Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911201975336960 |
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| author | Lupoian, Elvira Rawson, James |
| author_facet | Lupoian, Elvira Rawson, James |
| contents | 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 $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13577 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics Lupoian, Elvira Rawson, James Number Theory 11G30, 11G20 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 $p=2$. |
| title | Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics |
| topic | Number Theory 11G30, 11G20 |
| url | https://arxiv.org/abs/2401.13577 |