Quaternionic modular forms of any weight

TL;DR

This paper constructs a p-adic eigencurve for quaternionic modular forms of any weight, demonstrating their analytic family structure and properties of Hecke operators, including the compact U-operator.

Contribution

It introduces a new framework for p-adic quaternionic modular forms, defining them as sections of line bundles and establishing their analytic families and eigencurve construction.

Findings

Construction of the eigencurve for quaternionic modular forms.

Proof that the U-operator acts compactly on overconvergent forms.

Definition of p-adic modular forms of any weight as sections of line bundles.

Abstract

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 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 T_l, 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.