2023/10/09 by Oswal, Abhishek, Shankar, Ananth N., Zhu, Xinwen +1
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2310.06104
We prove that Shimura varieties of abelian type satisfy a p-adic Borel-extension property over discretely valued fields. More precisely, let D denote the rigid-analytic closed unit disc and D× = D ∖ \0\, let X be a smooth rigid-analytic variety, and let S(G,H)K denote a Shimura variety of abelian type with torsion-free level structure. We prove every rigid-analytic map defined over a discretely valued p-adic field D× × X → S(G,H)K^\textrman extends to an analytic map D × X → (S(G,H)K^\textrmBB)^\textrman, where S(G,H)K^\textrmBB is the Baily-Borel compactification of S(G,H)K. We also deduce various applications to algebraicity of analytic maps, degenerations of families of abeloids, and to p-adic notions of hyperbolicity. Along the way, we also prove an extension result for Rapoport-Zink spaces.