# About the Serre-Tate theorem

It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure parameterized by $x$: the deformation functor should be pro-represented by the formal completion of $X$ at $x$ ($X$ representable as assumed). In particular, if $x$ is a smooth geometric point, then the associated deformation functor is pro-represented by a ring formal power series of $d$ variables, $d=\dim_xX$.

What does one find when linking this principle with the Serre-Tate theorem? The theorem affirms that the deformation functor of an ordinary abelian variety of dimension $g$ over an algebraically closed field of char.$p$ is pro-represented by a formal torus of dimension $g^2$. does this functor corresponds to the formal completion of some muduli functor of abelian scheme? One cannot expect the Siegel moduli (with level structures) of genus $g>1$ to work as the latter is of rel.dimension $g(g+1)/2$. Does the dimension jump to $g^2$ because one is looking "purely locally" through the $p$-divisible group forgetting the polarization?

On the other hand, let $\mathcal{M}$ be the moduli functor of principally polarized $g$-dimensional abelian scheme with $\Gamma_0(p)$ level structure, $p$ being a rational prime (say large enough). It admits a coarse moduli scheme $M$, and according to a theorem of Chai and Norman ("Bad reduction of Siegel moduli scheme of genus two ...") the formal completion of $M$ at a geometric point $x$ corresponding to an ordinary abelian variety of char.$p$ is a formal torus of dimension $g(g+1)/2$, which agrees with the principle mentioned in the beginning.

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As for the $p$-divisible group situation (sans polarization), the main problem in having it be the deformation ring of some moduli problem is algebraicizability (is that how that's spelled?). When you have a polarization and you deform it as well, then the formal abelian scheme living over the deformation space can be algebraicized to an honest abelian scheme and thus situated within some moduli space, but this is not true for the bare $p$-divisible group. See Artin's great article 'Algebraization of formal moduli I'. – Keerthi Madapusi Pera Mar 13 '11 at 6:14

Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$. Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the universal deformation is isomorphic to $k[t_1, \dots, t_{g^2}]$.
Using the theory of p-divisble groups, one can show that in the case where $A$ is ordinary, the formal scheme representing the deformation functor can be equipped with the structure of a formal group scheme. This corresponds to a more canonical choice of parameters compared to the above one. In any case, the dimensions are the same.
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