Dear MO Community,

I am trying to understand Mazur's 1976 notes "Rational points on Modular Curves" (which can be found in Springer Lecture Notes in Mathematics 601).

Let N be a prime number, and let $X_0(N)$ be the usual modular curve over $\mathbb{Q}$. Say that a point on it is 'CM' if the elliptic curve corresponding to the point has Complex Multiplication. I am interested in the following sorts of questions (which are suggested by what happens over $\mathbb{Q}$):

  1. Let $K$ be a number field that is not $\mathbb{Q}$. For which N can I construct CM points on $X_0(N)(K)$ that are not defined over $\mathbb{Q}$? Are there finitely many such N?

  2. For such N, are there finitely many such CM points?

For example, let $K = \mathbb{Q}(\sqrt{-6})$, and let $R = O_K = \mathbb{Z}[\sqrt{-6}]$. Set $E = \mathbb{C}/R$. Choose $N$ such that $R/NR$ has nontrivial radical, or equivalently such that $R/NR = \mathbb{F}_N[\epsilon]$ for $\epsilon$ some nontrivial element in the radical. Then (E,ker $\epsilon$) determines a point $a_E(N)$ on $X_0(N)$ which "is defined over a subfield of index 2 in the ray class field of $R \otimes \mathbb{Q}$, with conductor equal to the conductor of $R$ (which in my example is 1). Is $a_E(N)$ defined over $K$? What is this index-2 subfield to which Mazur refers?

I think my second question is equivalent to asking: For such N, are there finitely many imaginary quadratic orders $R$ such that the aforementioned index 2 subfield is K, and such that R/NR has nontrivial radical?


Dear Barinder,

This is the subject of the arithmetic theory of complex multiplication. Given any CM elliptic curve $E$, and any kind of level structure on $E$, this theory determines the precise field of definition of $E$ with its given level structure. You can find a discussion of this in Silverman II ("Advanced topics") and in Shimura's book.

In your example: the $j$-invariant of $\mathbb C/R$ lives in the totally real subfield of the Hilbert class field of $K$ (which is $\mathbb Q(\sqrt{2})$, and is also the ray class field of conductor 1 that you mention). The $\Gamma_0(N)$ level structure ker $\epsilon$ is defined over this same field. So this is the field of definition of $a_E(N)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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