91É«ÇéƬ

Abstract

Grothendieck proposed a theory of regular polyhedra over finite fields.Ìý We will explain this proposal and how it relates to the modern theory of polytopes.Ìý Next we will relate these polyhedra to quaternion algebras and triangular modular curves, a variant of modular curves which are uniformized by triangle groups.Ìý Our main result answers a question of Grothendieck regarding the classification of these polyhedra, which simultaneously computes some new examples of triangular modular curves.Ìý