Abstract
<p> The purpose of this paper is to list the refined Humbert invariants for a given automorphism group of a smooth curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus 2 over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c h a r left-parenthesis upper K right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>char</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {char}(K) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This invariant is an algebraic generalization of the (usual) <italic>Humbert invariant</italic> . It is a positive definite quadratic form associated to the curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and it encodes many geometric properties of the curve. The paper has a special interest in the cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis upper C right-parenthesis asymptotically-equals upper D 4"> <mml:semantics> <mml:mrow> <mml:mi>Aut</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≃ </mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Aut}(C)\simeq D_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D 6"> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">D_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In these cases, several applications of the main results are discussed including the curves with elliptic subcovers of a given degree. </p>