Arithmetic on Modular Curves

Modular symbols, Eisenstein series, and congruences

“…The next few results stated below are generalizations of weight 2 results that can be found in [16], Chapter 2-in particular see Propositions 2.1.2 and 2.2.2. The proofs in higher weight are adaptations of the proofs used there; the modifications are simple and we omit them.…”

“…We define the sign of E to be −ε 1 (−1). (This is the same definition as in [16]. Note that it does not change with the weight of E.) Lemma 4.8.…”

“…Finally, we state need one last lemma, which is a generalization of [16], Lemma 3.1.1. Once again, the proof is standard.…”

“…We have excluded the case k = 2 from the result above, but it is treated in [16], Section 2.5. The non-exceptional cases yield the same formula as the above proposition when k = 2, but the exceptional cases are different (and when k = 2 we exclude the case x 1 = x 2 = 0 entirely).…”

“…His results were subsequently generalized by Stevens [16], and then refined by the second of the present authors, who also treated the case of congruences between cusp forms [18]. The paper [18] contains other congruence theorems for cusp forms of higher weight, but the case of congruences between higher-weight cusp forms and higher-weight Eisenstein series was left open, as was the case of congruences between cusp forms at primes for which the corresponding Galois representation is Eisenstein (reducible), and it is these gaps that we propose to close.…”