Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics
arXiv:2401.13577 · doi:10.1007/s40993-025-00672-4
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 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 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 -torsion subgroup can be used to compute local wild conductor exponents, including at .
Major changes: change of title, re-organised structure and added implementation. Comments are welcome! v4: Version appearing in Research in Number Theory