On equations defining fake elliptic curves par Pilar BAYER et Jordi GUÀRDIA

Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the case of CM-points on classical modular curves, CMfake elliptic curves play a key role in the construction of class fields by means of special values of automorphic functions (cf. [Sh67]).