An explicit Shimura canonical model for the quaternion algebra of discriminant 6 (Algebraic Number Theory and Related Topics 2016)

Abstract

According to K. Takeuchi ([Tku1], [Tku2]), all the arithmetic triangle groups are listed up (1977). They are classified in 19 commensurable classes (Table 2.1 below). It corresponds a quaternion algebra for each class. (a) The first class is of non-compact type, and it induces the usual elliptic modular function. (b) Among 18 remained classes, in 16 cases we have triangle unit groups. (c) For the rest 2 cases (the class II and XII) it appears quadrangle unit groups. Already we reported the result about the case (b) in the RIMS workshop of the previous year. There, we showed how to determine the Shimura canonical model modular function for them. As an application we exposed several defining equations of the Hilbert class fields of CM fields of higher degree. In this survey article we explain how to obtain the exact Shimura canonical model for the class II quadrangle case (that is for the quaternion algebra (-3, 2/Q)) using the modular function coming from the Appell's hypergeometric differential equation. We state only the framework of the argument together with several back grounds. Detailed proofs will be published elesewhere. This Shimura curve is studied by many mathematicians. See [Elk] for it.