Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of characteristic 0, and $A$, $B$ are of finite dimension). Let us further suppose that $A$, $B$ ''come from geometry'' (of course, there are much well-defined notion of geometric Galois representations, but, at the moment, I do not want to dive into there. But, I usually regard ''come from geometry $\sim$ motivic''). Here is the question

Let $E$ be an extension of $A$ by $B$ in the category of $G_K$-representations. Then does $E$ also come from geometry?

Since I did not give the precise meaning of ''come from geometry'', so it is welcome to start with your definition. Thank you very much in advance.

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    $\begingroup$ Conjecturally, the galois representations on $H^i(X_{\bar k}, \mathbb{Q}_\ell)$ are semisimple if $X$ is smooth and proper (this is part of the Tate conjecture). So if by "comes from geometry," you mean "is a subquotient of the cohomology of a smooth proper variety," the answer is "no" unless $E$ is the trivial extension, on the Tate conjecture. If you allow singular/non-proper varieties, there are still restrictions on the allowable extensions, arising from e.g. $p$-adic Hodge theory. $\endgroup$ – Daniel Litt Aug 9 '18 at 2:41
  • $\begingroup$ @DanielLitt Thank you very much, could you give me any reference on the last part of your comment? I mean, "there are still restrictions on the allowable extensions, arising from ...." $\endgroup$ – User0829 Aug 9 '18 at 2:50
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    $\begingroup$ The $p$-adic \'etale cohomology of any algebraic variety is always "de Rham" and hence "Hodge-Tate" at primes above $p$ -- if you google those phrases, along with "p-adic hodge theory" you should find some Galois cohomology computations showing that these properties are not closed under arbitrary extensions. IIRC there are extensions of the trivial representation by itself which are not de Rham. $\endgroup$ – Daniel Litt Aug 9 '18 at 3:01
  • $\begingroup$ Indeed, Brinon-Conrad's $p$-adic Hodge theory notes demonstrate a non-HT extension of the trivial one dimensional character by itself (which is, of course, geometric). $\endgroup$ – RKD Aug 15 '18 at 21:37
  • $\begingroup$ @Ravi: If I recall rightly, Brinon-Conrad only construct such an extension as a $G_{\mathbb{Q}_p}$-rep; strictly speaking, the question asks for an example over a number field, which probably requires $\epsilon$ more technique. $\endgroup$ – Daniel Litt Aug 21 '18 at 3:57

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