We investigate certain properties of (smooth) genus 2 curves. Our approach focuses on a special associated integral quadratic form, which is intrinsically attached to a curve C of genus 2. This form, known as the refined Humbert invariant, was introduced by Ernst Kani in 1994.
One of the nice features of the refined Humbert invariant is its role to translate geometric questions into arithmetic ones. This property enables to solve various intriguing geometric problems related to genus 2 curves, such as determining their automorphism groups and finding their elliptic subcovers. After illustrating the usefulness of this invariant through examples, we discuss its classification, which reveals interesting implications for Humbert surfaces.
Also, it is possible to derive numerous new formulas/expressions for the refined Humbert invariant. These formulas are particularly useful for establishing connections between the refined Humbert invariant and the quadratic forms associated with Shimura curves, as studied by Hashimoto, Runge, and Lin and Yang.
