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Filtrations of dc-weak eigenforms

Volume 180 / 2017

Nadim Rustom Acta Arithmetica 180 (2017), 297-318 MSC: Primary 11F80; Secondary 11F33. DOI: 10.4064/aa8491-8-2017 Published online: 9 October 2017

Abstract

The notions of strong, weak and dc-weak eigenforms mod $p^n$ were introduced and studied by Chen, Kiming and Wiese (2013). Here 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 (2016) which establishes a uniform weight bound on strong eigenforms mod $p^n$. As a step towards studying weight 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.

Authors

  • Nadim RustomNational Center for Theoretical Sciences
    National Taiwan University
    Room 203, Astronomy-Mathematics Building
    No. 1, Sec. 4, Roosevelt Rd.
    Taipei City 106, Taiwan
    e-mail

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