2019
DOI: 10.1007/s40993-018-0147-5
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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… Show more

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Cited by 2 publications
(5 citation statements)
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“…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].…”
Section: Definitionmentioning
confidence: 70%
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“…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].…”
Section: Definitionmentioning
confidence: 70%
“…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.…”
Section: Background Motivations Questionsmentioning
confidence: 99%
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“…But if it executes successfully, it provides the needed polynomial identities to prove the desired congruence. The code, written in Sage, can be found on the author's website ( [Rus17a]), along with computer-generated pdf files containing the propositions of the form Proposition 10.3, and the corresponding solutions stored as Sage objects.…”
Section: Theorem 101 ([Mer94]mentioning
confidence: 99%