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$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with $[L:\mathbb Q]=2g$, $\alpha: O_L\rightarrow End(A)$ is an embedding of $O_L$ into $A$'s endomorphism ring. Consider the deformation functor $Def(A,\alpha)$ on the category of $C_k$. Can one recommend any reference for the representability of such deformation functors that allow a possible extension of the base field? Thank you very much!

edited Dec 24 2010 at 15:43

asked Dec 24 2010 at 15:36

Taisong Jing
731●3●10

Without change of base field, using Serre-Tate and Grothendieck-Messing, this amounts to lifting the Hodge filtration on $\mathbb{D}(A)$ in a way such that $O_L$ will respect it. In particular, we can do this over the univ. deformation of $A$, and it appears that we get a closed condition cutting out the required (formal) sub-scheme for us. I don't know what happens when you allow change of base field, but some Galois cohomology might show up. – Keerthi Madapusi Pera Dec 24 2010 at 15:58

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Without specifying polarization, there is just a formal abelian scheme over the deformation ring, so the deformation functor is only pro-representable. (This really happens: there exist formal abelian schemes over $p$-adic dvrs that have "complex mult." but tangential action on generic fiber of Lie alg. is not a CM type and hence are not algebraizable. Such examples are in a forthcoming book I am writing with Chai and Oort on CM-lifting problems.) Maybe you really want to know more generally about interaction of deformation rings with change in $k$? The story works in char-free way. (cont'd) – BCnrd Dec 24 2010 at 16:24

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For complete local noetherian $\Lambda$, let $C_{\Lambda}$ be category of artin local $\Lambda$-algebras with same residue field $k$. (If $k$ perfect of char. $p > 0$ then $C_{W(k)}$ is same as $C_k$ in the question.) Let $\Lambda \rightarrow \Lambda'$ be local map between such rings, inducing $k \rightarrow k'$ on residue field. For a "structure" $X$ over $k$ whose formal def. functor $D_X$ on $C_{\Lambda}$ is pro-reptd by some $R$, is $D_{X_{k'}}$ on $C_{\Lambda'}$ pro-reptd by $\Lambda' \widehat{\otimes}_{\Lambda} R$? Yes, quite often. See Rim's expose in SGA7 for the technique. – BCnrd Dec 24 2010 at 16:46

In particular, for any proper $k$-scheme $X$ and the flat deformation theory of $X$, this all works out in the affirmative. For abelian varieties the same method works, or much much simpler is to make a direct analysis of tangent space, in view of the concrete structure of the formal deformation ring! Then as Keerthi said, by using separatedness (and representability -- as algebraic spaces in the absence of projectivity) of Hom-functors for abelian schemes (building on Hilbert functors) one can incorporate "closed" conditions involving endomorphisms to get an affirmative answer. – BCnrd Dec 24 2010 at 16:53

To Brian: Thanks for the explanation; however, in my question I am not requiring all the residue fields are the same extension of the base field! What about this case? – Taisong Jing Dec 24 2010 at 19:02

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deformation of abelian varieties

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