Automorphic Forms, Definite Quaternion Algebras, and Atkin–Lehner Theory on Trees Original Research Article

Associated to the multiplicative group of a definite quaternion algebra over image are two notions of automorphic forms. One is a complex-valued function on the idele group. The other, a p-adic-valued function on a p-adic “upper half-plane,” arises from considering differentials on a Mumford curve associated to the quaternion algebra. Explicit connections between these forms are explored. In particular, forms of the second type are seen to arise from certain newforms of the first type. In order to define newforms, the Atkin–Lehner theory is developed for certain spaces of functions on the tree associated to PGL2(imagep). As an application, Schneiderʹs proposed p-adic L-series associated to a differential on a Mumford curve is seen to be better interpreted as an adelic form twisted by a p-adic character.