A note on modular forms mod p

Abstract

In this paper we shall study relations among the spaces of modular forms mod p attached to Sk(Np, ψχ) and Sk,(N, ψ) by using certain identities between dimensions of these spaces.

On congruence divisors of cusp forms as factors of the special values of their zeta functions

“…. This morphism to be constructed is also deeply related to the results of Swinnerton-Dyer [37], Serre [26], Katz [13] and Koike [14], and our result may be viewed in part as an interpretation of some congruence relations among cusp forms discussed in those papers in terms of the Eichler-Shimura cohomology groups.…”

Modp Hecke operators and congruences between modular forms

“…Such relations may be verified by suitable trace formulas, as in Koike [10]. We leave to the reader the verification that the desired relation follows from [10, Th.…”

Theta series mod p and the basis problem for Nebentypus.

“…Since f £ ) M ( + 3)/2 has a basis < <p f = Υ a t *q J 9 i = l,..., t > with a y = d tj for 0 ^ j < t p > it is enough to Putting θ t = 0 Fi , i = l,..., 10, we obtain a relation where 0 15 ..., θ ί l correspond to the classes which represent l. As in the case of (2), there isn't Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/3/15 3:02 AM any obvious pattern to the coefficients Oj appearing in (3). We initially tested the hypothesis of Theorem 2, that the ff F span for p < 509.…”

“…Besides this characteristic zero relation there is an additional linear relation mod257 of the form 11(3) Besides this characteristic zero relation there is an additional linear relation mod257 of the form 11(3) …”