The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians
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)].
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