Counting two-step nilpotent wildly ramified extensions of function fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910843960033280 |
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| author | Gundlach, Fabian Seguin, Béranger |
| author_facet | Gundlach, Fabian Seguin, Béranger |
| contents | We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of nilpotency class at most $2$. Rather than the discriminant, we count extensions according to an invariant describing the last jump in the ramification filtration at each place. We prove a local-global principle relating the distribution of extensions over global function fields to their distribution over local fields, leading to an asymptotic formula for the number of extensions with a given global last-jump invariant. A key ingredient is Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize extensions and obtain bounds on the ramification of local extensions by estimating the number of solutions to certain polynomial equations over finite fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting two-step nilpotent wildly ramified extensions of function fields Gundlach, Fabian Seguin, Béranger Number Theory 11R45, 11N45, 11S15, 14G17 We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of nilpotency class at most $2$. Rather than the discriminant, we count extensions according to an invariant describing the last jump in the ramification filtration at each place. We prove a local-global principle relating the distribution of extensions over global function fields to their distribution over local fields, leading to an asymptotic formula for the number of extensions with a given global last-jump invariant. A key ingredient is Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize extensions and obtain bounds on the ramification of local extensions by estimating the number of solutions to certain polynomial equations over finite fields. |
| title | Counting two-step nilpotent wildly ramified extensions of function fields |
| topic | Number Theory 11R45, 11N45, 11S15, 14G17 |
| url | https://arxiv.org/abs/2502.18207 |