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arXiv · 2506.14026

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

Raymond van Bommel·Edgar Costa·Bjorn Poonen·Padmavathi Srinivasan
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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.

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Raymond van Bommel, Edgar Costa, Bjorn Poonen, Padmavathi Srinivasan. 2025-06-16. Curve equations from expansions of 1-forms at a nonrational point. https://arxiv.org/abs/2506.14026

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