Equations for arithmetic pointed tori

Abstract

In 1983, Kisao Takeuchi enumerated all 71 arithmetic (1;e)-groups. This is a special set of discrete subgroups of SL(2,R) of finite covolume and signature (1;e). The corresponding quotients of the upper half plane (called (1;e)-curves) have genus equal to 1 and a single elliptic point of order e. The additional arithmeticity demand comes down to asking that such a (1;e)-curve allows a natural finite cover by a Shimura curve associated with a quaternion algebra over a totally real field F. The field F equals the rational field for 7 subgroups in Takeuchi's list. Equations for the corresponding (1;e)-curves can be determined from work by Krammer and Rotger, However, apart from some pioneering numerical work by Chudnovsky and Chudnovsky, there seems to have been no other literature that explicitly computes arithmetic (1;e)-curves when F is not equal to the field of rationals. Exploiting the cover by a Shimura curve mentioned above, we construct explicit models for these (1;e)-curves over number fields. More precisely, it is known that a Shimura curve covering the (1;e)-curve has a canonical model over an abelian extension of F. We extend the notion of canonical model to the arithmetic (1;e)-curves themselves and use the arithmetic properties of the Shimura cover to compute conjectural equations for these models. In the majority of cases, we then prove the correctness of these candidate equations. The arithmetic methods mentioned above, and the concomitant algorithms, involve the calculation of Hecke operators, explicit p-adic uniformization, and the determination of Belyi maps of low degree. Though we restricted ourselves to the Shimura curves required to deal with arithmetic (1;e)-curves, these methods can be applied to furnish explicit information regarding more general Shimura curves.