On the component group of the algebraic monodromy group of a K3 surface

We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a K3 surface over a number field k. In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character CEJ, Definition 2.4.6 entirely in terms of the endomorphism field and the geometric Picard rank.