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I am wondering if there is some accessible reference to learn about product of elliptic curves and their 'properties'. For dimension 1, there is plenty to find. I think the dimension 2 case is done as follows:

Start with an elliptic curve over a field $K$, given by $$f(X,Y)=Y^2+a_1XY+a_3Y-X^3-a_2X^2-a_4X-a_6=0.$$ We know how the coordinate ring $K[E]$ and the funcion field on $E$ are defined. Now I would like to define, for instance, the coordinate ring on $E\times E$. This would be just $$\frac{K[X_1,Y_1,X_2,Y_2]}{(f(X_1,Y_1),f(X_2,Y_2))},$$ I assume.

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    $\begingroup$ What about products of elliptic curves would you like to find out? $\endgroup$ Dec 5, 2011 at 19:17
  • $\begingroup$ Good question. I would like to understand some properties on $K(E^n)$, the funcion field on $E^n$. The divisor of an element of $K(E^n)$, ... $\endgroup$
    – Nadori
    Dec 5, 2011 at 21:21

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Answered by comment from QiL.

You only get affine parts of an elliptic curve and of products of elliptic curves. The function field of $E^n$ is just $K(E)\otimes_K ...\otimes_K K(E)$

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