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Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and forgetting points.

If we look at the induced system on $l$-adic cohomology, we obtain some coherent systems of Gal($\bar{\mathbb{Q}}/\mathbb{Q}$) representations.

If we expect these Galois representations to be modular, where would we find the corresponding coherent systems of automorphic representations? Would the expected relations among them make it easier to identify or construct them?

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    $\begingroup$ Even better, you can add level structure... $\endgroup$ Apr 6, 2018 at 2:28
  • $\begingroup$ These Galois representations will not all come from Siegel modular forms. Indeed for $g=3$ Galois representations associated to Teichmuller modular forms start appearing in the cohomology of $M_{3,n}$ for sufficiently high $n$. I don't think these are automorphic forms or fit into any kind of automorphic framework, but I am not sure. Moreover, I don't know of any way to do Hecke operators for $M_{g,n}$. Even if we take $n=0$ and map to $A_g$ then (the image of) $M_g$ is not preserved by the Hecke operators. $\endgroup$ Apr 11, 2018 at 10:08

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