I'm reading a paper of Swinnerton-Dyer, "On l-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod $\ell$ modular forms:
If $\tilde{f} \in \tilde{M}$ (the mod $\ell$ modular forms) which is the sum of monomials in $M_k$ such that all $k$ are congruent modulo $\ell-1$, then the filtration of $\tilde{f}$, denoted $\omega(\tilde{f})$ is defined to be the least $k$ such that $\tilde{f} \in \tilde{M}_k$ (since the Eisenstein series $E_{\ell-1}$ reduces to $1$ modulo $l$, we can multiply by powers of it).
This notion turns out to be very useful for classifying the exceptional primes of a Galois representation attached to a modular form. The first question I'd like to ask is, what's the motivation for it?
Secondly, I'd like to ask about specific properties. If $f$ is a weight $k$ modular form, what can we say about $\omega(\tilde{f})?$ I am under the impression that the following are true -- am I right?
- $\omega(\tilde{f})$ must be congruent to $k$ modulo $\ell-1$ (an in particular, it is $k$ if $\ell > k$)
- $\omega(\tilde{f}) \leq k$.
I'm confused because on p. 29, Swinnerton-Dyer says $\omega(\tilde{f}) = k$ if $\ell > 2k$...