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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 ...

Let $p$ be an odd prime and $\mu_{p}$ be the group of $p^\text{th}$-roots of unity. Then, there exists a cup-product map which maps the product of $p$-copies of $H^{1}(\mathbb{Q}, \mu_{p})$ into $H^{p}...