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?