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

TL;DR

This paper classifies the monodromy groups of supersingular abelian surfaces over rac{ ext{Q}_p}{p} and shows they are generically isomorphic to a product of two rac{ ext{GL}_2}{ ext{GL}_1} groups, revealing their distribution in moduli space.

Contribution

It provides a parametrization of filtered rac{ ext{ ext{Q}_p}}{ ext{Q}_p} modules for supersingular abelian surfaces and determines the structure of their monodromy groups.

Findings

Neutral components are generically isomorphic to rac{ ext{ ext{GL}_2} imes ext{ ext{GL}_2}}{ ext{ ext{det}}}

Classification of filtered rac{ ext{ ext{Q}_p}}{ ext{ ext{Q}_p}}-modules for supersingular abelian surfaces

Distribution of monodromy groups in the moduli space analyzed

Abstract

For primes $p\ge 7$, we give a parametrization of the filtered $\varphi$-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 $\varphi$-modules. In particular, we prove that the neutral components are generically isomorphic to $\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$.