Let $E$ be a non-supersingular elliptic curve over $\mathbb{F}_p$, and $E^1$ be some elliptic curve, then $Hom_{\bar{\mathbb{F}_p}}(E, E^1)$ is free $\mathbb{Z}$ module of rank 2.
Can someone explain this please? I can't see how it is of rank 2.
7
-
$\begingroup$ This is true if the module is nonzero, but if $E$ and $E^1$ are nonisogenous than this module vanishes (more or less by definition). $\endgroup$– Will SawinCommented Jun 19, 2017 at 21:41
-
$\begingroup$ Yes, But, how it is of rank 2, please explain. I cant see that $\endgroup$– studentCommented Jun 19, 2017 at 21:53
-
$\begingroup$ See for example Silverman's The arithmetic of elliptic curves, Thm. V.3.1. $\endgroup$– R. van Dobben de BruynCommented Jun 20, 2017 at 0:22
-
$\begingroup$ See p.162 of Silverman's book the rank of $End(E)$ is $2$ or $4$, and rank $4$ is the definition of super-singular elliptic curve over a finite field. Using the dual isogeny, you get a ($\mathbb{Z}$-module) embedding of $Hom(E,E') $ into $End(E)$. $\endgroup$– reunsCommented Jun 20, 2017 at 1:14
-
$\begingroup$ p.91 , corollary III.7.5, of Silverman book says that $Hom(E,E^1)$ is free of rank 4. its not the rank for supersingular. $\endgroup$– studentCommented Jun 20, 2017 at 8:10
|
Show 2 more comments
1 Answer
$\begingroup$
$\endgroup$
5
Assume there is a non-zero isogeny $\phi:E\to E'$. Let $q$ be a power of $p$ so that both $E'$ and $\phi$ are defined over $\mathbb F_q$. Let $F_q:E'\to E'$ be the $q$-power Frobenius map. Then $\text{Hom}(E,E')$ at least contains $\{m\phi+n F_q\circ\phi: m,n\in\mathbb Z\}$, so the rank is at least 2. I'll let you figure out why the rank can't be greater than 2.
answered Jun 20, 2017 at 0:21
-
2$\begingroup$ Since you literally wrote the book on this I half expected that your answer would be a page reference to TAoEC . . . $\endgroup$ Commented Jun 20, 2017 at 2:09
-
$\begingroup$ @joe Thank you for the answer, What did you meant by $F_q$, I am sorry I can't see how it cant\'t be greater than 2. $\endgroup$– studentCommented Jun 20, 2017 at 8:23
-
-
$\begingroup$ @student Sorry, $F_q$ is the $q$-power Frobenius map. I'll edit the answer to fix that typo. As for TAoEC in Elkies answers, that's my book The Arithmetic of Elliptic Curves, so he was just joking that I could have simply given you a reference there, instead of writing out an answer. As for why the rank can't be 4 when E is ordinary, that should follow from the material in Chapter V (Sections 3 and 4) of the aforementioned book. $\endgroup$ Commented Jun 20, 2017 at 11:58
-
$\begingroup$ I am so happy that, I could communicate with you and prof. Elkies. @JoeSilverman , Professor, I am doing my thesis on "Algorithms for Elliptic Curves" at TU Kaiserslautern, Germany. Is there any way to find Atkin's "The Number of Points on an Elliptic Curve Modulo a Prime" article? Thank you very much. $\endgroup$– studentCommented Jun 21, 2017 at 21:56