Wild Galois representations: elliptic curves with wild cyclic reduction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911021961052160 |
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| author | Coppola, Nirvana |
| author_facet | Coppola, Nirvana |
| contents | In 1990, Kraus classified all possible inertia images of the $\ell$-adic Galois representation attached to an elliptic curve over a non-archimedean local field. In previous work, the author computed explicitly the Galois representation of elliptic curves having non-abelian inertia image, a phenomenon which only occurs when the residue characteristic of the field of definition is $2$ or $3$ and the curve attains good reduction over some non-abelian ramified extension. In this work, the computation of the Galois representation in all the remaining wild cases, i.e. when the residue characteristic is $p=2$ or $3$ and the curve attains good reduction over an extension whose ramification degree is divisible by $p$ (without assuming the condition on the image of inertia being non-abelian), is completed. This is based on Chapter V of the author's PhD thesis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wild Galois representations: elliptic curves with wild cyclic reduction Coppola, Nirvana Number Theory 11G07, 11F80 In 1990, Kraus classified all possible inertia images of the $\ell$-adic Galois representation attached to an elliptic curve over a non-archimedean local field. In previous work, the author computed explicitly the Galois representation of elliptic curves having non-abelian inertia image, a phenomenon which only occurs when the residue characteristic of the field of definition is $2$ or $3$ and the curve attains good reduction over some non-abelian ramified extension. In this work, the computation of the Galois representation in all the remaining wild cases, i.e. when the residue characteristic is $p=2$ or $3$ and the curve attains good reduction over an extension whose ramification degree is divisible by $p$ (without assuming the condition on the image of inertia being non-abelian), is completed. This is based on Chapter V of the author's PhD thesis. |
| title | Wild Galois representations: elliptic curves with wild cyclic reduction |
| topic | Number Theory 11G07, 11F80 |
| url | https://arxiv.org/abs/2506.20562 |