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This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.

In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any motivic local system of rank $2$ (on a curve over $\mathbb{C}$) with infinite monodromy must be a direct factor of a family of abelian varieties $p : \mathcal{A} \to X$. (By this I think the author means that it should be a direct factor (as a local system) of $R^1 p_* (\mathbb{C}_{\mathcal{A}})$.)

No reason is given for this claim and I'd be grateful for an explanation (or any other comments/suggestions). In Theorem 1.1 the local system is in fact an $\mathrm{SL}_2$-local system and this assumption might be needed for the statement to be true.

Edit: The determinant of any motivic local system must have finite monodromy, so any rank $2$ local system becomes an $\mathrm{SL}_2$-local system over a finite cover. I now think that the claim is probably false.

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    $\begingroup$ This follows from the arguments for Theorem 5 and Corollary 4.9 in Simpson's numdam.org/article/PMIHES_1992__75__5_0.pdf The local system there is required to be rigid, but this is used only to conclude that it and all of its Galois conjugates admit a complex VHS, the property which is satisfied by motivic local system. The key point of the proof is that Griffiths transversality allows you to pin down the a priori arbitrary Hodge type of the VHS as having weights 0,0 or 0,1. One then applies the observation that a polarized Z-VHS of weights 0,1 comes from an abelian scheme. $\endgroup$
    – SashaP
    Commented Jan 28, 2023 at 12:33
  • $\begingroup$ Thanks a lot for the reference! $\endgroup$
    – naf
    Commented Jan 30, 2023 at 2:13

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