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Simon Rose
Simon Rose
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No. In general, there are no elliptic curves on an Abelian surface.

Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $A$$\sqrt{-1}$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.

No. In general, there are no elliptic curves on an Abelian surface.

Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $A$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.

No. In general, there are no elliptic curves on an Abelian surface.

Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $\sqrt{-1}$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.

Source Link
Simon Rose
Simon Rose
  • 6.3k
  • 33
  • 53

No. In general, there are no elliptic curves on an Abelian surface.

Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $A$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.