Algebraic Curves Uniformized by Congruence Subgroups of Triangle Groups

We construct certain subgroups of hyperbolic triangle groups which we call “congruence” subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups. The rich arithmetic and geometric theory of classical modular curves, quotients of the upper half-plane by subgroups of SL2(Z) defined by congruence conditions, has fascinated mathematicians since at least the nineteenth century. One can see these curves as special cases of several distinguished classes of curves. Fricke and Klein [20] investigated curves obtained as quotients by Fuchsian groups which arise from the unit group of certain quaternion algebras, now called arithmetic groups. Later, Hecke [24] investigated his triangle groups, arising from reflections in the sides of a hyperbolic triangle with angles 0, π/2, π/n for n ≥ 3. Then in the 1960s, amidst a flurry of activity studying the modular curves, Atkin and Swinnerton-Dyer [1] pioneered the study of noncongruence subgroups of SL2(Z). In this paper, we consider a further direction: we introduce a class of curves arising from certain subgroups of hyperbolic triangle groups; these curves share many appealing properties in common with classical modular curves despite the fact that their uniformizing Fuchsian groups are in general not arithmetic groups. To motivate the definition of this class of curves, we begin with the modular curves. Let p be prime and let Γ(p) ⊆ PSL2(Z) = Γ(1) be the subgroup of matrices congruent to (plus or minus) the identity modulo p. Then Γ(p) acts on the completed upper half-plane H∗, and the quotient X(p) = Γ(p)\H∗ can be given the structure of Riemann surface, a modular curve. The subgroup G = Γ(1)/Γ(p) ⊆ Aut(X(p)) satisfies G ∼= PSL2(Fp) and the natural map j : X(p) → X(p)/G ∼= PC is a Galois cover ramified at the points {0, 1728,∞}. In this paper, we will be interested in the class of (algebraic) curves X over C with the property that there exists a subgroup G ⊆ Aut(X) with G ∼= PSL2(Fq) or G ∼= PGL2(Fq) (for some prime power q) such that the map X → X/G ∼= P is a Galois cover ramified at exactly three points. On the one hand, Bely̆ı [3, 4] proved that a curve X over C admits a Bely̆ı map, a map X → PC ramified at exactly three points, if and only if X can be defined Date: August 3, 2011.