p-adic modular forms of non-integral weight over Shimura curves

TL;DR

This paper develops a comprehensive theory of p-adic modular forms of non-integral weight over Shimura curves, including the construction of sheaves, families, Hecke operators, and the eigencurve.

Contribution

It introduces a new framework for p-adic modular forms of arbitrary weights over Shimura curves, extending classical theories to non-integral weights and constructing the eigencurve.

Findings

Defined sheaves of p-adic differential invariants for non-integral weights

Constructed p-adic families of modular forms over a weight space

Built the eigencurve parametrizing overconvergent modular forms

Abstract

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.