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David Loeffler
David Loeffler
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Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.

Then $\mathbb{C} / \mathfrak{f}$ is an elliptic curve over $\mathbb{C}$ with CM by $\mathcal{O}_K$, and $1 \in \mathbb{C}$ maps to a primitive $\mathfrak{f}$-torsion point; and it's a standard theorem that there is a pair $(E, \alpha)$, consisting of an elliptic curve with CM by $\mathcal{O}_K$ and a primitive $\mathfrak{f}$-torsion point, defined over the ray class field $K(\mathfrak{f})$ which becomes isomorphic to $(\mathbb{C} / \mathfrak{f}, 1)$ over $\mathbb{C}$, and $(E, \alpha)$ is unique up to unique isomorphism.

Here's the question: can we find a model for $(E, \alpha)$ over $K(\mathfrak{f})^+ = K(\mathfrak{f}) \cap \mathbb{R}$? I'm pretty sure we can descend $E$ to $K(\mathfrak{f})^+$, but will the torsion point $\alpha$ be rational over this smaller field too?

(If the complex conjugation on $E(\mathbb{C})$ arising from the $K(\mathfrak{f})^+$ model of $E$ coincides with the natural complex conjugation on $\mathbb{C} / \mathfrak{f}$ this is immediate, but it's not completely clear to me that this is the case.)

Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.

Then $\mathbb{C} / \mathfrak{f}$ is an elliptic curve with CM by $\mathcal{O}_K$, and $1 \in \mathbb{C}$ maps to a primitive $\mathfrak{f}$-torsion point; and it's a standard theorem that there is a pair $(E, \alpha)$, consisting of an elliptic curve with CM by $\mathcal{O}_K$ and a primitive $\mathfrak{f}$-torsion point, defined over the ray class field $K(\mathfrak{f})$ which becomes isomorphic to $(\mathbb{C} / \mathfrak{f}, 1)$ over $\mathbb{C}$, and $(E, \alpha)$ is unique up to unique isomorphism.

Here's the question: can we find a model for $(E, \alpha)$ over $K(\mathfrak{f})^+ = K(\mathfrak{f}) \cap \mathbb{R}$? I'm pretty sure we can descend $E$ to $K(\mathfrak{f})^+$, but will the torsion point $\alpha$ be rational over this smaller field too?

(If the complex conjugation on $E(\mathbb{C})$ arising from the $K(\mathfrak{f})^+$ model of $E$ coincides with the natural complex conjugation on $\mathbb{C} / \mathfrak{f}$ this is immediate, but it's not completely clear to me that this is the case.)

Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.

Then $\mathbb{C} / \mathfrak{f}$ is an elliptic curve over $\mathbb{C}$ with CM by $\mathcal{O}_K$, and $1 \in \mathbb{C}$ maps to a primitive $\mathfrak{f}$-torsion point; and it's a standard theorem that there is a pair $(E, \alpha)$, consisting of an elliptic curve with CM by $\mathcal{O}_K$ and a primitive $\mathfrak{f}$-torsion point, defined over the ray class field $K(\mathfrak{f})$ which becomes isomorphic to $(\mathbb{C} / \mathfrak{f}, 1)$ over $\mathbb{C}$, and $(E, \alpha)$ is unique up to unique isomorphism.

Here's the question: can we find a model for $(E, \alpha)$ over $K(\mathfrak{f})^+ = K(\mathfrak{f}) \cap \mathbb{R}$? I'm pretty sure we can descend $E$ to $K(\mathfrak{f})^+$, but will the torsion point $\alpha$ be rational over this smaller field too?

(If the complex conjugation on $E(\mathbb{C})$ arising from the $K(\mathfrak{f})^+$ model of $E$ coincides with the natural complex conjugation on $\mathbb{C} / \mathfrak{f}$ this is immediate, but it's not completely clear to me that this is the case.)

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David Loeffler
David Loeffler
  • 37k
  • 3
  • 89
  • 194

Torsion points of CM elliptic curves

Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.

Then $\mathbb{C} / \mathfrak{f}$ is an elliptic curve with CM by $\mathcal{O}_K$, and $1 \in \mathbb{C}$ maps to a primitive $\mathfrak{f}$-torsion point; and it's a standard theorem that there is a pair $(E, \alpha)$, consisting of an elliptic curve with CM by $\mathcal{O}_K$ and a primitive $\mathfrak{f}$-torsion point, defined over the ray class field $K(\mathfrak{f})$ which becomes isomorphic to $(\mathbb{C} / \mathfrak{f}, 1)$ over $\mathbb{C}$, and $(E, \alpha)$ is unique up to unique isomorphism.

Here's the question: can we find a model for $(E, \alpha)$ over $K(\mathfrak{f})^+ = K(\mathfrak{f}) \cap \mathbb{R}$? I'm pretty sure we can descend $E$ to $K(\mathfrak{f})^+$, but will the torsion point $\alpha$ be rational over this smaller field too?

(If the complex conjugation on $E(\mathbb{C})$ arising from the $K(\mathfrak{f})^+$ model of $E$ coincides with the natural complex conjugation on $\mathbb{C} / \mathfrak{f}$ this is immediate, but it's not completely clear to me that this is the case.)