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Let $R$ be a complete discrete valuation ring, with fraction field $L$. Let $E$ be an elliptic curve over $L$, and $W$ a minimal Weierstrass model over $R$. Why is $W(R) \simeq E(L)$? We have a map $W(R) \to W(L) \to E(L)$, and my question is why is it surjective. Given an $L$-point of $E$, we get an $L$-point of $W$, and how do I get an $R$-point of $W$?

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The surjectivity of this map follows from $E$ being proper (the "valuative criterion" of properness).

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    $\begingroup$ Concretely, if you want to specifically construct the lift for a projective variety using equations, given a $\mathbb{Q}_p$ point in projective space, we can always multiply the homogeneous coordinates by an appropriate multiple of $p$ until all coordinates are integers and at least one is a unit. $\endgroup$ – hunter Oct 11 '18 at 13:09
  • $\begingroup$ Oh, yeah. Would the same argument involving homogenous coordinates work in global case, such as going from a $\mathh{Q}$-point to a $\mathbb{Z}$-point? There I don’t have valuative criteria. $\endgroup$ – usr0192 Oct 11 '18 at 13:14

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