ALGORITHMS FOR HYPERELLIPTIC MUMFORD CURVES:
p-ADIC UNIFORMIZATION, p-ADIC INTEGRALS AND p-ADIC HEIGHTS

ENIS KAYA, MARC MASDEU, J. STEFFEN MÜLLER, AND MARIUS VAN DER PUT

Mumford curves generalize the Tate uniformization of elliptic curves with split multiplicative reduction and provide p-adic analogues of the uniformization of Riemann surfaces. In this paper, we present several algorithms for hyperelliptic Mumford curves. For a given hyperelliptic Mumford curve X defined over a finite extension of the field of p-adic numbers for some p≠2, we first describe how to compute a p-adic Schottky group W that uniformizes X; this is based on our extension to Kadziela's approximation theorem. As applications, we explain how to use this uniformization in order to compute p-adic Abelian integrals and p-adic Schneider heights on X; the latter uses Werner's formula expressing the p-part of the Schneider height in terms of theta functions. We illustrate our algorithms with numerical examples computed using the computer algebra system SageMath.