Non-archimedean hyperbolicity of the moduli space of curves

Publication date: 28 September 2020

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^{m a t h r m a n}

of

S

to the rigid analytification

X^{m a t h r m a n}

of

X

is algebraic. Using the Viehweg-Zuo construction and the

K

-analytic big Picard theorem of Cherry-Ru, we show that, for

N g e q 3

and

g g e q 2

, the fine moduli space

m a t h c a l M_{g, K}^{[N]}

over

K

of genus

g

curves with level

N

-structure is

K

-analytically Borel hyperbolic.

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