Nowell, Sarah Catherine; (2022) Models of hyperelliptic curves over p-adic fields. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Let C : y² = f(x) be a hyperelliptic curve, with tame potentially semistable reduction, over a local field with algebraically closed residue field. The p-adic distances between the roots of f(x) can be described by a purely combinatorial object known as a cluster picture. We show that the cluster picture of C, along with the leading coefficient of f, completely determines the dual graph of the special fibre of the minimal strict normal crossings (SNC) model of C. In particular, we give an explicit description of the special fibre in terms of this data. Further to this, we define open quotient BY trees, showing there is a one-to-one correspondence between these and cluster pictures of hyperelliptic curves with tame reduction. Using these trees we introduce a way of classifying reduction types of hyperelliptic curves. As a demonstration of our results we give a complete classification in genus 2 using cluster pictures and open quotient BY trees.

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