Galois representations with big image in the general symplectic group $\operatorname{GSp}_4(\mathbb{Z}_p)$

TL;DR

This paper constructs Galois representations with large images in the symplectic group over finite fields, under specific conditions relating to prime irregularity, and demonstrates their liftability to characteristic zero with controlled ramification.

Contribution

It introduces a method to construct Galois representations with big images in $ ext{GSp}_4(Z_p)$ under certain irregularity conditions, expanding understanding of Galois representations with large symplectic images.

Findings

Constructed Galois representations with images in the diagonal torus of $ ext{GSp}_4(F_p)$

Lifted these representations to characteristic zero with unramified properties

Established conditions on prime irregularity index for the construction

Abstract

Let $p$ be an odd prime and $e_p$ be its irregularity index. If $4e_p+8 <\frac{p-1}{2}$ we construct a Galois representation with image in the diagonal torus of $\op{GSp}_4(\Fp)$ that lifts to a characteristic $0$ representation unramified at all primes $\ell\neq p$ with image containing a finite index subgroup of $\Symp$.