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

Let (Formula presented.) be a cuspidal automorphic representation of (Formula presented.), whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If (Formula presented.) is not CAP or endoscopic, then we show that its associated (Formula presented.) -adic Galois representations are irreducible and crystalline for (Formula presented.) of primes (Formula presented.). If, moreover, (Formula presented.) is neither an automorphic induction nor a symmetric cube lift, then we show that, for (Formula presented.) of primes (Formula presented.), the image of its mod (Formula presented.) Galois representation contains (Formula presented.).