What Is The Definition Of The J-invariant Of An Elliptic Curve?

Definition of the j-invariant of an elliptic curve

• It seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of course, for moduli reasons, it is clear why one might want such an invariant, but the actual formula has always seemed quite mysterious to me. Does anyone know of a nice self-contained source that explains the definition of the j-invariant?

• Answer:

  Actually, Ravi Vakil's notes give a great reason (modulo the strange constant out front). This explanation is given somewhere in the Foundations of Algebraic Geometry notes here http://math.stanford.edu/~vakil/preprints.html#coursenotes. I'm just going off memory, so don't attribute any errors I make to him. Notice that once you prove that every elliptic curve has an affine model given by $y^2=x(x-1)(x-\lambda)$, then you know the $j$-invariant has to be independent of the different $\lambda$ you get by permuting. You can just explicitly work it out that the six choices of lambda then are $\lambda, \frac{1}{\lambda}, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda}{\lambda -1}, \frac{\lambda -1}{\lambda}$. Some obvious first choices for an invariant with respect to all of these is to multiply them all together. You get $1$, oops, that isn't a good invariant since it not only doesn't depend on the choice of $\lambda$, but also is independent of curve. So you could also try adding them all, oops again, they come in pairs each adding to $1$, so they add to $3$. So let's try the next best thing which is to sum the squares. If you check, this is exactly the $j$ invariant (but without the constant which is stuck in for characteristic $2$ reasons).