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?
Elliptic curve terminology confusion
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terminology
elliptic-curves
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0Even 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. – 2012-12-20