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Let $E$ be an elliptic curve over $\mathbb{Q}$. Then it has a model $W$ satisfying $W(\mathbb{Z})=E(\mathbb{Q})$. This means $E$ only has integer points. But this is not true. What went wrong?

Mikko Pitkonen
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    What do you call integer points for $E$? Solutions $(x,y)$ where $x,y$ are integers? But that’s not what the existence of $W$ says… Note that, by the valuative criterion, any proper integral $\mathbb{Z}$-scheme $S$ should verify $S(\mathbb{Z})=S(\mathbb{Q})$. – Aphelli Oct 10 '22 at 07:46
  • Great. I see it now. – Mikko Pitkonen Oct 10 '22 at 09:34

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