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

Abstract

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.

Revised version corresponding to the published version in Int. J. Number Theory. This version includes significant corrections and additional applications of the main classification results, including the study of intersections of Humbert surfaces