Resolution of Degree $\leq 6$ Algebraic Equations by Genus Two Theta Constants

Abstract

We adjoin complete first kind Abelian integrals of genus two to resolve the general sextic equation $c_0 z^6 + c_1 z^5 + \cdots + c_6 = 0$ with simple zeros by genus two theta constants (Thetanullwerten). Using the same formulas, we also resolve each algebraic equation of degree five, four or three. It is shown that the monodromy group of a sextic is isomorphic to the second congruence sub-group $\Gamma (2)$ of the symplectic group ${\rm Sp}_4 ({\mathbb Z})$.