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Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.

Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is semisimple. This is the famous theorem of Faltings (Invent. Math. 73).

Now let $X/K$ be a smooth projective variety and $0\le q\le 2\dim(X)$, and define $\overline{X}=X_{\overline{K}}$.

Question. Is it known that the representation of $G$ on $H^q(\overline{X}, \mathbb{Q}_\ell)$is semisimple?

Remark. The answer is yes for $q=1$, because $H^1(\overline{X}, Q_\ell)$ is dual to $V_\ell(A)$ where $A$ is the Albanese variety of $X$.

I would also be interested in the case where the number field $K$ is replaced by a global function field (say), and $\ell$ is assumed to be coprime to the characteristic.

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  • $\begingroup$ I strongly suspect that the answer is "no" in the number field case, and it is surely "no" over finite fields (already). $\endgroup$
    – Mikhail Bondarko
    Feb 23, 2011 at 10:54
  • $\begingroup$ Thx for your comment! I somehow expected a "not known" in the number field case as well. Why is the answer a definite "no" over finite fields? I do not know how to prove this. Can you give me a view details on that? $\endgroup$
    – Sebastian Petersen
    Feb 23, 2011 at 12:09
  • $\begingroup$ Sorry, just to be sure: Do you mean "not known" or "false" in the case of a finite ground field? $\endgroup$
    – Sebastian Petersen
    Feb 23, 2011 at 12:21
  • $\begingroup$ One more comment: My question is exactly conjecture $SS^i(X)$ in Tate's article "Conjectures on algebraic cycles on l-adic cohomology", Proceedings of Symposia in Pure Mathematics 55 (the motives volume I). So my question is, whether there has been progress on this conjecture since this article was written. $\endgroup$
    – Sebastian Petersen
    Feb 23, 2011 at 12:32
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    $\begingroup$ Joel Bellaiche's Hawaii notes people.brandeis.edu/~jbellaic/BKHawaii4.pdf (page 5) say: "This is sometimes called “conjecture of Grothendieck-Serre”. This is known for abelian varieties, by a theorem that Faltings proved at the same times he proved the Mordell’s conjecture, and in a few other cases (some Shimura varieties, for example)." (But that's all he says.) $\endgroup$
    – fherzig
    Feb 23, 2011 at 15:56

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This semi-simplicity is a part of what is called the Tate conjecture. It is generally believed to be true, but little is known about it outside the case of $H^1$, in either the finite field or global field case. Searching on mathscinet for "Tate conjecture" (or googling) should turn up the relevant literature.

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In https://arxiv.org/pdf/1709.04489.pdf, Moonen proves that for finitely generated fields of characteristic $0$, the Tate conjecture (surjectivity of the cycle class map $\mathrm{CH}^r(X) \otimes \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell)^{G_K}$) implies the semisimplicity conjecture.

(For the Tate conjecture, see e.g. http://www.math.columbia.edu/~chaoli/docs/TateConjecture.html or Tate's article in the Motives volume.)

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