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

2025/06/16 by van Bommel, Raymond, Costa, Edgar, Poonen, Bjorn +1
#14G35 (Secondary) #14Q05 (Primary) 11G18 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.14026

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 ℚ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database.

Citations

• Modular forms invariant under non-split Cartan subgroups
• Finiteness results for modular curves of genus at least 2

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• Constructing curves over finite fields with many points by solving linear equations