Algorithms for Mumford curves. (May 2015) - Morrison, Ralph; Ren, Qingchun
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.