Faraggi, Omri;
(2021)
Models of Curves over Local Fields.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
Cluster pictures are a recent innovation which have been developed to study the arithmetic of hyperelliptic curves. The cluster picture of such a curve $C:y^2=f(x)$ over a local field $K$ is a completely combinatorial object containing the data of the $p$-adic distances between the roots of $f$. It determines many invariants associated to $C$, and most pertinently for us it was used in \cite{DDMM18} to calculate the minimal regular model of $C$ when it has semistable reduction. We extend their results to the case where $C$ has tame reduction, calculating its minimal snc model in terms of its cluster picture. As an application we state a condition in terms of the cluster picture for $C$ to have a $K$-rational point. In addition we use a generalisation of the cluster picture, the chromatic cluster picture, to work out the minimum regular model of a bihyperelliptic curve with semistable reduction.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Models of Curves over Local Fields |
| Event: | UCL (University College London) |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10140699 |
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