Semistable types of hyperelliptic curves

Celine Maistret, Tim Dokchitser, Vladimir Dokchitser, Adam Morgan

Research output: Chapter in Book/Report/Conference proceedingChapter in a book

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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.
Original languageEnglish
Title of host publicationAlgebraic Curves and Their Applications
EditorsLubjana Beshaj, Tony Shaska
PublisherAmerican Mathematical Society
Number of pages64
ISBN (Electronic)9781470451530
ISBN (Print)9781470442477
Publication statusPublished - 22 Jan 2019

Publication series

NameContemporary Mathematics

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