Filtrations of dc-weak eigenforms

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 $n\ge 2$. 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.

26 pages. This is a revised version with several corrections. The statement of the main result is now more precise

References in corpus (4)

• On certain finiteness questions in the arithmetic of modular forms
• Level lowering modulo prime powers and twisted Fermat equations
• The Hecke Algebra T_k has Large Index
• Computing level one Hecke eigensystems (mod p)

Cited by in corpus (2)

• Congruences modulo prime powers of Hecke eigenvalues in level $1$
• Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4