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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.

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) 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 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.

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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naf
naf
  • 10.5k
  • 1
  • 45
  • 63

Rank $2$ motivic local systems on a curve

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) 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 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.