Non-archimedean hyperbolicity and applications

Abstract

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

Urata's theorem in the logarithmic case and applications to integral points

“…We follow [20, 22 29] and say that a variety $$V$$ over $$k$$ is groupless over $$k$$ if, for every finite type connected group scheme $$G$$ over $$k$$, every morphism $$G\rightarrow V$$ is constant. Note that Lang refers to such varieties as being ‘algebraically hyperbolic’; see [37].…”

“…Indeed, if $$X$$ is a projective arithmetically hyperbolic variety over $$k$$, then every subvariety of $$X$$ is of general type by this conjecture. Then, it follows that every subvariety of $$X_L$$ is of general type (see, for instance, [29] for a proof of this well‐known fact), so that (again by Lang–Vojta's conjecture), the variety $$X_L$$ is arithmetically hyperbolic over $$L$$.…”

Construction of Arithmetic Teichmuller Spaces and some applications

“…This assumption will be in force throughout the rest of this section and let X/L be a geometrically connected, smooth projective variety over L. § 6.3 Let E ⊃ Q be any complete valued field, which is either an archimedean or a nonarchimedean with a rank one valuation inducing a p-adic valuation on Q, and E be its completed algebraic closure. I will say that X/E is a hyperbolic variety if the analytic space X an / E, is a (Brody) hyperbolic variety (see [Lang, 1986] for the archimedean case, [Javanpeykar and Vezzani, 2018] for the non-archimedean case). § 6.4 If dim(X) = 1 then X/E is hyperbolic (in the above sense) if and only if X × E C is a hyperbolic Riemann surface.…”

A non-Archimedean analogue of Campana's notion of specialness

“…Since understanding the arithmetic properties of such varieties is difficult, we seek other ways to describe being of general type and special. There exist (conjectural) complex analytic characterizations of these notions (see, for example, [Lan86,Kob98] and [Cam04]), and recently, there has been work on providing a (conjectural) non-Archimedean characterization of general type (see, for example, [Che94,Che96,JV21,Mor21,Sun20]).…”