Let $k$ be a finitely generated field of positive characteristic p. Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural action of $Gal(k^{sep}|k)$ over $T_p(A)$ is semisimple?
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$\begingroup$ mathoverflow.net/questions/275355/… $\endgroup$– user19475Commented Jun 29, 2018 at 10:46
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$\begingroup$ @TKe thank you for your answer. In that answer, k is a finite field, while here k is a finitely generated field. For the moment i'm not able to deduce the case when k is finitely generated from the situation when k is finite. $\endgroup$– Emiliano AmbrosiCommented Jun 29, 2018 at 15:46
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$\begingroup$ I tried to see if one could do something following either (1) Deligne's proof of semisimplicity of geometric monodromy for ell-adic etale cohomology or (2) the Arakelov-Faltings proof of the Tate conjecture for endomorphisms of abelian varieties over global fields. I don't think (1) works because the p-adic analogue would be in the setting of crystalline cohomology or something but the relationship between crystalline monodromy and monodromy of the p-adic Tate module is not obvious. But (2) might work. $\endgroup$– Will SawinCommented Dec 12, 2021 at 17:47
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$\begingroup$ Given a subspace of the p-adic Tate module which, consider the quotient of the abelian variety by the $p^n$-torsion image of that subspace for all $n$. If these are isomorphic for infinitely many $n$ then I think you can check that the subspace is complemented. Then maybe you can bound the height and prove finiteness and deduce that infinitely many are isomorphic. $\endgroup$– Will SawinCommented Dec 12, 2021 at 17:51
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