The Constant in Thomae-Type Formulas for Eight Points on the Complex Projective Line

Abstract

We consider the family of cyclic fourfold covers $w^4 = \prod_{j=1}^{7}(z-x_j)$ of the complex projective line branched at the eight points $x_1,\ldots,x_7,\infty$. The period map identifies the configuration space $X(2,8)$ of the branch points with a Zariski open subset of a quotient of the five-dimensional complex ball. The inverse of the period map is expressed projectively by $105$ automorphic forms $f_J$, which are proportional to the signed branch-point polynomials $\hat x_J$ with a common scalar factor. We determine this factor for the period $\eta$ of the differential $dz/w$: it is the product of the constant $-1/(2^{12}\Gamma(3/4)^{16})$ and the square of a quadratic form in $\eta$.