On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914234353319936 |
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| author | Ishii, Shun |
| author_facet | Ishii, Shun |
| contents | In this paper, we compare a certain field arising from the pro-$p$ outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field $K$ with the maximal pro-$p$ Galois extension of the mod-$p$ ray class field $K(p)$ of $K$ unramified outside $p$. We prove that these two fields coincide for every prime $p$ which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro-$p$ outer Galois representation associated to the projective line minus three points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves Ishii, Shun Number Theory 11G05, 11G15, 11R32, 12E30 In this paper, we compare a certain field arising from the pro-$p$ outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field $K$ with the maximal pro-$p$ Galois extension of the mod-$p$ ray class field $K(p)$ of $K$ unramified outside $p$. We prove that these two fields coincide for every prime $p$ which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro-$p$ outer Galois representation associated to the projective line minus three points. |
| title | On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves |
| topic | Number Theory 11G05, 11G15, 11R32, 12E30 |
| url | https://arxiv.org/abs/2312.04196 |