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?

  • 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, 2016 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? $\endgroup$ Apr 20, 2016 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, 2016 at 16:10


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