Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the formula $g.x=\chi(g)^nx$ where $\chi:G\to \mathbb{Z}_\ell^{\times}$ is the cyclotomic character.

Question: Is it true that the ext groups $\mathrm{Ext}^*(\mathbb{Z}_\ell(0),\mathbb{Z}_\ell(n))$ vanish for $n$ negative ?

The reason I am asking this is that this would imply that the triangulated subcategory of the derived category of Galois representations spanned by the objects $\mathbb{Z}_\ell(n)$ has a weight filtration, as constructed for instance in Lemma 1.2. of this paper of Marc Levine:



No, life is not so easy I'm afraid. For instance, the group $\mathrm{Ext}^1_{G_{\mathbf{Q}}}(\mathbf{Z}_\ell, \mathbf{Z}_\ell(n)) = H^1(\mathbf{Q}, \mathbf{Z}_\ell(n))$ has positive rank for all odd integers $n$, whatever the sign; this is easy to see from Tate's global Euler characteristic formula.

The point is that if you have two irreducible Galois representations $V_1, V_2$, and you know that $V_1$ and $V_2$ arise in geometry (as the realisations of motives $M_1, M_2$), then there are in general many more extensions of $V_1$ by $V_2$ in the category of Galois reps than there are extensions of $M_1$ by $M_2$ in the category of mixed motives.

But all is not lost: there is a beautiful and deep theory that seeks to characterise in terms of local properties at $\ell$ those extensions which come from geometry. You might like to read Bloch and Kato's article in the Grothendieck Festschrift. The upshot is that for a geometric Galois representation $V$, one defines a group $H^1_\mathrm{f}(K, V) \subseteq H^1(K, V)$, which parametrises those extensions of the trivial rep by $V$ which are expected to arise in geometry. It is expected that $H^1_\mathrm{f}(\mathbf{Q}, V)$ is zero if the Hodge--Tate weights of $V$ are $\le -1$, and this is known for the representations $V = \mathbf{Q}_\ell(n)$ by a theorem of Soule.


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