Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic

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 $(\phi ,D)$-ampleness. Generalizing what was known for the Hodge line bundle, we also show that many automorphic vector bundles on the Siegel modular variety are $(\phi ,D)$-ample.