A remark on non-integral p-adic slopes for modular forms

TL;DR

This paper introduces a condition called 'Buzzard irregularity' that guarantees the existence of cuspidal eigenforms with non-integral p-adic slopes, advancing understanding of modular forms' p-adic properties.

Contribution

It establishes a sufficient condition for the existence of non-integral p-adic slopes in cuspidal eigenforms, providing new insights into modular forms.

Findings

Buzzard irregularity implies non-integral p-adic slopes.

Existence of cuspidal eigenforms with non-integral slopes under this condition.

Enhances understanding of p-adic properties of modular forms.

Abstract

We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.