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.