What's the difference between $E(K)$ and $E/K$ in elliptic curves?

Question

For a field $K$, sometimes I see elliptic curves denoted as $$E(K) : y^2 = x^3 + ax+b$$ and other times as $$E/K :y^2 = x^3 + ax+b$$

What is the difference? I believe that $E/K$ denotes that the coefficients $a,b$ are pulled from the field $K$ only, whereas in $E(K)$ they may be pulled from the algebraic closure of $K$. Is this correct? I.e. if $\bar{K}$ is the algebraic closure, then $E(K) = E/\bar{K}$ ? Also, is the $E/K$ notation at all related to the notation used for extension fields or quotient groups?

Answer 1

The notation $E/K$ means that $E$ is an elliptic curve defined over the field $K$.

The notation $E(K)$ refers to the group formed by the points on $E$ with coordinates in the field $K$.

Answer 2

$E/K$ means that $E$ is, depending on how sophisticated you want to get, a variety or a scheme defined over $K$. What this means exactly again depends on how sophisticated you want to get. The simplest way to say it is that it makes sense to consider the set of solutions $E(L)$ to the defining equations of $E$ over, not just $K$, but over all field extensions $L$ of $K$, in particular over the algebraic closure $\overline{K}$. (This is a simplified version of the functor of points approach to algebraic geometry.)