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Bernie
Bernie
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Let us work over $\mathbb{C}$ for the moment.

Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$.

$\mathbf{Q1:}$$\mathbf{Question:}$ Is there a smooth projective curve $C$ of genus $g=2$ such that $End(Jac(C))$ is a non-maximal order in $K$, that is $End(Jac(C))=\mathcal{O}_{K,f}=\mathbb{Z}+f\mathcal{O}_K$ for some $f>1$, especially $\neq \mathcal{O}_K$? If yes can we find such a curve for every $K$ and every $f>1$?

$\mathbf{Q2:}$ If there are such curves, where is the principally polarized Jacobian $(Jac(C),\theta_C)$ located in $\mathcal{A}_2$? It should be in some Humbert surface, but is it on $H_{\Delta_K}$ or $H_{f^2\Delta_K}$ where $\Delta_K$ is the discriminant of $K$ and $f^2\Delta_K$ is the discriminat of $\mathcal{O}_{K,f}$?

Let us work over $\mathbb{C}$ for the moment.

Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$.

$\mathbf{Q1:}$ Is there a smooth projective curve $C$ of genus $g=2$ such that $End(Jac(C))$ is a non-maximal order in $K$, that is $End(Jac(C))=\mathcal{O}_{K,f}=\mathbb{Z}+f\mathcal{O}_K$ for some $f>1$, especially $\neq \mathcal{O}_K$? If yes can we find such a curve for every $K$ and every $f>1$?

$\mathbf{Q2:}$ If there are such curves, where is the principally polarized Jacobian $(Jac(C),\theta_C)$ located in $\mathcal{A}_2$? It should be in some Humbert surface, but is it on $H_{\Delta_K}$ or $H_{f^2\Delta_K}$ where $\Delta_K$ is the discriminant of $K$ and $f^2\Delta_K$ is the discriminat of $\mathcal{O}_{K,f}$?

Let us work over $\mathbb{C}$ for the moment.

Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$.

$\mathbf{Question:}$ Is there a smooth projective curve $C$ of genus $g=2$ such that $End(Jac(C))$ is a non-maximal order in $K$, that is $End(Jac(C))=\mathcal{O}_{K,f}=\mathbb{Z}+f\mathcal{O}_K$ for some $f>1$, especially $\neq \mathcal{O}_K$? If yes can we find such a curve for every $K$ and every $f>1$?

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Bernie
Bernie
  • 1k
  • 5
  • 8

Are there curves of genus 2 with real multiplication by a non-maximal order?

Let us work over $\mathbb{C}$ for the moment.

Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$.

$\mathbf{Q1:}$ Is there a smooth projective curve $C$ of genus $g=2$ such that $End(Jac(C))$ is a non-maximal order in $K$, that is $End(Jac(C))=\mathcal{O}_{K,f}=\mathbb{Z}+f\mathcal{O}_K$ for some $f>1$, especially $\neq \mathcal{O}_K$? If yes can we find such a curve for every $K$ and every $f>1$?

$\mathbf{Q2:}$ If there are such curves, where is the principally polarized Jacobian $(Jac(C),\theta_C)$ located in $\mathcal{A}_2$? It should be in some Humbert surface, but is it on $H_{\Delta_K}$ or $H_{f^2\Delta_K}$ where $\Delta_K$ is the discriminant of $K$ and $f^2\Delta_K$ is the discriminat of $\mathcal{O}_{K,f}$?