Monodromy Groups of Supersingular Abelian Surfaces over ${\mathbb{Q}}_p$

For primes $p\ge 7$, we give a parametrization of the filtered $\phi$-modules attached to the $p$-adic Tate modules of abelian surfaces over ${\mathbb{Q}}_p$ with supersingular good reduction. We use this classification to determine the neutral components of the monodromy groups of the associated $p$-adic representations up to ${\bar{\mathbb{Q}}}_p$-isomorphism. Furthermore, we analyze the $p$-adic distribution of these groups in the moduli space of filtered $\phi$-modules. In particular, we prove that the neutral components are generically isomorphic to ${\mathbf{G}\mathbf{L}}_2{\times }_{\det }{\mathbf{G}\mathbf{L}}_2$.