A rigid analytic proof that the Abel-Jacobi map extends to compact-type models

Abstract

Let $K$ be a non-Archimedean valued field with valuation ring $R$. Let $C_{\eta }$ be a $K$-curve with compact type reduction, so its Jacobian $J_{\eta }$ extends to an abelian $R$-scheme $J$. We prove that an Abel-Jacobi map $\iota :C_{\eta }\to J_{\eta }$ extends to a morphism $C\to J$, where $C$ is a compact-type $R$-model of $J$, and we show this is a closed immersion when the special fiber of $C$ has no rational components. To do so, we apply a rigid-analytic "fiberwise" criterion for a finite morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_{\eta }$.

6 pages, comments welcome