Two-Torsion Subgroups of some Modular Jacobians
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866915566839660544 |
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| author | Lupoian, Elvira |
| author_facet | Lupoian, Elvira |
| contents | We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the $2$-torsion subgroup. Using $p$-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton-Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the $2$-torsion points. We demonstrate the practicality of our method by computing the $2$-torsion of the modular Jacobians $J_{0}\left( N \right)$ for $N = 42, 55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg conjecture for these values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13017 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two-Torsion Subgroups of some Modular Jacobians Lupoian, Elvira Number Theory 11G30 We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the $2$-torsion subgroup. Using $p$-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton-Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the $2$-torsion points. We demonstrate the practicality of our method by computing the $2$-torsion of the modular Jacobians $J_{0}\left( N \right)$ for $N = 42, 55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg conjecture for these values. |
| title | Two-Torsion Subgroups of some Modular Jacobians |
| topic | Number Theory 11G30 |
| url | https://arxiv.org/abs/2205.13017 |