Local to Global Compatibility on the Eigencurve (l not equal p)

TL;DR

This paper extends Coleman's Hecke operator construction to define GL_2(Q_l) actions on overconvergent p-adic modular forms, linking eigencurve points to Weil-Deligne representations and local Langlands correspondence.

Contribution

It generalizes the classical construction to the case l not equal p and connects eigencurve points with local Galois representations via the Langlands correspondence.

Findings

GL_2(Q_l) action on overconvergent forms is well-defined

Associates Weil-Deligne representations to eigencurve points

Shows local-global compatibility away from a discrete set

Abstract

We generalise Coleman's construction of Hecke operators to define an action of GL_2(Q_l) on the space of finite slope overconvergent p-adic modular forms (l not equal p). In this way we associate to any C_p-valued point on the tame level N Coleman-Mazur eigencurve an admissible smooth representation of GL_2(Q_l) extending the classical construction. Using the Galois theoretic interpretation of the eigencurve we associate a 2-dimensional Weil-Deligne representation to such points and show that away from a discrete set they agree under the Local Langlands correspondence.