### 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 < mori@dm.unito.it>