Frobenius Morphism on Elliptic Curves

Question

I am having some confusion concerning the Frobenius morphism of an elliptic curve over a finite field $\mathbb{F}_q$ with $q = p^r$ and $p$ prime. I am working with Silverman's "Arithmetic of Elliptic Curves" and currently on the following example:

My question is, if the Frobenius endomorphism fixes exactly $E(\mathbb{F}_q)$ and further $E^{(q)}=E$, what does this endomorphism do? Isn't it then just the identitly, leaving every point fixed? Or shouldn't it fix the points $E(\mathbb{F}_p)$? I just don't see what the frobenius morphism does on an elliptic curve over a finite field, if all the points in questions are left fixed.

I hope someone understands my, a little messy and unclear, question and can help me to free the knot in my head.

Answer 1

The Frobenius $\phi_q$ fixes $E({\Bbb F}_q)$ as shown but doesn't fix the points in $E({\Bbb F})$ where ${\Bbb F}_q\subset{\Bbb F}$ is a field extension.

As an analogy, you should think of an elliptic curve defined over the real field $\Bbb R$ (Weierstrass equation with real coefficients). Then the complex conjugation fixes the real points $E({\Bbb R})$ but acts as a non-trivial involution on the set of complex points $E({\Bbb C})$.

Answer 2

Assume that $E$ is defined over $\Bbb{F}_q$.

A point here is that while $\phi_q$ maps the curve $E$ to itself, and fixes all the points of $E(\Bbb{F}_q)$, it does not fix all the points of $E(\Bbb{F}_{q^r})$ for $r>1$. In fact, its fixed points are exactly the points in $E(\Bbb{F}_q)$. But there is more to an elliptic curve defined over $\Bbb{F}_q$ than its rational points! When we consider "all of the curve", we want to include the points of $E(\overline{\Bbb{F}_q})$ with coordinates in an algebraic closure $\overline{\Bbb{F}_q}$ of $\Bbb{F}_q$, i.e. over the union of all the extension fields $\Bbb{F}_{q^r}$.

Viewing $\phi_q$ as a mapping from $E(\overline{\Bbb{F}_q})$ to itself brings with it a lot of tools. Largely because algebraic geometry really should be done over an algebraic closed field. The set of points in $E(\Bbb{F}_q)$ as such is just a finite collection of points not worthy of being called a curve.

As already indicated in the quoted passage, study of the action of $\phi_q$ on $E(\overline{\Bbb{F}_q})$ is at the heart of many a further development. The Hasse bound on the number of the points is just the tip of that iceberg. The celebrated point counting algorithm due to Schoof-Elkies-Atkin depends on the study of the action of $\phi_q$.

I am the wrong person to describe all the connections in detail, but I think that use of $\phi_q$ here is an analogue of the Lefschetz' fixed point counting formula that you may have seen in algebraic topology.