11
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I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{p,q}( \overline{\mathcal M}_{g,n})\neq 0$, $p\neq q$, and $p+q$ is small, or to finding non-algebraic cohomology classes in $H^i(\overline{\mathcal M}_{g,n},\mathbb Q)$ for small $i$.

My question is: What's the smallest weight / degree that can occur?

For $g=0$, this never happens. For $g=1$, the cohomology comes from modular forms of level $1$, and the lowest weight one is weight $11$, arising from the modular form $\Delta(q)$. For $g\geq 2$, I don't know if you can do better than $11$.

I'd also be intersted in knowing about especially low weight Galois representations that don't arise from classical modular forms, even if their weight is larger than $11$. I think the lowest one I know about is weight $19$ on $\overline{\mathcal M}_{2,16}$ from a Siegel modular form. Should I expect the weights to keep increasing as $g$ goes to infinity, so that these are the smallest ones?

I became interested in these because they are natural examples of everywhere unramified Galois representations arising from geometry, and the lowest weight ones seem like the simplest in some sense. I'd also like to know the answer purely to get a better intuition about the cohomology of these spaces.

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  • $\begingroup$ I think that Gaëtan Chenevier is thinking about these things as well, you could ask him directly. $\endgroup$ Jul 13, 2014 at 9:03
  • $\begingroup$ math.polytechnique.fr/~chenevier/levelone.html If I read the tables correctly, for $g=3$ the answer is weight $23$, and for $g=4$ is weight $25$. $\endgroup$ Jul 13, 2014 at 10:01
  • $\begingroup$ @NAME_IN_CAPS is that the moduli space of curves, or abelian varieties? For $g>3$ they're substantially different. I think only the second one is directly connected to automorphic forms. But I may be wrong? $\endgroup$
    – Will Sawin
    Jul 13, 2014 at 14:05
  • $\begingroup$ Linear in $g$ is the range of dimensions where $\overline{\mathcal M}_{g,n}$ has the same cohomology as ${\mathcal M}_{g,n}$, as is the range of dimensions that is generated by tautological classes. So, yes, $i$ must increase with $g$. $\endgroup$ Jul 13, 2014 at 18:40
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    $\begingroup$ I'm late to the party. Will, you probably know this already, but it is conjectured that $11$ is in fact the optimal bound on the weight of an effective motive which is everywhere unramified. @BenWieland It is in fact not true that $H^k(M_{g,n}) \cong H^k(\overline M_{g,n})$ for $g \gg k$. A paper of Pikaart shows e.g. that $H^{33}(\overline M_{g,n}) \neq 0$ for all $g \gg 0$. $\endgroup$ Jan 29, 2018 at 8:18

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