The refined Humbert invariant for an automorphism group of a genus 2 curve

The purpose of this paper is to list the refined Humbert invariants for a given automorphism group of a smooth curve C / K C/K of genus 2 over an algebraically closed field K K with char ⁡ ( K ) = 0 \operatorname {char}(K) = 0 . This invariant is an algebraic generalization of the (usual) Humbert invariant . It is a positive definite quadratic form associated to the curve C C , and it encodes many geometric properties of the curve. The paper has a special interest in the cases where Aut ⁡ ( C ) ≃ D 4 \operatorname {Aut}(C)\simeq D_4 or D 6 D_6 . In these cases, several applications of the main results are discussed including the curves with elliptic subcovers of a given degree.