# Unramified Galois representations not from smooth and proper stacks

Are there any irreducible representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ over $\mathbb Q_l$ that are unramified away from $l$ and crystalline at $l$ but are not known to arise from the etale cohomology of smooth and proper stacks over $\operatorname{Spec}\mathbb Z$?

Thee simplest examples of unramified crystalline Galois representations known, other than powers of the cyclotimc character, are the two-dimensional Galois representations arising from level $1$, weight $n+1$ modular forms. These live inside the etale cohomology of the stack $\overline{M}_{1,n}$, which is indeed smooth and proper. More examples come from other $\overline{M}_{g,n}$, but these are the only examples I am aware of!

The situation for schemes is very different where, as far as I can tell, no Galois representation other than powers of the cyclotomic character is known to arise from smooth and proper schemes over $\mathbb Z$.

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You probably want to restrict to irreducible representations, since otherwise there are simply too many Gal reps to come from any kind of geometry, just for cardinality reasons. –  David Loeffler Feb 16 '13 at 20:09
Alright, fixed. –  Will Sawin Feb 16 '13 at 20:26
When I think of Galois representations that are not known to be motivic, I think of those automorphic ones constructed as $p$-adic limits. The fact that you want "no" ramification is imposing a "high" weight condition on the automorphic representations involved. So, maybe one could find some automorphic representations for GL(n) (n>2). –  Rob Harron Feb 17 '13 at 0:13

Here's a class of examples which isn't in your list as far as I know. Take an even unimodular lattice, e.g. the $E_8$ lattice. The corresponding orthogonal group is a reductive group over $\mathbf{Q}$ which is split at every finite place and compact at $\infty$ (see Gross, "Groups over $\mathbf{Z}$", Inventiones 124 (1996)). There will be lots of algebraic automorphic representations of this group $G$ if you take the weight large enough, and these will give you automorphic representations of $GL(n)$ which are self-dual and unramified at all finite places. It's known that these have Galois representations attached, which will be crystalline at $\ell$ and unramified everywhere else; but I don't think these are known to come from smooth proper stacks over $\mathbf{Z}$.