[Math] CM rational points on modular curves

complex multiplicationelliptic-curvesnt.number-theory

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?

Best Answer

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 $a_E(N)$.

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