Elliptic curves on abelian surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:43Z http://mathoverflow.net/feeds/question/79944 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79944/elliptic-curves-on-abelian-surface Elliptic curves on abelian surface fds 2011-11-03T16:00:42Z 2011-11-04T11:56:36Z <p>Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?</p> http://mathoverflow.net/questions/79944/elliptic-curves-on-abelian-surface/79946#79946 Answer by Simon Rose for Elliptic curves on abelian surface Simon Rose 2011-11-03T16:15:10Z 2011-11-03T16:22:13Z <p>No. In general, there are no elliptic curves on an Abelian surface. </p> <p>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.</p> http://mathoverflow.net/questions/79944/elliptic-curves-on-abelian-surface/79951#79951 Answer by Sándor Kovács for Elliptic curves on abelian surface Sándor Kovács 2011-11-03T17:04:32Z 2011-11-03T17:04:32Z <p>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 <a href="http://books.google.com/books?id=MOW2gEP7HIkC&amp;lpg=PP1&amp;dq=birkenhake%2520lange&amp;pg=PA125#v=onepage&amp;q=poincare%2520reducibility&amp;f=false" rel="nofollow">Poincaré's reducibility theorem</a>. (See also Poincaré's complete reducibility theorem, same book, next page). </p> <p>An abelian variety is called <em>simple</em> 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. </p> http://mathoverflow.net/questions/79944/elliptic-curves-on-abelian-surface/80040#80040 Answer by Qfwfq for Elliptic curves on abelian surface Qfwfq 2011-11-04T11:56:36Z 2011-11-04T11:56:36Z <p>You may have a look to:</p> <p><a href="http://www.mast.queensu.ca/~kani/papers/hum-msm.pdf" rel="nofollow">Ernst Kani, <em>Elliptic curves on Abelian surfaces</em></a></p> <p>(the credit for this reference goes to Dan Petersen who already suggested it in a comment to <a href="http://mathoverflow.net/questions/21439/which-curves-can-be-found-on-abelian-varieties" rel="nofollow">this</a> question) </p>