Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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, – Zev Chonoles Nov 19 '11 at 18:35
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.