Level raising and completed cohomology

TL;DR

This paper applies Poincaré duality to completed homology spaces to analyze level raising in p-adic modular forms, providing a new perspective on the p-adic Jacquet-Langlands correspondence and the structure of eigencurves.

Contribution

It introduces a novel approach using Poincaré duality for completed homology to describe level raising and the image of Chenevier's p-adic Jacquet-Langlands map.

Findings

New description of the image of Chenevier's p-adic Jacquet-Langlands map

Identification of level raising points as intersections of old and new components

Application of duality to understand eigencurve structures

Abstract

We describe an application of Poincar\'e duality for completed homology spaces (as defined by Emerton) to level raising for p-adic modular forms. This allows us to give a new description of the image of Chenevier's p-adic Jacquet-Langlands map between an eigencurve for a definite quaternion algebra and an eigencurve for GL(2). The points on the eigencurve at which we "raise the level" are (non-smooth) points of intersection between an "old" and a "new" component.