The classification of the refined Humbert invariant for curves of genus 2

The refined Humbert invariant is a positive definite quadratic form intrinsically attached to a curve $C$ of genus 2. This invariant is an algebraic generalization of the (usual) Humbert invariant. This invariant is useful because many geometric properties of $C$ are reflected in the arithmetic properties of this invariant. The purpose of this paper is to complete the classification of this invariant when the Jacobian $J_C$ of $C$ is isogenous to a product of an elliptic curve with complex multiplication.