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

TL;DR

This paper presents uniformizations of the Klein quartic and Fermat septic curves as Shimura curves, which parametrize Abelian 6-folds with specific endomorphism structures, linking algebraic curves to moduli spaces.

Contribution

It introduces new uniformizations of these classical curves as Shimura curves associated with Abelian 6-folds having endomorphisms by []7.

Findings

Uniformizations of Klein quartic and Fermat septic as Shimura curves

Parametrization of Abelian 6-folds with []7 endomorphisms

Connection between algebraic curves and moduli spaces

Abstract

We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.