Split degenerate superelliptic curves and $\ell$-adic images of inertia
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908316518580224 |
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| author | Yelton, Jeffrey |
| author_facet | Yelton, Jeffrey |
| contents | Let $K$ be a field with a discrete valuation, and let $p$ and $\ell$ be (possibly equal) primes which are not necessarily different from the residue characteristic. Given a superelliptic curve $C : y^p = f(x)$ which has split degenerate reduction over $K$, with Jacobian denoted by $J / K$, we describe the action of an element of the inertia group $I_K$ on the $\ell$-adic Tate module $T_\ell(J)$ as a product of powers of certain transvections with respect to the $\ell$-adic Weil pairing and the canonical principal polarization of $J$. The powers to which the transvections are taken are given by a formula depending entirely on the cluster data of the roots of the defining polynomial $f$. This result is demonstrated using Mumford's non-archimedean uniformization of the curve $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Split degenerate superelliptic curves and $\ell$-adic images of inertia Yelton, Jeffrey Number Theory 11G10, 11G20, 14K25, 11G07 Let $K$ be a field with a discrete valuation, and let $p$ and $\ell$ be (possibly equal) primes which are not necessarily different from the residue characteristic. Given a superelliptic curve $C : y^p = f(x)$ which has split degenerate reduction over $K$, with Jacobian denoted by $J / K$, we describe the action of an element of the inertia group $I_K$ on the $\ell$-adic Tate module $T_\ell(J)$ as a product of powers of certain transvections with respect to the $\ell$-adic Weil pairing and the canonical principal polarization of $J$. The powers to which the transvections are taken are given by a formula depending entirely on the cluster data of the roots of the defining polynomial $f$. This result is demonstrated using Mumford's non-archimedean uniformization of the curve $C$. |
| title | Split degenerate superelliptic curves and $\ell$-adic images of inertia |
| topic | Number Theory 11G10, 11G20, 14K25, 11G07 |
| url | https://arxiv.org/abs/2504.00205 |