Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL\_2(K)$, and let $X\_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid analytic space over $K$, but also a proper algebraic curve (due to Mumford's theorem). Let $J\_\Gamma$ be the Jacobian variety of degree 0 divisors on $X\_\Gamma$, and let $a: X\_\Gamma \to J\_\Gamma$ be the Abel-Jacobi map which sends a point $x$ to the class of the divisor $[x-x\_0]$ for some $x\_0 \in X\_\Gamma$ fixed in advance. 


It is known that $J\_\Gamma$ admits a uniformisation as the quotient $\mathbb{G}\_m^g/\Gamma'$ where $\Gamma'$ is the abelianisation $\Gamma/[\Gamma,\Gamma]$ of the group $\Gamma$, and $g$ is the genus of $X\_\Gamma$.


Is there a way to describe the Abel-Jacobi map $a$ in analytic terms, similarly to the description of this map over $\mathbb{C}$ in terms of Abelian integrals?



Abel-Jacobi map for Mumford curves analytically
This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262\_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put's book on Mumford curves.


