, extending previous methods and applicable to hyperelliptic and nonhyperelliptic curves.

Contribution

It introduces a new algorithm for computing algebraic curve equations from local expansions at nonrational points, generalizing prior work focused on rational points.

Findings

Successfully computed equations for hyperelliptic modular curves over $\\mathbb{Q}$.

Provides explicit models as double covers of conics or projective lines.

Extends algorithmic techniques to nonrational points on algebraic curves.

Abstract

We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, Gonz\'alez-Jim\'enez, Gonz\'alez, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database.