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The Constant in Thomae-Type Formulas for Eight Points on the Complex Projective Line
Abstract
We consider the family of cyclic fourfold covers w4 = ∏j=17(z−xj) of the complex projective line branched at the eight points x1,…,x7,∞. 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 fJ, which are proportional to the signed branch-point polynomials x̂J with a common scalar factor. We determine this factor for the period η of the differential dz/w: it is the product of the constant −1/(212Γ(3/4)16) and the square of a quadratic form in η.
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BibTeX
@misc{nakano2026constant,
title = {{The Constant in Thomae-Type Formulas for Eight Points on the Complex Projective Line}},
author = {Ryunosuke Nakano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06439},
eprint = {2609.06439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06439},
}
RIS
TY - UNPB
TI - The Constant in Thomae-Type Formulas for Eight Points on the Complex Projective Line
AU - Ryunosuke Nakano
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06439
UR - https://arxiv.org/abs/2609.06439
AB - We consider the family of cyclic fourfold covers w^4 = ∏_j=1^7(z−x_j) of the complex projective line branched at the eight points x_1,…,x_7,∞. 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 x̂_J with a common scalar factor. We determine this factor for the period η of the differential dz/w: it is the product of the constant −1/(2^12Γ(3/4)^16) and the square of a quadratic form in η.
ER -