Quaternionic modular forms of any weight

In this work we construct an eigencurve for p-adic modular forms attached to an indefinite quaternion algebra over ℚ. Our theory includes the definition, both as rules on test objects and sections of a line bundle, of p-adic modular forms, convergent and overconvergent, of any p-adic weight. We prove that our modular forms can be put in analytic families over the weight space and we introduce the Hecke operators U and Tₗ, that can also be put in families. We show that the U-operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of overconvergent eigenforms of finite slope with respect to the U-operator.

International Journal of Number Theory