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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 $\ell > 2k$...

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Dear Tony, I don't have my copy of SwD with me, but are you sure that he defines filtration for $\ell$-adic modualr forms, rather than for mod $\ell$ modular forms. My memory is that it is a notion that applies in the latter context, not the former. Regards, –  Matt E Jun 17 '12 at 4:47
Matt - sorry, yes you're right! I've got these two terms jumbled around in my head. Will edit. –  Tony Jun 17 '12 at 4:52
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1 Answer

If $M_k$ denotes the space of weight $k$ modular forms with coefficients in $\mathbb Z$, then there is an embedding $M_k \hookrightarrow \mathbb Z[[q]]$ given by taking $q$-expansion. This induces a map $$\bigoplus_k M_k \rightarrow \mathbb Z[[q]]$$ which is again an embedding (although not longer quite obviously so).

If we let $\widetilde{M}_k$ denote the image of $M_k$ in $\mathbb F_{\ell}[[q]]$, then there is similarly an induced map $$\bigoplus_k \widetilde{M}_k \rightarrow \mathbb F_{\ell}[[q]],$$ but this is no longer an embedding. If we denote its image by $\widetilde{M}$, then this is the ring of mod $\ell$-modular forms. It is the sum of the various $\widetilde{M}_k$s, but is not their direct sum.

Assuming $\ell \geq 5$, its kernel is generated by $E_{\ell} - 1$. Note that this is not a homogenous element of the source. Thus the image is not naturally graded. However, any graded ring is also naturally filtered --- in our case we filter the direct sum by the subobjects $\bigoplus_{i = 0}^k \widetilde{M}_i$ --- and the image of a filtered ring is naturally filtered --- in our case we define $\widetilde{M}_{\leq k}$ to be the image of $\bigoplus_{i = 0}^k \widetilde{M}_i$ in $\mathbb F_{\ell}[[q]]$.

Then $\widetilde{M}$ is the union of the $\widetilde{M}_{\leq k}$. We say that an element of $\widetilde{M}$ has filtration $k$ if it lies in $\widetilde{M}_{\leq k}$, but not in $\widetilde{M}_{\leq k-1}$.

So, regarding motivation: it is the what replaces the notion of weight for an element of $\widetilde{M}$. In short, if we are handed a $q$-expansion in $\mathbb F_{\ell}[[q]]$ and told that it is the $q$-expansion of a modular form mod $\ell$, the weight is not intrinsically determined by the $q$-expansion (unlike in the case with char. $0$ coeffients), since e.g. the $q$-expansion $1$ is the $q$-expansion of the wt. $0$ modular form $1$, the weight $\ell-1$ modular forms $E_{\ell -1} $, and more generally the weight $(\ell-1)i$ module forms $E_{\ell -1}^i$ for any $i$. But the filtration of the $q$-expansion is well-defined. When you sort it out, it is essentially the minimal weight of a modular form having that given $q$-expansion.

As for our more specific question: since the kernel of the $q$-expansion map is generated by $E_{\ell -1} - 1$, any non-zero element of $\widetilde{M}_k$ must have filtration congruent to $k$ mod $\ell - 1$. (In short, the grading mod $\ell -1$ is well-defined.) So if $k < \ell -1$ then any non-zero element of $\widetilde{M}_k$ must have filtration equal to $k$.

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