Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture

TL;DR

This paper examines the ghost conjecture for p-adic modular forms, identifies a formulation error related to reducible Galois representations, and proposes a correction to improve its accuracy.

Contribution

The authors identify an oversight in the ghost conjecture concerning reducible Galois representations and suggest a corrected formulation to address this issue.

Findings

The original ghost conjecture is not correctly formulated for reducible Galois representations.

A detailed explanation of the formulation error is provided.

A revised version of the ghost conjecture is proposed to fix the identified issue.

Abstract

The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.