Congruences modulo prime powers of Hecke eigenvalues in level $1$

Abstract

We continue the study of strong, weak, and dc-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level 1 modulo 128, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level 1 modulo 9, reducing the question to the finiteness of a single eigenvalue. Furthermore, we answer the question of comparing weak and dc-weak eigenforms and provide the first known examples of non-weak dc-weak eigenforms.

Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

“…An affirmative answer for all m was conjectured in [18, Conjecture 1] that also gave some (weak) evidence in favor of it. Further evidence (for N = 1) has been obtained in [24].…”

“…The background for our study of the form f is the theory of modular forms modulo prime powers that has attracted some attention in recent years, cf. for instance the papers [6,7,8,18,23,24,26,27,28]. Before reviewing some of the questions that have arisen recently, let us set up some notation.…”

“…It is not necessary for the purposes of this paper to give the precise definition which can be found in [8, Section 1]. At a fixed level N , we have {strong eigenforms mod p m } ⊆ {weak eigenforms mod p m } ⊆ {dc-weak eigenforms mod p m }, and the inclusions can in fact be strict: the first examples of dc-weak eigenforms that are not weak can be found in [24] (Theorem 5.4 and the consequences of it; the simplest example is the form ∆ + 2d∆ modulo 4 where d := (E 4 − 1)/16. )…”

“…[8,Proposition 19] which is based on work of Hida.) However, the result of this kind of level-lowering seems naturally to be a dcweak eigenform at the lowered level (though, it seems, this can be strengthened by utilizing [24,Theorem 4.3]. )…”