Failure of the Local to Global Principle in the Eigencurve

TL;DR

This paper investigates instances where the local to global compatibility principle fails on the p-adic eigencurve, linking these failures to intersections of specific geometric components, thus extending Ribet's level raising and lowering concepts.

Contribution

It proves that points of failure are intersections of principal series and special components on the eigencurve under mild hypotheses, providing a geometric perspective on known level raising/lowering phenomena.

Findings

Failures occur at intersections of principal series and special components.

Such points are characterized by degenerations in Satake parameters or monodromy.

The results extend classical level raising and lowering theorems to a geometric setting.

Abstract

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away from p) under one of two conditions: on a generically principal series component where monodromy vanishes; or on a generically special component where the ratio of the Satake parameters degenerates. We prove, under mild restrictive hypotheses, that such points are the intersection of generically principal series and special components. This is a geometric analogue of Ribet's level raising and lowering theorems.