Deformation space of non-ordinary abelian varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:36:46Z http://mathoverflow.net/feeds/question/88648 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88648/deformation-space-of-non-ordinary-abelian-varieties Deformation space of non-ordinary abelian varieties Cyrus 2012-02-16T16:23:42Z 2013-02-24T12:28:23Z <p>It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian variety over $W$, the ring of Witt vectors has a group structure. This prompts the following question: Does this group structure extend to the formal moduli of non-ordinary abelian varieties? and if not, what is the biggest class of abelian varieties, for which, a group structure on the formal moduli exists?</p> http://mathoverflow.net/questions/88648/deformation-space-of-non-ordinary-abelian-varieties/122792#122792 Answer by Ulrich Goertz for Deformation space of non-ordinary abelian varieties Ulrich Goertz 2013-02-24T12:28:23Z 2013-02-24T12:28:23Z <p>I cannot comment yet, so I will add the following remark as an answer. I do not think that this answers your question, but at least it is a case where a group structure does not exist, yet some generalization does:</p> <p>In <a href="http://dx.doi.org/10.1016/j.ansens.2003.04.004" rel="nofollow">Serre-Tate theory for moduli spaces of PEL-type</a>, Ann. scient. de l'Ec. Norm. Sup. 37 (2004), 223-269, (arXiv:math/0203288v2), Ben Moonen looks at this question for abelian varieties (equivalently: $p$-divisible groups) with additional structure in the $\mu$-ordinary case.</p>