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arXiv · 2401.13577

Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics

Elvira Lupoian·James Rawson
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Abstract

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve with a marked rational point. We describe $3-$torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree $728$ scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the $3$-torsion subgroup can be used to compute local wild conductor exponents, including at $p=2$.

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Elvira Lupoian, James Rawson. 2024-01-24. Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics. https://doi.org/10.1007/s40993-025-00672-4

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