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 $S^{\mathrm{an}}$ of $S$ to the rigid analytification $X^{\mathrm{an}}$ of $X$ is algebraic. Using the Viehweg-Zuo construction and the $K$-analytic big Picard theorem of Cherry-Ru, we show that, for $N\ge 3$ and $g\ge 2$, the fine moduli space ${\mathcal{M}}_{g,K}^{[N]}$ over $K$ of genus $g$ curves with level $N$-structure is $K$-analytically Borel hyperbolic.

10 pages, comments welcome