The Shimura canonical model for the quaternion algebra of discriminant 6 on the Ihara-Kurihara conic

Abstract

In this paper we construct the Shimura canonical model for the quaternion algebra B of discriminant 6 over Q in the sense of the original paper [SmrB] (see also [Shg2]). The “unit group” of B becomes to be a quadrangle group Γ=□(0;2,2,3,3) acting on the complex upper half space H. For our purpose we use the defining equation of the Shimura curve V=H/□(0;2,2,3,3) given by Y. Ihara which found in the work of A. Kurihara (1979). We are requested to find a modular function ψ defined on H with respect to Γ which realizes V satisfying several arithmetic conditions stated later. Basically our ψ is the modular function for the family of abelian surfaces with quaternion multiplication by B, so called false elliptic curves, with some arithmetic conditions. Our main tool is the hypergeometric modular function given by M. Petkova and H. Shiga (2011). This is the full paper of the survey report [Shg2] (2018).