Semistable types of hyperelliptic curves
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866915721010741248 |
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| author | Dokchitser, Tim Dokchitser, Vladimir Maistret, Celine Morgan, Adam |
| author_facet | Dokchitser, Tim Dokchitser, Vladimir Maistret, Celine Morgan, Adam |
| contents | In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand $L$-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1704_08338 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Semistable types of hyperelliptic curves Dokchitser, Tim Dokchitser, Vladimir Maistret, Celine Morgan, Adam Number Theory Algebraic Geometry Primary 11G20, secondary 14H45, 05C22 In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand $L$-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians. |
| title | Semistable types of hyperelliptic curves |
| topic | Number Theory Algebraic Geometry Primary 11G20, secondary 14H45, 05C22 |
| url | https://arxiv.org/abs/1704.08338 |