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

Abstract

Let $\pi$ be a cuspidal automorphic representation of ${\mathrm{GSp}}_4({\mathbf{A}}_{\mathbf{Q}})$, 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 ${\mathrm{Sp}}_4({\mathbf{F}}_{\ell })$.

25 pages. Final version. To appear in Journal of the LMS

References in corpus (5)

• Pseudocharacters of Classical Groups
• Arthur's multiplicity formula for GSp${}_4$ and restriction to Sp${}_4$
• On the Bloch-Kato conjecture for the Asai L-function
• Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms
• The Fontaine-Mazur conjecture in the residually reducible case

Cited by in corpus (3)

• Serre weights for $GS{p}_4$ over totally real fields
• Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
• On the images of certain $G_2$-valued automorphic Galois representations