# Semisimplicity of étale cohomology representations

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.

• I strongly suspect that the answer is "no" in the number field case, and it is surely "no" over finite fields (already). Feb 23, 2011 at 10:54
• 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? Feb 23, 2011 at 12:09
• Sorry, just to be sure: Do you mean "not known" or "false" in the case of a finite ground field? Feb 23, 2011 at 12:21
• 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. Feb 23, 2011 at 12:32
• 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.) Feb 23, 2011 at 15:56

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.
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.