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gsvr
gsvr
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Let $C$ be a generic(very) general genus 1 curve embedded in $P^1\times P^1$$\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $P^1$$\mathbb{CP}^1$ and induces an involution $\tau_i:C\to C$.

  Let $G\simeq \mathbb Z/2 * \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

My question is: areAre there points on $C$ with finite orbit under $G$?

Let $C$ be a generic genus 1 curve embedded in $P^1\times P^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $P^1$ and induces an involution $\tau_i:C\to C$.

  Let $G\simeq \mathbb Z/2 * \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

My question is: are there points on $C$ with finite orbit under $G$?

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an involution $\tau_i:C\to C$. Let $G\simeq \mathbb Z/2 * \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

Are there points on $C$ with finite orbit under $G$?

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gsvr
gsvr
  • 235
  • 2
  • 9

Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a generic genus 1 curve embedded in $P^1\times P^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $P^1$ and induces an involution $\tau_i:C\to C$.

Let $G\simeq \mathbb Z/2 * \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

My question is: are there points on $C$ with finite orbit under $G$?