We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo... more We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo , like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate 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 is a Mersenne prime.
We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo... more We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo , like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate 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 is a Mersenne prime.
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Papers by Neil Dummigan