Unirational algebraic groups and tame ramification
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910149087592448 |
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| author | Overkamp, Otto Vanni, Ismaele |
| author_facet | Overkamp, Otto Vanni, Ismaele |
| contents | Let $\mathcal{O}_K$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k.$ Let $G$ be a smooth connected commutative algebraic group over $K$ which does not contain a copy of $\mathbf{G}_{\mathrm{a}}.$ For each $d$ prime to $p:=\mathrm{char}\, k,$ let $K(d)$ be the unique extension of $K$ of degree $d.$ We investigate how the Néron lft-model of $G$ behaves under base change to the ring of integers $\mathcal{O}_{K(d)}.$ Information about this behaviour is encoded in the "jumps" of Edixhoven's filtration on the special fibre of the Néron lft-model of $G,$ as well as in Halle-Nicaise's motivic zeta function of $G.$ If $G$ is unirational (e. g. an algebraic torus), we show that the jumps of $G$ are rational numbers and that the motivic zeta function of $G$ is a rational function. We also deduce analogous results for Abelian varieties with potentially totally multiplicative reduction. This answers a question of Halle-Nicaise and partially one of Edixhoven. Along the way, we answer a question of Oesterlé about the structure of unipotent algebraic groups over function fields in positive characteristic. Under stronger conditions on $G,$ we obtain rationality of jumps even for separably closed but imperfect $k.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18436 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unirational algebraic groups and tame ramification Overkamp, Otto Vanni, Ismaele Algebraic Geometry Number Theory 14L15, 14G10, 11G35 Let $\mathcal{O}_K$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k.$ Let $G$ be a smooth connected commutative algebraic group over $K$ which does not contain a copy of $\mathbf{G}_{\mathrm{a}}.$ For each $d$ prime to $p:=\mathrm{char}\, k,$ let $K(d)$ be the unique extension of $K$ of degree $d.$ We investigate how the Néron lft-model of $G$ behaves under base change to the ring of integers $\mathcal{O}_{K(d)}.$ Information about this behaviour is encoded in the "jumps" of Edixhoven's filtration on the special fibre of the Néron lft-model of $G,$ as well as in Halle-Nicaise's motivic zeta function of $G.$ If $G$ is unirational (e. g. an algebraic torus), we show that the jumps of $G$ are rational numbers and that the motivic zeta function of $G$ is a rational function. We also deduce analogous results for Abelian varieties with potentially totally multiplicative reduction. This answers a question of Halle-Nicaise and partially one of Edixhoven. Along the way, we answer a question of Oesterlé about the structure of unipotent algebraic groups over function fields in positive characteristic. Under stronger conditions on $G,$ we obtain rationality of jumps even for separably closed but imperfect $k.$ |
| title | Unirational algebraic groups and tame ramification |
| topic | Algebraic Geometry Number Theory 14L15, 14G10, 11G35 |
| url | https://arxiv.org/abs/2604.18436 |