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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 $.

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

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

Primes Dividing Invariants of CM Picard Curves

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2 results

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

Primes Dividing Invariants of CM Picard Curves