Triangle groups and Hilbert modular varieties

Abstract

In this paper we study the Hilbert series of triangle groups Δ(𝑝, 𝑞, 𝑟). The 11 groups in this series are, conjecturally, the only cocompact triangle groups that admit matrix models over totally real fields.

We provide evidence for this conjecture, along with explicit integral models for every group in the Hilbert series. The most remarkable among them, Δ(14, 21, 42), is the only known triangle group with a split invariant quaternion algebra.

Using this special group, we construct the first example of a compact Kobayashi geodesic curve 𝑉 on a Hilbert modular variety (aside from those that reside on proper Shimura subvarieties). For comparison, there are no compact Kobayashi geodesic curves in the moduli space ℳ_𝑔.