I have a problem of some fundamental definitions of an elliptic curve over a field $K$.
An elliptic curve $E$ is the graph of an equation of the form (the so called generalized Weierstrass equation) $y^2=a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$, see Washington (2008). Its coefficients lie in the field $K$: $a_i\in K$ $(i=1,2,3,4,6)$, see also Silverman (2009). Silverman also says $E$ is a subset of the squared projective space: $E\subset\mathbb{P}^2$.
My understanding is that $E$ is allowed to contain points with coordinates in some larger field $L\supseteq K$ and we denote this point set $E(L)=\{O\}\cup\{(x,y)\in L^2:y^2=a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\}$. This (large) set of points forms an abelian group, where the point $O$ at infinity acts as the identity element.
My understanding is that the set of $K$-rational points $E(K)$ is a subgroup of $E(L)$.
Now my confusion: As more I read literature, I see that an elliptic curve over $K$ comprises only points whose coordinates lie in $K$. For example Wiki writes "An elliptic curve is defined over a field $K$ and describes points in $K^2$, the Cartesian product of $K$ with itself."
Is the relevant trick here that Silverman says $E\subset\mathbb{P}^2$ and not $E\subset\mathbb{P}^2(K)$? And the wiki definition is a bit too strong?
I would be very grateful for a brief help in separating the definitions.
Best Answer
If you're following Silverman, a bit of a cheat sheet is as follows.
We have $E/K$ denoting the underlying elliptic curve- naively this is "$E$, which is cut out in some projective space by some equations, and a marked $K$-point $O$", or even more simply you could take it to be "the curve cut out in $\mathbb{P}^2$ by a Weierstrass equation with the understanding that $(0:1:0)$ is $O$".
Suppose that $L/K$ is an extension of fields. He uses $P \in E(L)$ to denote a $L$-rational point on $E$ (i.e., the defining equations for $E$ vanish at $P$, and the coordinates of $P$ are in $L$).
Where he says $P \in E$ he means that $P$ is a geometric point, i.e., $P \in E(\bar{K})$.
To address your confusion. You say "as I read more, I see an elliptic curve over $K$ comprises only of points over $K$". This is definitely not the case, that is the $K$-points on an elliptic curve over $K$. The curve itself "is" the equations and the point $O$ - which in turn describe an awful lot more information (e.g., the points over every extension $L/K$).
Of course, different authors will have different conventions with respect to these things.