Ordinary Modular Forms and Companion Points on the Eigencurve

TL;DR

This paper provides a new proof connecting the splitting behavior of p-adic Galois representations associated with p-ordinary modular forms to the existence of overconvergent p-adic companion forms, enhancing understanding of the eigencurve structure.

Contribution

It offers a novel proof of a key relationship between Galois representation splitting and companion forms, deepening the theoretical framework of modular forms and eigencurves.

Findings

Established the link between Galois splitting and companion forms

Provided a new proof of Breuil and Emerton's result

Enhanced understanding of p-adic modular forms and eigencurve structure

Abstract

We give a new proof of a result due to Breuil and Emerton which relates the splitting behavior at p of the p-adic Galois representation attached to a p-ordinary modular form to the existence of an overconvergent p-adic companion form for f.