Dummigan, N. and Fretwell, D. (2014) Ramanujan-style congruences of local origin. Journal of Number Theory, 143. pp. 248-261. ISSN 0022-314X
Abstract
We prove that if a prime ℓ>3 divides pk−1, where p is prime, then there is a congruence modulo ℓ, like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate ℓ to primes like 691 by viewing it as a divisor of a partial zeta value, and see how a construction of Ribet links the congruence with the Bloch–Kato conjecture (theorem in this case). This viewpoint allows us to give a new proof of a recent theorem of Billerey and Menares. We end with some examples, including where p=2 and ℓ is a Mersenne prime.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2014 Elsevier. This is an author produced version of a paper subsequently published in Journal of Number Theory. Uploaded in accordance with the publisher's self-archiving policy. Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) |
Keywords: | Congruences of modular forms |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 25 Jan 2017 12:35 |
Last Modified: | 23 Mar 2018 19:55 |
Published Version: | https://doi.org/10.1016/j.jnt.2014.04.008 |
Status: | Published |
Publisher: | Elsevier |
Refereed: | Yes |
Identification Number: | 10.1016/j.jnt.2014.04.008 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:110211 |