Difference between $E(\overline{K})$ and $E$ for elliptic curve $E/K$

Question

Let $K$ be a field, and consider an elliptic curve $E/K$. Let $\overline{K}$ be algebraic closure of $K$. It is common to abbreviate or identify $E(\overline{K})$ with $E$.

Can this abbreviation or identification be explained in the context of scheme theory or category theory?

Your insights would be greatly appreciated.

Answer 1

I'll try my best to give a more elaborate answer to a question which I believe diracdeltafunk has already tried and answered in the comments.

Firstly, I don't really think you need an elliptic curve in your question, because, for any variety (i.e. nice scheme) $X$, over a field $k$, the space $X(\bar{k})$ has most of the information need. To understand this, I cite a remark made in [Milne, 1.19]:

Let $X$ be an algebraic scheme over $k$ in the sense of EGA, and let $X_0$ be the set of closed points. The map $S\mapsto S\cap X_0$ is an isomorphism from the lattice of closed (resp. open, constructible) subsets of $X$ onto the lattice of similar subsets of $X_0$... Thus, the ringed spaces $(X,\mathcal{O}_X)$ and $(X_0,\mathcal{O}_X|_{X_0})$ have the same lattice of open subsets and the same $k$-algebra for each open subset; they differ only in the underlying set... The functor $(X,\mathcal{O}_X)\rightarrow (X_0,\mathcal{O}_X|_{X_0})$ is an equivalence from the category of algebraic schemes over $k$ to the category of ultraschemes over $k$.

Some personal remarks I have on this are:

1. Honestly, I don't know if I have seen anyone identify an elliptic curve with its $\bar{k}$-points (does Silverman do it? I don't remember anymore), but as we have just seen, it is completely harmless to do so. In relation to this, the first thing you should probably check is that the points of $X(\bar{k})$ are exactly the closed points in $X_{\bar{k}}$. I'm guessing they want the algebraic closure because a lot of properties are best checked after base change to $\bar{k}$.

2. My reference is his notes on reductive groups, which does not prove this statement. But, I am guessing (although I am not sure) that his notes on Algebraic geometry does prove it, because Milne loves to work with these "ultraschemes" in his different notes.

3. I would strongly recommend you read the entirety of 1.19, because it tells you just how strong this identification can be, and that lots of properties can actually be checked on these ultraschemes.