The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians
type: Item
instance of: scholarly article
title: The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians
zbMATH Open document ID: 1390.14070
author: Kenji Koike
published in: Hokkaido Mathematical Journal
publication date: 2018-03-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1505.02471
full work available at URL: https://projecteuclid.org/euclid.hokmj/1520928062
review text: The paper under review concerns uniformizations of Fermat curves related to triangle groups. The main emphasis lies on the Fermat septic, and on the family of genus 3 curves given affinely as \[ X_t: \;\; y^7 = x(x-1)(x-t). \] The author computes explicit period matrices for these curves, and as an application, the inverse of the Schwarz map in terms of theta constants. In particular, this leads to interpretations of the Fermat septic and of the Klein curve as Shimura varieties. The paper concludes by deriving a certain 1-dimensional family of \(K3\) surfaces admitting a non-synplectic automorphism of order 7. By exploiting the relation with to \(X_t\) (arising from the construction as a product quotient surface), it is shown that the period function is the Schwarz map, so its inverse can be regarded as a ``\(K3\) modular function'' in the terminology of \textit{H. Shiga} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 609--635 (1979; Zbl 0427.14013); ibid. 8, 157--182 (1981; Zbl 0501.14019)].
reviewed by: Matthias Schütt
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6853593
zbMATH Keywords: Schwarz map
zbMATH Keywords: triangle group
zbMATH Keywords: Fermat curve
zbMATH Keywords: period matrix
zbMATH Keywords: theta constant
zbMATH Keywords: Shimura curve
MaRDI profile type: Publication
OpenAlex ID: W2963388674
arXiv ID: 1505.02471
Wikidata QID: Q130135063
DOI: 10.14492/HOKMJ/1520928062
description: scientific article