Computing CM Points on Shimura Curves Arising from Cocompact Arithmetic Triangle Groups

Let PSL2(R) be a cocompact arithmetic triangle group, i.e. a triangle Fuchsian group that arises from the unit group of a quaternion algebra over a totally real number eld. We introduce CM points de ned on the Shimura curve quotient XC = nH, and we algorithmically apply the Shimura reciprocity law to compute these points and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how these methods work in practice. As motivation, we begin with a description of the classical situation. The subgroup 0(N) of matrices in SL2(Z) which are upper triangular modulo N 2 Z>0 acts on the completed upper half-plane H by linear fractional transformations; the quotient X0(N)C = 0(N) n H can be given the structure of a compact Riemann surface. The complex curve X0(N)C itself is a moduli space for (generalized) elliptic curves equipped with a cyclic subgroup of order N , and consequently it has a model X0(N)Q de ned over Q. There exist \special" points on X0(N)Q, known as CM points, where the corresponding elliptic curves have complex multiplication by quadratic imaginary elds K. CM points are de ned over abelian extensions H of K, and the Shimura reciprocity law explicitly describes the action of the Galois group Gal(H=K) on them. The image of a CM point under the elliptic modular j-function is known as a singular modulus. Gross and Zagier give a formula for the norm of the di erence of two singular moduli [5]; the traces of singular moduli arise as the coe cients of modular forms (see e.g. [17]). In this article, we generalize this situation by replacing the modular curve X0(N) by a Shimura curve X0(N), associated to a quaternion algebra de ned over a totally real number eld F . The curves X0(N) we will consider similarly come equipped with a map j : X0(N)! P1 and CM points de ned over abelian extensions H of a totally imaginary extension K of F . Developing ideas of Elkies [4], we can compute these points to high precision as complex numbers, and we generalize his methods by using the Shimura reciprocity law to recognize them as putative algebraic numbers by also computing their conjugates under Gal(H=K). We may then compute the norms, traces, and other information about these CM points, with a view towards a generalized Gross-Zagier formula in this setting.