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arXiv · 2607.02160

Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

Enis Kaya·Marc Masdeu·J. Steffen Müller·Marius van der Put
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Abstract

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 $p\neq 2$, 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.

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Enis Kaya, Marc Masdeu, J. Steffen Müller, Marius van der Put. 2026-07-02. Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights. https://arxiv.org/abs/2607.02160

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