On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves

Shun Ishii

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arXiv · 2312.04196

On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves

Shun Ishii

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Abstract

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.

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Shun Ishii. 2023-12-07. On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves. https://arxiv.org/abs/2312.04196

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