# The cardinality of the preimage of a point under a nonzero isogeny equals the separable degree of the isogeny

Let $f:E_1\rightarrow E_2$ be a nonzero isogeny between elliptic curves.

Take a point $Q \in E_2$.

I am looking for a reference to a proof, or a proof, of the following fact:

$|f^{-1}(Q)|=\text{deg}_s(f)$

where $\text{deg}_s(f)$ is the separable degree of $f$.

In fact, it's enough to show that this is true for all but a finite set of points $Q$. The desired claim would then follow because $f$ is a group homomorphism.

EDIT:

1. I'm working over an algebraically closed field.
2. If you prefer to, you may assume that $f$ is a separable isogeny.
• (1). Isogenies of elliptics curves are always non-zero. (2). You need to work over an algebraically closed field.(3) You can easily reduce to the case $f$ is separable. – Cantlog Jun 3 '14 at 13:46
• You can check III.4 of Silverman's "The Arithmetic of Elliptic Curves", or II-6.8 of Harthsorne's "Algebraic Geometry" . – DonAntonio Jun 3 '14 at 13:53
• @DonAntonio: III.4.10 in Silverman refers to II.2.6b in Silverman, which refers to II.6.8 in Hartshone. So, the part of the proof I'm interested in is probably in Hartshone's book. I'll grab it now. – Olnek Jun 3 '14 at 13:59
• Indeed so, @Olnek...if I get another reference I'll write it down, since Hartshorne's book is, imo, written as if it was assumed the reader already knows almost all... – DonAntonio Jun 3 '14 at 14:01
• @DonAntonio: Thanks! I indeed don't understand exactly how II.6.8 gives what I need. I guess that the part I need is related to the proof of the fact that $f$ is a finite morphism, is that right? – Olnek Jun 3 '14 at 14:03