Computations with hyperelliptic Mumford Curves: Uniformization, $p$-adic integrals, $p$-adic heights

Mumford curves generalize the Tate uniformization of elliptic curves and provide $p$-adic analogues of the uniformization of Riemann surfaces. In this talk, we present algorithms for hyperelliptic Mumford curves. More precisely, for a given hyperelliptic Mumford curve $X$ over the field of $p$-adic numbers for some $p\neq 2$, we describe how to compute a Schottky group $W$ that uniformizes $X$, and using this uniformization, we explain how to compute $p$-adic Abelian integrals and $p$-adic Schneider heights on $X$. We illustrate these algorithms with numerical examples computed in SageMath. This talk is based on a joint work with Marc Masdeu and J. Steffen Müller.