# Structure on the set of elliptic curves via $j$-invariant

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

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 $cL_1=L_2$.