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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2102.09684 |
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| _version_ | 1866912377006456832 |
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| author | Adams, Ophelia |
| author_facet | Adams, Ophelia |
| contents | We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_09684 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Dynamical Analogue of Sen's Theorem Adams, Ophelia Number Theory Dynamical Systems 37P05 11S15 We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194). |
| title | A Dynamical Analogue of Sen's Theorem |
| topic | Number Theory Dynamical Systems 37P05 11S15 |
| url | https://arxiv.org/abs/2102.09684 |