Let $E_1$ and $E_2$ be two ordinary elliptic curves. It is well known that the group $Home(E_1,E_2)$ is a $\mathbb{Z}$ module of rank at most four. In the case where $E_1=E_2$, this module has rank two. Could we say anything about the rank of this module in general cases?
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5$\begingroup$ What do you call the general case? If $E_1$ and $E_2$ are general, there is no isogeny betwwen them. $\endgroup$ – abx Apr 18 '16 at 14:25

$\begingroup$ I mean if they are isogenous what can we say about the number of generators of $Hom(E_1,E_2)$. Is it can be greater than two? $\endgroup$ – somayeh didari Apr 20 '16 at 7:27

1$\begingroup$ If they are isogeny, since you want the number of generators, is the same as taking $E_1 = E_2$ (just compose with the dual isogeny), so no. The rank is either 0 or 2 in the ordinary case $\endgroup$ – A. Pacetti Apr 25 '16 at 16:10