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A rigid analytic proof that the Abel–Jacobi map extends to compact-type models
Published online by Cambridge University Press: 09 January 2024
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 \colon 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 morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_\eta $.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
Dupuy was supported by the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013)/ERC Grant agreement no. 291111/MODAG. Rabinoff was supported by NSF DMS-1601842.