Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
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2$\begingroup$ So as explained by Sandor, if $Y$ is a simple abelian surface, the anwser is no. If $Y$ contains an elliptic curve $E$, then by any point $y\in Y$, it passes a curve $y+E$ of genus $1$. If we want a real elliptic curve (abelian subvariety) passing through a general $P$, then it is easy to see that $Y$ is isogeneous to the square of an elliptic curve. I think that even when $Y=E^2$, the answer is no for a general $P$. $\endgroup$– Qing LiuNov 3, 2011 at 20:20
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2$\begingroup$ @Qing Liu: your last statement is true at least over the complex numbers. An abelian variety has at most countably many subvarieties, hence by Baire's theorem the union of all the abelian subvarieties cannot be all the space. $\endgroup$– ritaNov 3, 2011 at 20:49
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$\begingroup$ @Rita: the same arguments hold over uncountable fields. $\endgroup$– Qing LiuNov 3, 2011 at 21:07
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2$\begingroup$ @rita: "An abelian variety has at most countably many subvarieties" - I'm missing something: what about the translates of a fixed subvariety? $\endgroup$– QfwfqNov 4, 2011 at 12:01
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2$\begingroup$ Qfwfq: insert the word "abelian" before the word "subvarieties" in the quote. $\endgroup$– user5117Nov 4, 2011 at 12:41
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3 Answers
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In general, if an abelian variety $A$ contains an abelian subvariety $B\subseteq A$, then $A$ contains another abelian subvariety $B'\subseteq A$ such that $A$ is isogenous to $B\times B'$. This is Poincaré's reducibility theorem. (See also Poincaré's complete reducibility theorem, same book, next page).
An abelian variety is called simple if it does not contain any nontrivial abelian subvarieties. Simon's argument shows that there exist simple complex tori of dimension 2. One could also count moduli and conclude that not every abelian surface (or abelian variety of arbitrary dimension $>1$) can be isogenous to a product.
answered Nov 3, 2011 at 17:04
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$\begingroup$ @anon: If you don't say why $\mathrm{End}(A)=\mathbb Z$, then this statement is as good as saying that "a general abelian variety is simple". $\endgroup$ Nov 4, 2011 at 4:01
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2$\begingroup$ @anon: It is not true that on an open subset of moduli space $End(A) = \mathbb{Z}$; for example, in characteristic zero, abelian varieties isogenous to a fixed abelian variety form a dense subset. $\endgroup$– nafNov 4, 2011 at 12:38
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$\begingroup$ @anon: What do you mean by "the correct condition"? The question was about elliptic curves on surfaces. We're not "working", we're "discussing". $\endgroup$ Nov 4, 2011 at 17:27
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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 $\sqrt{-1}$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true.
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5$\begingroup$ Of course, most lattices do not give abelian varieties. $\endgroup$– AngeloNov 3, 2011 at 17:41
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3$\begingroup$ Note that this is true even if $A$ is algebraic (which, unlike the case for elliptic curves, is not the case for the quotient of ${\bf C}^2$ by a generic lattice). $\endgroup$ Nov 3, 2011 at 17:42
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You may have a look to:
Ernst Kani, Elliptic curves on Abelian surfaces
(the credit for this reference goes to Dan Petersen who already suggested it in a comment to this question)
