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Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics

Elvira Lupoian, James Rawson

TL;DR

This work develops an explicit algorithm to compute the $3$-torsion subgroup $J[3]$ of the Jacobian of a smooth plane quartic genus $3$ curve with a rational point by relating $3$-torsion to cubics intersecting the curve with triple multiplicity, and then reconstructs exact algebraic expressions from high-precision numeric approximations. The authors solve a high-degree zero-dimensional system (degree $728$) in two steps: first via homotopy continuation and Newton-Raphson to obtain complex approximations, then via lattice reduction or continued fractions to recover minimal polynomials and exact coefficients. They extend these torsion results to compute local Galois representations and wild conductor exponents, especially at $p=2$, and illustrate with Fermat and Klein quartics as well as generic examples. The paper provides practical methods and code for using genus 3 torsion data to determine conductors, enabling broader applications to arithmetic questions on plane quartics and their Jacobians.

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 $p=2$.

Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics

TL;DR

This work develops an explicit algorithm to compute the -torsion subgroup of the Jacobian of a smooth plane quartic genus curve with a rational point by relating -torsion to cubics intersecting the curve with triple multiplicity, and then reconstructs exact algebraic expressions from high-precision numeric approximations. The authors solve a high-degree zero-dimensional system (degree ) in two steps: first via homotopy continuation and Newton-Raphson to obtain complex approximations, then via lattice reduction or continued fractions to recover minimal polynomials and exact coefficients. They extend these torsion results to compute local Galois representations and wild conductor exponents, especially at , and illustrate with Fermat and Klein quartics as well as generic examples. The paper provides practical methods and code for using genus 3 torsion data to determine conductors, enabling broader applications to arithmetic questions on plane quartics and their Jacobians.

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 .
Paper Structure (24 sections, 3 theorems, 5 equations, 1 table)

This paper contains 24 sections, 3 theorems, 5 equations, 1 table.

Table of Contents

  1. Introduction
  2. Three-Torsion Points and Sections of $K_{C}$
  3. Computing Three Torsion Points
  4. Schemes of $3$-torsion points
  5. Complex Approximations
  6. Lattice Reduction and Precise Expressions
  7. Precise Expressions and Continued Fractions
  8. Examples
  9. The $3$-torsion Subgroup of the Fermat Quartic
  10. The $3$-torsion subgroup of the Klein Quartic
  11. Generic Examples
  12. Local Galois Representations at $3$
  13. Splitting Field and Galois Group
  14. Galois groups and $\text{GSp}_6(\mathbb{F}_{3})$
  15. Inertia and Ramification Groups
  16. ...and 9 more sections

Key Result

Theorem 1

Let $C$ be as above. Three-torsion points of the Jacobian of $C$ correspond to cubics of the form which intersect $C$ with multiplicity divisible by $3$ at each point.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2
  • proof
  • Proposition 3
  • ...and 3 more