5
$\begingroup$

I have a question about exercise 6.15 of Silverman's book AEC.

Suppose that $E$ is a nonsingular elliptic curve over $\mathbb{C}$ given by the equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Then we can define the division polynomials $\psi_n(x,y)$ as usual. Exercise 6.15 of Silverman's book AEC says that we consider $\psi_n$ as a function on $\mathbb{C}/\Lambda$. Because I want to do the latter explicitely, I started with an admissible change of variables such that $E(\mathbb{C})$ is isomorphic to an elliptic curve $\bar{E}(\mathbb{C})$ given by $$y^2=4x^3-g_2x-g_3.$$ Let $\Lambda$ be the lattice corresponding to the above elliptic curve. Then $\mathbb{C}/\Lambda$ is isomorphic to $\bar{E}(\mathbb{C})$ and hence $\mathbb{C}/\Lambda$ is isomorphic to $E(\mathbb{C})$. We can summarize with the following group isomorphisms $$\mathbb{C}/\Lambda\ \xrightarrow{z\ \mapsto\ [\wp(z),\,\wp'(z),\,1]}\ \bar{E}(\mathbb{C})\ \xrightarrow{\text{admissible change of variables }}\ E(\mathbb{C}) .$$ How exactly can we consider $\psi_n$ as a function on $\mathbb{C}/\Lambda$ from the above descriptions?

$\endgroup$
1
  • $\begingroup$ Nadori, to bump your question back to the front page, so that it might come to more people's attention, you can make a trivial edit to the text (e.g. add some spaces, change one word, etc.). I have done this for you. If bumping it a few times doesn't help, another approach to attract attention and answers to your question would be to place a bounty on it. Best, $\endgroup$ Nov 19, 2011 at 18:35

1 Answer 1

5
$\begingroup$

I am not sure I understand your question correctly, so here are two different hints.

If you are looking for an inverse map to the isomorphism $\mathbb{C}/\Lambda\to E(\mathbb{C})$ that sends $z \mapsto [\wp(z),\wp'(z),1]$, this inverse map is given in Silverman, Proposition 5.2. It is given by the map $E(\mathbb{C})\to \mathbb{C}/\Lambda$ and sends $P=(x_0,y_0)$ to $\int_O^P \frac{dx}{y} \bmod \Lambda$, where $O$ is the zero element in $E$, i.e., the point at infinity.

If you are confused about how to think about $\psi_n$ defined over $\mathbb{C}/\Lambda$, notice that the division polynomials $\psi_n(x,y)$ are defined so that, if $P=(x,y)\in E$, then $\psi_n(x,y)=0$ if and only if $[n]P = O$. So $\psi_n$ on $\mathbb{C}/\Lambda$ is a function $\psi_n:\{\mathbb{C}/\Lambda\}-\{ 0\bmod \Lambda\} \to \mathbb{C}$ such that $\psi_n(z)=0$ if and only if $n\cdot z \equiv 0 \bmod \Lambda$. Moreover, $\psi_n$ has a pole of order $n^2$ at infinity (Silverman, Exercise 3.7(f)). Hence, you know exactly what the divisor of $\psi_n$ is, i.e., $\operatorname{div}(\psi_n) = (\sum_{T\in E[n]} T)-n^2\cdot O$. If you calculate the divisor of $\frac{\sigma(nz)}{\sigma(z)^{n^2}}$, you'll find that both functions have the same divisor, and therefore their quotient is a constant. Then follow Silverman's hint to find the constant.

I hope this helps.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .