Equations of Shimura Curves of Genus Two

LetBD be the indefinite quaternion algebra overQ of reduced discriminantD=p1· · · · ·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown by Shimura [23], XD is the coarse moduli space of abelian surfaces with quaternionic multiplication by BD. Let W = {ωm : m | D} ⊆ Aut Q(XD) be the group of Atkin-Lehner involutions. For any m | D, we will denote X D = XD/〈ωm〉, the quotient of the Shimura curve XD, by ωm. The importance of the curves X D is enhanced by their moduli interpretation as curves embedded in Hilbert-Blumenthal surfaces and Igusa’s threefold A2 (cf. [21, 22]). The classical modular case arises when D = 1. In this case, automorphic forms of these curves admit Fourier expansions around the cusp of infinity and we know explicit generators of the field of functions of such curves. Also, explicit methods are known to determine bases of the space of their regular differentials, which are used to compute equations for quotients of modular curves. When D = 1, the absence of cusps has been an obstacle for explicit approaches to Shimura curves. Explicit methods to handle functions and regular differential forms on these curves are less accessible and we refer the reader to [3] for progress in this regard. For this reason, at present, few equations of Shimura curves are known, all of them of genus 0 or 1 (cf. [6, 11, 13]). In addition, in a later work, Kurihara conjectured equations for all Shimura curves of genus two and for several curves of genera three and five, though he was not able to give a proof for his guesses (cf. [14]).