On Congruences Mod
نویسنده
چکیده
Given a prime p and cusp forms f1 and f2 on some Γ1(N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod p (where p is a fixed prime of K over p) for Hecke operators Tl at all primes l ∤Np. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod p in the above sense. The prime motivation for this study is the connection to modular mod p Galois representations, and we also explain this connection.
منابع مشابه
Ramanujan’s Elementary Method in Partition Congruences
Page 182 in Ramanujan’s lost notebook corresponds to page 5 of an otherwise lost manuscript of Ramanujan closely related to his paper providing elementary proofs of his partition congruences p(5n + 4) ≡ 0 (mod 5) and p(7n+ 5) ≡ 0 (mod 7). The claims on page 182 are proved and discussed, and further results depending on Ramanujan’s ideas are established.
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