Let $k$ be an algebraically closed of characteristic $\neq 2,3$. The $j$-invariant induces a bijection


$\{\text{elliptic curves over } k\}/\cong \longrightarrow k.$


Perhaps this is a silly question: I'm curious if there is any meaningful geometric interpretation of the induced ring structure on the set of isomorphism-classes of elliptic curves. The resulting operations can be written down in Weierstrass forms, but the formulas are, of course, not illuminating at all. The zero element is $y^2+y=x^3$. It would be even more interesting what structures there are on the category of elliptic curves over $k$, so without modding out isomorphisms.



Structure on the set of elliptic curves via $j$-invariant
The normalization of the j-invariant is a historical accident: it was chosen to have a zero of order 3 at a cube root of 1. There is no really good reason for this: one could equally well normalize it to have a double zero at i, or a zero at some integer of an imaginary quadratic field of class number 1. So it is unlikely that the ring structure on isomorphism classes of elliptic curves has any meaning. In any case j is not really a bijection: there is really only 1/2 or 1/3 of an elliptic curve with j invariant 1728 or 0. One can mumble something about stacks at this point.


