Consider an abelian variety $A$ over a number field, and look at the representation of its MumfordTate group on $H^1(A)$, restricted to the commutator subgroup. Is it possible that every element of this commutator subgroup preserves some element of $H^1(A)$, without this representation containing a copy of the trivial representation (i.e. for it to look like the standard representation of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$)?
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