Ramanujan-style congruences of local origin

Neil Dummigan*, Daniel Fretwell

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)248-261
Number of pages14
JournalJournal of Number Theory
Early online date4 Jun 2014
Publication statusPublished - Oct 2014
Externally publishedYes


  • 11F33
  • Congruences of modular forms


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