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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?

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2 Answers 2

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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.

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You can also find an interpretation using integrals, this time multiplicative, and mesures, in the preprint "The Abel Jacobi map for Mumford Curves via Integration": https://arxiv.org/abs/1609.09285 which generalizes the construction given by Darmon and Longhi in the local case.

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