Algorithms for Mumford curves

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 nonarchimedean 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.