On the Algebraic Curves Uniformized by Arithmetical Automorphic Functions

Abstract

The purpose of this note is to prove a theorem on algebraic curves uniformized by automorphic functions with respect to a discontinuous group obtained from an indefinite quaternion algebra, and to study some related problems. The theorem may be viewed as an addition to a recent work of Shimura [8] (quoted hereafter as [C]). To state it we repeat here the notation of [C]. Let B be a quaternion algebra over a totally real algebraic number field F of finite degree. We assume that B is unramified at the real archimedean prime of F corresponding to the identity mapping of F, and ramified at other archimedean primes of F. Here we consider F as a subfield of the real number field R. We denote by D(B/F) the product of all prime ideals of F which are ramified in B. Let B+ denote the set of all elements a of B such that NB!F(a) is totally positive. Then the elements of B+ act naturally on the upper half complex plane &. For every maximal order o in B and an integral ideal c in F, we put