Local Kummer theory for Drinfeld modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914677579055104 |
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| author | Mornev, Maxim Pink, Richard |
| author_facet | Mornev, Maxim Pink, Richard |
| contents | Let $ϕ$ be a Drinfeld $A$-module of finite residual characteristic $\bar{\mathfrak{p}}$ over a local field $K$. We study the action of the inertia group of $K$ on a modified adelic Tate module $\smash{T^\circ_{\text{ad}}}(ϕ)$ which differs from the usual adelic Tate module only at the $\bar{\mathfrak{p}}$-primary component. After replacing $K$ by a finite extension we can assume that $ϕ$ is the analytic quotient of a Drinfeld module $ψ$ of good reduction by a lattice $M\subset K$. The image of inertia acting on $T^\circ_{\text{ad}}(ϕ)$ is then naturally a subgroup of $\operatorname{Hom}_A(M,T^\circ_\text{ad}(ψ))$.
This subgroup is described by a canonical local Kummer pairing that we study extensively in this article. In particular we give an effective formula for the image of inertia up to finite index, and obtain a necessary and sufficient condition for this image to be open. We also determine the image of the ramification filtration |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08254 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Kummer theory for Drinfeld modules Mornev, Maxim Pink, Richard Number Theory 11G09, 11F80 Let $ϕ$ be a Drinfeld $A$-module of finite residual characteristic $\bar{\mathfrak{p}}$ over a local field $K$. We study the action of the inertia group of $K$ on a modified adelic Tate module $\smash{T^\circ_{\text{ad}}}(ϕ)$ which differs from the usual adelic Tate module only at the $\bar{\mathfrak{p}}$-primary component. After replacing $K$ by a finite extension we can assume that $ϕ$ is the analytic quotient of a Drinfeld module $ψ$ of good reduction by a lattice $M\subset K$. The image of inertia acting on $T^\circ_{\text{ad}}(ϕ)$ is then naturally a subgroup of $\operatorname{Hom}_A(M,T^\circ_\text{ad}(ψ))$. This subgroup is described by a canonical local Kummer pairing that we study extensively in this article. In particular we give an effective formula for the image of inertia up to finite index, and obtain a necessary and sufficient condition for this image to be open. We also determine the image of the ramification filtration |
| title | Local Kummer theory for Drinfeld modules |
| topic | Number Theory 11G09, 11F80 |
| url | https://arxiv.org/abs/2402.08254 |