  We study hyperbolicity properties of the moduli space of polarized abelian
varieties (also known as the Siegel modular variety) in characteristic $p$. Our
method uses the plethysm operation for Schur functors as a key ingredient and
requires a new positivity notion for vector bundles in characteristic $p$
called $(\varphi,D)$-ampleness. Generalizing what was known for the Hodge line
bundle, we also show that many automorphic vector bundles on the Siegel modular
variety are $(\varphi,D)$-ample.
