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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.

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I don't see why there should be. Over $k$, there's no reason to single out the $j$-invariant over $\frac{aj + b}{cj + d}$ for $ad - bc \neq 0$, is there? The singling out only occurs for integrality reasons and even then I don't think that implies the addition or multiplication is meaningful. – Qiaochu Yuan May 31 '11 at 14:40
up vote 11 down vote accepted

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.

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The $j$-invariant actually induces a bijection in all characteristics, you don't need characteristic greater than $3$. But as Qiaochu said, there's no reason why $j(E_1)+j(E_2)$ should have any significance. Working over $\mathbb{C}$, the space of elliptic curves is the space of lattices in $\mathbb{C}$, with isomorphism being the relation $L_1\sim L_2$ if there is a $c\in\mathbb{C}^*$ with $cL_1=L_2$.

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