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

Abstract

AbstractLet be a totally real field and a natural number. We study the monodromy groups of any ‐dimensional strictly compatible system of ‐adic representations of with distinct Hodge–Tate numbers such that is irreducible for some . When , , and is fully symplectic, the following assertions are obtained. The representation is fully symplectic for almost all . If in addition the similitude character of is odd, then the system is potentially automorphic and the residual image has a subgroup conjugate to for almost all .