Non-archimedean hyperbolicity of the moduli space of curves

Abstract

Let K be a complete algebraically closed non-archimedean valued field of characteristic zero, and let X be a finite type scheme over K. We say X is K-analytically Borel hyperbolic if, for every finite type reduced scheme S over K, every rigid analytic morphism from the rigid analytification S an of S to the rigid analytification X an of X is algebraic. Using the Viehweg-Zuo construction and the K-analytic big Picard theorem of Cherry-Ru, we show that, for N ≥ 3 and g ≥ 2, the fine moduli space M[N] g,K over K of genus g curves with level N -structure is K-analytically Borel hyperbolic.

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 e.g., [Lan86,Kob98] and [Cam04]), and recently, there has been work on providing a (conjectural) non-Archimedean characterization of general type (see e.g., [Che94,Che96,JV18,Mor21,Sun20]).…”