On the jacobian origin of CM abelian varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:52:16Z http://mathoverflow.net/feeds/question/104945 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104945/on-the-jacobian-origin-of-cm-abelian-varieties On the jacobian origin of CM abelian varieties Hugo Chapdelaine 2012-08-17T20:48:00Z 2012-08-18T06:10:07Z <p>Let $K$ be a CM field of degree $2n$ over $\mathbf{Q}$ and let $\mathcal{O}_K$ be its ring of integers. Let $\Phi=(\phi_1,\ldots,\phi_n)$ be a CM type of K. Then it is known that complex torus $A:=\mathbf{C}^n/\Phi(\mathcal{O}_K)$ is algebraic. For instance one may construct a Riemann form on $A$ in the following way: choose $\xi\in K$ such that $\xi^2\in K^+$ is totally negative and $\Im(\phi_i(\xi))>0$ for all $i$. Then for a suitable integer $m$ one has that the alternating form $$E(z,w):=m\sum_i \phi_i(\xi)(\overline{z}_iw_i-z_i\overline{w}_i)$$ is a Riemann form of $A$.</p> <p><strong>Q</strong>: Having an abelian varietey $A$ as above is it reasonable to expect it to be the isogeneous to the Jacobian of a curve? </p> <p>Of course if $n=1$ then the answer is always true but what about $n>1$? Are there some obvious obstructions that prevent $A$ to have a Jacobian origin? </p> http://mathoverflow.net/questions/104945/on-the-jacobian-origin-of-cm-abelian-varieties/104966#104966 Answer by Ari Shnidman for On the jacobian origin of CM abelian varieties Ari Shnidman 2012-08-18T03:08:59Z 2012-08-18T06:10:07Z <p>When $n &lt; 4$, $A$ is isogenous to the Jacobian of a stable curve because the Torelli locus is dense. This should imply that $A$ is isogenous to the Jacobian of a smooth curve because $A$ is simple (at least if the CM-type is primitive). </p> <p>If $n \geq 4$, then conditional results of Chai-Oort (see their recent Annals paper) imply that $A$ is not necessarily isogenous to the Jacobian of a curve. Tsimerman then proved this unconditionally. I would guess that the set of CM points whose isogeny orbit is disjoint from the Torelli locus is dense in $\mathcal{A_g}$, but I don't know. </p>