We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of mixed motives then we should have $$A(k)\otimes \mathbb{Q}\cong \text{Ext}^1_{\mathcal{MM}_k}(1,h^1(A)(1))$$ (unless I got the indices wrong). My question how we can describe the map $$A(k)\to \text{Ext}^1_{\mathcal{MM}_k}(1,h^1(A)(1)),$$ by which I mean there should be a representative of the Ext class in terms of the section $x\in A(k)$.
$\begingroup$
$\endgroup$
4
-
$\begingroup$ As far as I understand these matters, the extension class you are interested in should be an extension of 1-motives. It appears that "everything is known" about 1-motives; you may try to read the book of Barbieri-Viale and Kahn. Yet the answer may be rather tautological. $\endgroup$– Mikhail BondarkoCommented Mar 20, 2023 at 12:01
-
$\begingroup$ @MikhailBondarko I'm sorry, but why "should it be" an extension of $1$-motives as opposed to mixed motives? $\endgroup$– curious math guyCommented Mar 20, 2023 at 18:52
-
$\begingroup$ 1-motives form a very explicit and non-conjectural exact subcategory of conjectural mixed motives. $\endgroup$– Mikhail BondarkoCommented Mar 20, 2023 at 18:53
-
1$\begingroup$ I think the idea should be that $\mathrm{Ext}^{1}(A,\mathbb{G}_{m})\otimes\mathbb{Q}\simeq\mathrm{Ext}^{1}_{\mathcal{MM}_{k}}(\mathbb{Q}(-1),h^{1}(A))\simeq\mathrm{Ext}^{1}_{\mathcal{MM}_{k}}(\mathbb{Q},h^{1}(A)(1))$, and $\mathrm{Pic}^{0}(A)\simeq\mathrm{Ext}^{1}(A,\mathbb{G}_{m})$. Note that this also gives $\mathrm{Hom}_{\mathcal{MM}_{k}}(\mathbb{Q},h^{2}(A)(1))\simeq\mathrm{NS}(A)_{\mathbb{Q}}$. $\endgroup$– Oli GregoryCommented Mar 23, 2023 at 9:27
Add a comment
|