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

TL;DR

This paper completes the classification of the refined Humbert invariant for genus 2 curves whose Jacobian is isogenous to a product of an elliptic curve with complex multiplication, linking geometric and arithmetic properties.

Contribution

It provides a comprehensive classification of the refined Humbert invariant in a specific case involving complex multiplication.

Findings

Classification of the invariant is completed for Jacobians isogenous to CM elliptic curves.

The invariant reflects geometric properties of genus 2 curves.

The work extends understanding of algebraic invariants related to genus 2 curves.

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.