An expansion principle for quaternionic modular forms, by Andrea Mori

Let $D$ be an indefinite quaternion division algebra with a choice of an Eichler order $R$ of level prime to the discriminant. By using the fact that the associated Shimura curve is the set of complex points of the moduli space of abelian surfaces with $R$ in the endomorphism ring, it is possible to endow the spaces of modular forms with a canonical integral structure. The goal of this paper is to give an explicit characterization of the $v$-adic integral structure of the spaces of even weight, for places $v$ of good reduction. The characterization is given in terms of the values at CM points of the iterates of the Maass operator $\delta_k=-{1\over\pi}\bigl(2i{d\over dz}+{k\over\Im(z)\bigr)$, and resembles the classical $q$-expansion principle.

Andrea Mori <>