Published In

Bulletin of the Australian Mathematical Society

Document Type

Article

Publication Date

1-1-2026

Subjects

Hecke eigenvalues, period polynomials

Abstract

We study a conjecture motivated by Coleman and Stein’s work on approximating eigenforms of infinite slope by those of finite slopes, which was recast by Rustom [‘Congruences modulo prime powers of Hecke eigenvalues in level 1’, Res. Number Theory 5(1) (2019), Article no. 10, 27 pages] as a congruence for the second Fourier coefficient. We establish the weight condition required for this coefficient to be divisible by nine and obtain the full conjecture under a natural ramification hypothesis. The proof uses period polynomials, especially Zagier’s description of the Hecke action on them.

Rights

Copyright (c) 2026 The Authors

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

10.1017/S0004972726101889

Persistent Identifier

https://archives.pdx.edu/ds/psu/45244

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