Monodromy of four‐dimensional irreducible compatible systems of Q$\mathbb {Q}$

Let F $F$ be a totally real field and n ⩽ 4 $n\leqslant 4$ natural number. We study the monodromy groups of any $n$ -dimensional strictly compatible system { ρ λ } $\lbrace \rho _\lambda \rbrace _\lambda$ $\lambda$ -adic representations with distinct Hodge–Tate numbers such that 0 $\rho _{\lambda _0}$ is irreducible for some $\lambda _0$ . When = Q $F=\mathbb {Q}$ , $n=4$ fully symplectic, following assertions are obtained. (i) The representation symplectic almost all (ii) If in addition similitude character μ $\mu odd, then potentially automorphic residual image ¯ ( Gal ) $\bar{\rho }_\lambda (\operatorname{Gal}_\mathbb {Q})$ has subgroup conjugate to Sp ℓ $\mathrm{Sp}_4(\mathbb {F}_\ell )$