Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics

2024/01/24 by Elvira Lupoian, Lupoian, Elvira, James Rawson +1
Engineering · Mathematics · Computer Science · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2401.13577

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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