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

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, González-Jiménez, González, 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.