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

Let π $\pi$ be a cuspidal automorphic representation of GSp 4 ( A Q ) $\operatorname{GSp}_4(\mathbf {A}_\mathbf {Q})$ , whose archimedean component is holomorphic discrete series or limit representation. If not CAP endoscopic, then we show that its associated ℓ $\ell$ -adic Galois representations are irreducible and crystalline for 100 % $100\%$ primes . If, moreover, neither an induction nor symmetric cube lift, that, the image mod contains Sp F $\operatorname{Sp}_4(\mathbf {F}_\ell )$