An explicit computation of the Hecke operator and the ghost conjecture

TL;DR

This paper explores the Hecke operator at p=5, revealing a ghost pattern in its matrix minors, and proposes a conjecture that allows slope computation using a modified ghost series, leading to an upper bound consistent with Gouvea's conjecture.

Contribution

It introduces a new conjecture linking the Hecke operator at p=5 to the ghost series, providing a novel method for slope computation and an upper bound aligned with existing conjectures.

Findings

Hecke operator at p=5 exhibits a ghost pattern in matrix minors.

A conjecture relates the Hecke slopes to a ghost series variant.

Derived an upper bound for slopes matching Gouvea's conjecture.

Abstract

In this paper, we investigate the Hecke operator at p = 5 and show that the upper minors of the matrix have non zero corank and, interestingly, follow the same ghost pattern in the Ghost conjecture of Bergdall and Pollack. Due to this facts, we conjecture that the slope of Hecke action in this case can be computed using an appropriate variant of ghost series. Assume this result, we achieve an upper bound for the slopes that is similar to the Gouvea's (k-1)/(p+1) conjecture.