Congruence covers of triangular modular curves and their Galois groups

Abstract

Abstract. This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.

Algebraic curves uniformized by congruence subgroups of triangle groups

“…First, we note that whenever P ∤ β, the order O is maximal at P. Second, even for a ramified prime P (or p), we still can consider the natural map to the completion; however, instead of PGL 2 (F P ) we instead obtain the units of an order in a division algebra over F P , a prosolvable group. Our interest remains in the groups PSL 2 and PGL 2 , but this case also bears further investigation: see Takei [73] for some results in the case where b = c = ∞.…”