Ramanujan-style congruences of local origin
type: Item
instance of: scholarly article
title: Ramanujan-style congruences of local origin
zbMATH Open document ID: 1304.11027
author: Neil Dummigan
author: Daniel Fretwell
published in: Journal of Number Theory
publication date: 2014-09-02T00:00:00Z
review text: Let \(p\) be a prime and \(k\geq4\) an even integer. Further suppose that \(\ell>3\) is a prime such that \(\mathrm{ord}_{\ell}\left((p^k-1)B_k/2k\right)>0\), where \(B_k\) is the \(k\)-th Bernoulli number. Then the authors show that there exists an eigenform in \(S_k(\Gamma_0(p))\) whose Fourier coefficients satisfy a ``Ramanujan-style'' congruence modulo \(\ell\) similar to the famous 691 congruence for the Ramanujan \(\tau\)-function. The technique used is the construction of explicit Eisenstein series of level \(p\) and weight \(k\) which vanish modulo \(\ell\) at both cusps, together with the Deligne-Serre theory of \(\ell\)-adic Galois representations attached to modular forms. The authors further describe the relation of their work to the \(\ell\)-part of the Bloch-Kato conjecture, and supply new proofs of theorems of \textit{N. Billerey} and \textit{R. Menares} [``On the modularity of reducible mod \(l\) Galois representations'', Preprint, arXiv:1309.3717] concerning the modularity of reducible mod \(\ell\) Galois-representations. The authors then conclude with a detailed description of several illuminating examples.
reviewed by: Larry Rolen
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 6339007
zbMATH Keywords: congruences of modular forms
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Galois representations
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.04.008
OpenAlex ID: W2007841059
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DOI: 10.1016/J.JNT.2014.04.008
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description: scientific article; zbMATH DE number 6339007