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.