Algorithms for Mumford curves. (May 2015)

• Record Type: Journal Article
• Authors: Morrison, Ralph; Ren, Qingchun
• Is Part Of: Journal of symbolic computation. Volume 68:Part 2(2015)
• Page Start: 259
• Page End: 284
• Publication Date: 2015-05
• Subjects: Mumford curve -- Tropical geometry; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2014.09.013
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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.