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I've been reading a paper that says "Let $E(K)$ be an elliptic curve..." where $E(K)$ means the $K$-rational points of $E$ (where $K$ is a number field). I've seen phrases like "Let $E/K$ be an elliptic curve" and I know that one considers $E(K)$ to be a group and such. But why can I say $E(K)$ is an elliptic curve?

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This seems imprecise to me. –  Qiaochu Yuan Dec 20 '12 at 0:58
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This of course depends on the author, but it is my experience that authors with a reputation for being incredibly careful with notation and terminology would always use $E(K)$ to mean the set of $K$-rational points, i.e. it is shorthand for the functor of points evaluated on a particular variety: $h_E(Spec(K))$. Thus if $L/K$ is an extension it doesn't make sense to talk about the $L$-valued points of $E(K)$ (since it is just a set with no extra structure) which should make sense if $E(K)$ is an elliptic curve. Given the set $E(K)$ there is no natural way to recover $E/K$ the elliptic curve. –  Matt Dec 20 '12 at 1:29
    
@Matt In the vast majority of cases, nothing is recoverable from just knowing the set! But you would have a more than decent chance of recovering $E$ if you had $E (K)$ as an algebraic variety (in the classical sense) and $K$ an algebraically closed field... –  Zhen Lin Dec 20 '12 at 16:43
    
Even authors who are sloppy with notation usually don't write $E(K)$, they just write $E$ and forget to mention what $K$ is. The notation $E(K)$ is designed for people who want to be precise, so it is particularly sad to misuse it this way. –  David Speyer Dec 20 '12 at 17:42

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