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?

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    $\begingroup$ This question is not that well-defined. I mean, for a supersingular elliptic curve, the deformation space is non-canonically isomorphic to the formal spectrum of $W[[t]]$. If the question is "can this be given a group structure" then the answer is "sure -- it's the open unit disc". If this question is "can this be given a canonical group structure" then my answer is "can you define canonical?". I have problems with the rest of the question for similar reasons. But let me try and say something a bit more helpful. In the ordinary case, the universal deformation space has a canonical point... $\endgroup$ Feb 16, 2012 at 20:28
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    $\begingroup$ ...in it, namely the canonical lift. In the supersingular case I find it very very hard to tell one lift from the other, especially if the lifts are $W$-valued; I spent ages staring at such things once and found it really difficult to tell any of them apart. If there's a natural group structure on the deformation space, then whatever could the origin be? I can't guess. So I am skeptical about there being any natural sort of group structure -- a vague answer to a vague question but I hope it helps. $\endgroup$ Feb 16, 2012 at 20:30
  • $\begingroup$ Dear Kevin.Thank you very much for your answer. Yes, my question was not very well-posed.I try to make it clearer. The Serre-Tate group structure implies that if $\mathcal{X/U}$ is the universal deformation of $A$, then $H^{1}(\mathcal{X/U})$, can be equipped with a basis ${a_{i}, b_{j}}$ such that $\nabla(a_{i})=0$ and $\nabla(b_{j})= \Sigma a_{i}\otimes \eta_{ij}$ , where $\eta_{ij}= log(q_{ij})$ and $q_{ij}$ are the coordinates of the Serre-Tate.does such a basis exist for non-ordinary abelian varieties? or what is the best class of abelian varieties,equipped with such nice basis? $\endgroup$
    – Cyrus
    Feb 16, 2012 at 22:30

1 Answer 1


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:

In Serre-Tate theory for moduli spaces of PEL-type, 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.


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