This question is inspired by the question: [Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?](https://mathoverflow.net/questions/104627/example-of-non-projective-variety-with-non-semisimple-frobenius-action-on-etale)


Let $K$ be a number field (or finitely generated field of characteristic $0$). If I am not mistaken, it is not expected for general varieties to have semisimple Galois representations on their $\ell$-adic cohomology.



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> Is there a known example of a variety $X/K$ such that the Galois representation on the $\ell$-adic cohomology is not semisimple?
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If $X$ is not proper, I guess we should work with cohomology with compact support.


Two subquestions:



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> * Are there examples where $X$ is smooth?
> * Are there examples where $X$ is proper?
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Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology
Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a nontrivial class in $H^1(K, H^1(E\_{\bar K}, \mathbf{Q}\_\ell)(1))$ and thus corresponds to a non-split extension of $H^1(E\_{\bar K}, \mathbf{Q}\_\ell)(1)$ by $\mathbf{Q}\_\ell$; and this non-semisimple Galois representation is realized in the etale cohomology of the (smooth, non-proper) variety $E - \{x, \infty\}$.


EDIT: There are two ways of seeing that we obtain a class in $H^1(K, H^1(E\_{\bar K}, \mathbf{Q}\_\ell)(1))$. On the one hand, identifying $H^1(E\_{\bar K}, \mathbf{Z}\_\ell)(1)$ with the $\ell$-adic Tate module $T\_\ell(E)$, this is simply the Kummer map (the inverse limit of the boundary maps associated to the sequence $0 \to E[\ell^n] \to E(\overline{K}) \to^{\times \ell^n} E(\overline{K}) \to 0$). The more high-powered approach is that we have an etale cycle class 
$$ CH^1(E) \to H^2\_{et}(E, \mathbf{Q}\_\ell(1)) $$
and the composite with the edge-map of the Hochschild--Serre exact sequence, going into $H^2\_{et}(E\_{\overline{K}}, \mathbf{Q}\_\ell)(1))^{G\_K} = \mathbf{Q}\_\ell$, is just the degree of the divisor, which is 0; so one lands in the next step of the filtration of $H^2\_{et}(E, \mathbf{Q}\_\ell(1))$ induced by the spectral sequence, which is 
$ H^1(K, H^1(E\_{\bar K}, \mathbf{Q}\_\ell(1)))$.


To see how this class is manifested in the cohomology of the open variety $E \setminus Z$, where $Z = \{x, \infty\}$, consider the excision exact sequence
$$ 0 \to H^1(E\_{\bar K}) \to H^1(E\_{\bar K} - Z\_{\bar K}) \to H^2\_Z(E\_{\bar K}) \to H^2(E\_{\bar K}) \to \dots $$


The cycle class of the divisor $\{x\} - \{\infty\}$ gives an element of $H^2\_Z(E\_{\bar K})(1)$ whose image in $H^2(E\_{\bar K})(1)$ is trivial, so by pullback we get a short exact sequence
$$ 0 \to H^1(E\_{\bar K})(1) \to W \to \mathbf{Q}\_\ell \to 0 $$
where $W$ is a subspace of $H^1(E\_{\bar K} - Z\_{\bar K})(1)$. This gives a geometric realization of the extension of Galois representations associated to the class in $ H^1(K, H^1(E\_{\bar K}, \mathbf{Q}\_\ell(1)))$ coming from the spectral sequence.


I've worked with non-compact supports here, but everything is smooth of dimension 1, so the compactly-supported cohomology $H^i\_c$ is just the dual of the non-compact $H^{2-i}$, by Poincare duality.


