Abelian surfaces with supersingular good reduction and non semisimple Tate module

TL;DR

This paper demonstrates the existence of abelian surfaces over  with supersingular good reduction whose associated p-adic Galois modules are not semisimple, challenging previous assumptions about their structure.

Contribution

It provides the first explicit example of abelian surfaces with supersingular reduction and non-semisimple Tate modules over .

Findings

Existence of abelian surfaces with supersingular good reduction

Associated p-adic Galois modules are not semisimple

Challenges previous expectations about Tate module structure

Abstract

We show the existence of abelian surfaces $A$ over $\mathbb{Q}_p$ having good reduction with supersingular special fibre whose associated $p$-adic Galois module $V_p(A)$ is not semisimple.