Finite Flat Group Schemes for Modular Forms of Higher Weight

Question

Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, hence the local representation at $\ell \nmid N$ arises from an $\ell$-divisible group, or finite flat group scheme of order $\ell^2$ if we look at the mod $\ell$ reduction. My understanding is that this is true in the case $k>2$, but I'm having trouble seeing why or finding a reference. I understand the construction of the representation using $H^1_{\acute{e}t}(X,\mathrm{Sym}^{k-2}(R^1_{\acute{e}t}f_* \underline{\mathbb{Z}_\ell}))$.

The reason for this is to use Raynaud's results to get a good local picture of the mod $\ell$ representation at primes dividing $\ell$.

I imagine one could just use the crystalline property in p-adic Hodge theory, but I'd like to see it in the more elementary way (and I understand that better).

Answer 1

David, the $\ell$-adic representation doesn't come from an $\ell$-divisible group if $k>2$. This can be seen using $p$-adic Hodge theory: The Hodge-Tate weights of the $\ell$-adic representation attached to a modular of forms of weight $k$ are $0$ and $k-1$ (or $0$ and $1-k$, depending or you sign convention). On the other hand, the Hodge-Tate weights of the representations attached to an $\ell$-divisible group are $0$ and $1$: cf. for instance the section 3 of http://math.uchicago.edu/~lxiao/files/notes/p-Divisble%20Groups.pdf

So there is no way to avoid using p-adic Hodge theory and all the Fontaine's stuff if $k>2$.