# Rational points on elliptic curve over a local field correspond to integral points on minimal Weierstrass model

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 $$W$$?

The surjectivity of this map follows from $$E$$ being proper (the "valuative criterion" of properness).
• 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. – hunter Oct 11 '18 at 13:09
• 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. – usr0192 Oct 11 '18 at 13:14