How does the Frobenius work on the Torsionpoints of an ellitptic curve with CM

Question

I am trying to understand the main theorem ov CM for elliptic curves. I work with the version stated in the second chapter of Silvermans "Advanced Topics in the Arithmetics of Elliptic Curves". I would like to make an example or something what illustrates clearly the statement.

Edit: What I did up to now is:

Let $K=\mathbb{Q}(i)$ and $E: y^2 = x^3 +x$. I computed the m-torsion points for $m=1,2,3,4$. Now I want to see how the Gaolisgroup act an this points. And then check that there is a Frobenius who acts the same. I don't konw a lot about this Frobenius. How con I get it or describe it?

Answer 1

This is more a long comment :

• Take an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ and a fractional ideal $\mathfrak{a} \in K$, ie. $\mathfrak{a} = \lambda(\mathbb{Z}+\tau \mathbb{Z})$ is a lattice in $\mathbb{C}$ for some $\lambda,\tau \in K$. Look at the complex torus $\mathbb{C}/\mathfrak{a}$. Note its torsion subgroup is $K/\mathfrak{a}$.

• Take a sequence $b_n \in \mathcal{O}_K$, for example $b_n = \prod_{c \in \mathcal{O}_K} c^{\lfloor n/N(c)\rfloor}$ such that $$K = \bigcup_n \frac{\mathcal{O}_K}{b_n}, \qquad K/\mathfrak{a}=\lim_{n \to \infty}\frac{\mathcal{O}_K}{b_n}/\mathfrak{a}$$ where the last limit is understood in term of inclusion of finite subgroups of the complex torus.

• Note for any $s \in \mathcal{O}_K$ then $z \mapsto sz$ is an endomorphism of $\mathbb{C}/\mathfrak{a}$ and $K/\mathfrak{a}$. And if $s=\frac{u}{v} \in K, u,v \in \mathcal{O}_K$ then $z \mapsto sz$ is an homomorphism $K/\mathfrak{a} \to K/v^{-1}\mathfrak{a}$.

• For a sequence $s_m \in K$, you can then ask whenever $\lim_{n \to \infty} s_n$ defines an homomorphism $K/\mathfrak{a} \to K/v^{-1}\mathfrak{a}$ for some $v \in K$. The answer is : when $s_m$ stabilizes modulo $b_n$ for every $n$, ie. when it stabilizes modulo $\mathfrak{p}^k$ for every prime ideal power $\mathfrak{p}^k$, ie. when $\lim_{m \to \infty} s_m$ converges as an idele of $K$.

• The main theorem of complex multiplication says that (using the Weierstrass function $\wp(z)$) such an homormorphism on the complex torus side has an elliptic curve counterpart $E(\mathbb{C}) \to E^\sigma(\mathbb{C})$ for some $\sigma \in \text{Aut}(\mathbb{C})$.