Slopes of modular forms and the Ghost conjecture

TL;DR

This paper presents an algorithm to compute the slope sequence of modular forms with fixed Galois components, refining previous conjectures and exploring symmetries that may impact understanding of Coleman-Mazur eigencurves.

Contribution

It introduces a refined algorithm for slope computation of modular forms based on initial data, building on recent results related to the ghost conjecture.

Findings

Algorithm effectively computes slope sequences from initial entries.

Identifies symmetries in slope sequences with potential implications for eigencurve symmetries.

Builds on and refines the conjecture of Buz05 using recent theoretical results.

Abstract

We give an algorithm to compute the slope sequence of modular forms with fixed Galois components from its first few entries, which is a refined version of the conjecture of [Buz05]. We use the results of arXiv:2302.07697 on the ghost conjecture from axXiv:1710.01572. These symmetries in slope sequences have potential implication to unexplained symmetries in many Coleman-Mazur eigencurves.