On the images of Galois representations attached to low weight Siegel modular forms

TL;DR

This paper proves that for certain low weight Siegel modular forms, the associated Galois representations are irreducible, crystalline, and have large images for almost all primes, advancing understanding of their arithmetic properties.

Contribution

It establishes irreducibility, crystallinity, and large image properties of Galois representations attached to low weight Siegel modular forms under specific conditions, for almost all primes.

Findings

Galois representations are irreducible and crystalline for 100% of primes.

The mod $\ell$ Galois image contains $ ext{Sp}_4( extbf{F}_\ell)$ for 100% of primes under certain conditions.

Results apply to non-CAP, non-endoscopic automorphic representations.

Abstract

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $\pi$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $\pi$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.