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TitleAlgorithms for Mumford curves
Author(s) Ralph Morrison, Qingchun Ren
TypeArticle in Journal
AbstractAbstract Mumford showed that Schottky subgroups of PGL ( 2 , K ) give rise to certain curves, now called Mumford curves, over a non-archimedean field K. Such curves are foundational to subjects dealing with non-archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p-adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings. We also discuss specific methods and future work for hyperelliptic Mumford curves.
KeywordsMumford curve, Tropical geometry
URL http://www.sciencedirect.com/science/article/pii/S0747717114000819
JournalJournal of Symbolic Computation
Volume68, Part 2
Pages259 - 284
NoteEffective Methods in Algebraic Geometry
Translation No
Refereed No