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On rigid-analytic Picard varieties. (English) Zbl 1044.14007
Let $$R$$ be a complete discrete valuation ring with field of fractions $$K$$, and let $$X$$ be a smooth, proper, connected rigid analytic space over $$K$$ which possesses a $$K$$-rational point. Under the hypothesis that $$X_K$$ has a strict semi-stable formal model over $$R$$, the authors construct a rigid analytic variety, $$\text{Pic}_{X/K}$$, representing the rigid analytic Picard functor of $$X$$, and the Poincaré line bundle on $$\text{Pic}_{X/K} \times X$$. They show that the connected component of $$\text{Pic}_{X/K}$$ is, after a finite base extension, an extension of a smooth abeloid variety (a smooth proper rigid group variety) by an affine torus. They also show that the rigid Néron-Severi group is finitely generated.

##### MSC:
 14G22 Rigid analytic geometry 14K30 Picard schemes, higher Jacobians 14C22 Picard groups
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