Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations

Francesco Baldassarri; Velibor Bojković

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arXiv · 1901.07644

Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations

Francesco Baldassarri·Velibor Bojković

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Abstract

We consider a finite étale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $Γ_f=(Γ_Y,Γ_X)$ of the morphism $f$. We prove that $Γ_f$ radializes $f$ if and only if $Γ_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\mathcal{O}_Y,d_Y)$. Furthermore, when $f$ is a finite étale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depends on the radius of the point $x$.

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BibTeX RIS

Francesco Baldassarri, Velibor Bojković. 2019-01-22. Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations. https://arxiv.org/abs/1901.07644

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