Filtrations of dc-weak eigenforms

TL;DR

This paper investigates the properties of dc-weak eigenforms mod p^n, demonstrating the absence of a uniform weight bound for these forms at fixed level when n ≥ 2, contrasting with known results for strong eigenforms.

Contribution

It proves the non-existence of a uniform weight bound for dc-weak eigenforms mod p^n at fixed level for n ≥ 2 and offers a criterion to distinguish weak eigenforms among dc-weak eigenforms.

Findings

No uniform weight bound exists for dc-weak eigenforms mod p^n when n ≥ 2.

A criterion is provided to detect weak eigenforms among dc-weak eigenforms.

Contrasts with the uniform weight bound for strong eigenforms mod p^n.

Abstract

The notions of strong, weak and dc-weak eigenforms mod $p^n$ was introduced and studied by Chen, Kiming and Wiese. In this work, we prove that there can be no uniform weight bound (that is, depending only on $p$, $n$) on dc-weak eigenforms mod $p^n$ of fixed level when $n \geq 2$. This is in contrast with the result of Kiming, Rustom and Wiese which establishes a uniform weight bound on strong eigenforms mod $p^n$. As a step towards studying weights bounds for weak eigenforms mod $p^n$, we provide a criterion which allows us to detect whether a given dc-weak eigenform mod $p^n$ is weak.