On Local Constants Associated to Arithmetical Automorphic Function

Abstract

This article deals with the diophantine nature of the local constants associated to automorphic functions with respect to Fuchsian groups obtained from quaternion algebras. We focus on the canonical model of those curves arising from the quaternion Q-algebra of discriminant D = 6.

Computing Power Series Expansions of Modular Forms

“…We follow Bayer [2], Bayer and Travesa [3,4], and Baba and Granath [1]; for further detail, we refer to these articles.…”

“…The Dirichlet fundamental domain D(p) is then given in Figure 2. (A slightly different fundamental domain, emphasizing the relationship to the triangle group ∆ (2,4,6), is considered by the other authors.) We have ρ = 0.447213 .…”

“…In this case, power series expansions for a uniformizing function at the vertices of the fundamental triangle are obtained as the inverse of the ratio of 2 F 1 -hypergeometric functions. This case was also of great classical interest, and has been taken up again more recently by Bayer [2], Bayer and Travesa [3,4], the first author [45], and Baba and Granath [1]; this includes the well-studied case where the Fuchsian group arises from the quaternion algebra of discriminant 6 over Q, corresponding to the (2, 4, 6)-triangle group.…”

Differential equations and expansions for quaternionic modular forms in the discriminant 6 case

“…In [5], the authors compute the Schwarzian differential equation for the Shimura curve of discriminant 6, and use it to deduce some results about the hauptmodul j. In [4], these results are extended to give some terms of the Taylor expansions of the j-function at the elliptic points.…”