The Mordell-Weil theorem states that for an abelian variety $A$ over a number field $K$ the group of $K$-rational points of $A$ is finitely generated and abelian.
What if $K$ is not a number field, e. g. Q$_p$ or a transcendental extension of $\mathbb Q$?
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FusRoDah
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For $K=\Bbb Q_p$ there will be uncountably many $K$-rational points. The exact structure of $K$-rational points will depend on the reduction type of the curve, but points close to the base-point $O$ are in essence parameterised by the formal group of the curve. See Silverman's book for much more detail.
Angina Seng
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What is the group structure if the variety is an elliptic curve? – FusRoDah Sep 03 '17 at 20:51
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@FusRoDah Over finite extensions of $\mathbf{Q}_p$ if $|j|_p > 1$ then see the Tate curve, the elliptic curve stays isomorphic to a ($p$-adic) torus : $L^\times / q^\mathbb{Z}$ for some finite extension $L/\mathbf{Q}_p,q \in L^\times$. This is chapter V of Silverman, Advanced Topics in the Arithmetic of Elliptic Curves – reuns Sep 03 '17 at 22:06