Let be an abelian variety of dimension over a field of characteristic . In the previous posts, we saw that to give a deformation of over a local artinian ring with residue field was the same as giving a continuous morphism of rings : that is, the local deformation space is smooth on parameters.

There is another description of the deformation space in terms of the -divisible group, though. Given , we can form the formal scheme

where each is a finite group scheme of rank over . As a formal scheme, this is *smooth*: that is, given a small extension in and a morphism , there is an extension

In fact, we start by finding an extension (as is smooth), and then observe that this extension must land in some if landed in .

As a result, we can talk about deformations of this formal (group) scheme.

Definition 16Adeformationof a smooth, formal group scheme over over a ring is a smooth, formal group scheme over which reduces mod the maximal ideal (i.e., when restricted to -algebras) to .

Suppose . Using Schlessinger’s criterion, we find that (under appropriate finiteness hypotheses), must be prorepresentable by a power series ring for some . In other words, corresponds to a *formal group* over .

Suppose for instance that was a supersingular elliptic curve. By definition, each of the is a thickening of the zero point, and consequently is the formal completion of at zero. This data is equivalent to the *formal group* of , and as we just saw a deformation of this formal group over a ring is a formal group over which reduces to the formal group of mod the maximal ideal.

The main result is:

Theorem 17 (Serre-Tate)Let . There is an equivalence of categories between abelian schemes over and pairs where is an abelian variety over and a deformation of over .

So, in particular, deformations of over are equivalent to deformations of over .

In this post, I’ll describe an argument due to Drinfeld for this result (which I learned from Katz’s article).

**1. Lifting**

The key step in Drinfeld’s argument is the following idea. Let with maximal ideal , and let be a fppf sheaf of abelian groups on -algebras, satisfying certain assumptions below. Let for sufficiently large (depending on ). The claim is that for any -algebra , there exists a natural map of groups

which lifts the map . In other words, there is a functorial lift of multiplication by on the reduction mod to the thickened version. As we’ll see, this implies that any two deformations of an abelian variety are *isogeneous* (though not necessarily isomorphic).

We need the following assumptions:

- is formally smooth (as a sheaf on -algebras).
- The “formal completion ” defined via is a formal group scheme, represented by a formal group over (i.e., of the form ).

This is satisfied by the sheaf of abelian groups associated to an abelian scheme over , and ditto for a deformation of if is an abelian variety over .

Proposition 18Let be a functor satisfying the above two conditions (where has maximal ideal ). Then, there exists an (depending on ) with the following property: for any -algebra , there is a map of abelian groups

lifting multiplication by on . This is natural in homomorphisms of such functors.

*Proof:* The idea of the proof is simple: given , we use formal smoothness to lift this to an element , and then take to be the required image. The whole point is that the choice of doesn’t depend on the choice of . In fact, given two choices lifting , we find that

But now is a formal Lie group, so is represented by an element in coordinates where the . Multiplication by now annihilates these if is appropriate, because is artinian and (e.g. we could take ).

Let’s now use this to see that any two deformations of an abelian variety over are isogeneous. Let be two such. The claim is that there is a morphism of abelian schemes

which lifts multiplication by on . For this we can work at the level of abelian sheaves for the flat topology, and then we define, for an -algebra , the map

where is the morphism of the previous lemma, and we note that are the same on -algebras (they are ). By construction, this is a lifting of multiplication by on , and is consequently an isogeny.

Note that this morphism is necessarily *unique* among lifts of multiplication by . We could see this by the rigidity lemma, or we could note that two lifts would have their difference . Since multiplication by is a surjection on (when considered as a fppf sheaf on -algebras) while multiplication by is zero on , it follows that must be zero.

In fact, we can state:

Lemma 19Let be two fppf abelian sheaves satisfying the two conditions above (i.e., formally smooth with formal completions represented by a formal Lie group). Suppose multiplication by is a surjection on each, moreover. Then reduction mod gives an injection

and, moreover, there is a map lifting multiplication by : if , lifts .

In other words, we cannot necessarily lift a morphism to a morphism , but we can always lift times that. This lift is necessarily *unique*, because multiplication by is a surjection and reduction mod the maximal ideal is injective.

*Proof:* This is now proved in exactly the same way as for abelian schemes. Given an -algebra and , we define

where we use the multiplication by lift mentioned earlier.

**2. Full faithfulness**

Now, we can prove the Serre-Tate theorem. The theorem states that a certain forgetful functor is an equivalence of categories between abelian schemes over and abelian schemes over together with a deformation of their -divisible group. In order to see this, we’ll have to show that it is fully faithful and essentially surjective.

Let’s start by showing that it is fully faithful.

Lemma 20Given abelian schemes over with -reductions , a morphism of abelian varieties, and a lift lifting , then there is a unique morphism of abelian -schemes lifting these.

*Proof:* By the previous analysis, we know that there is a morphism of abelian schemes

which lifts , and which is unique with this property. We need to show that can be “divided by ,” which amounts (by descent theory) to saying that it annihilates the kernel of multiplication by .

To see this, note that this morphism induces a map which lifts . There is another morphism which has this property: namely, (here this is honest multiplication by !). By the uniqueness assertions of the previous analysis, we find that

because both lift the same morphism . Anyway, the conclusion is that is actually (when restricted to the -divisible group) a multiple of , which means that it has to annihilate . It now follows that itself can be divided by , which means that we can lift and .

**3. Essential surjectivity**

The last step in the proof of the Serre-Tate theorem is the essential surjectivity. That is, given an abelian variety , and a deformation of over , we need to produce an abelian scheme over lifting all this data. The first step is to lift itself over to an abelian scheme: we’ve seen that deformations of abelian varieties are unobstructed, so this is not an issue. So, we get a lifting , which induces a new lifting of the -divisible group . This is not the same as most likely, but we can use the same analysis to see that it is isogeneous to .

In fact, we have a map

and this induces a lift of ,

(Technically, the should be in quotes as well, since it does not make sense.) Similarly, we can lift to a map , and the composite either way must be (by uniqueness). It follows now that the map is an isogeny, and the kernel is a finite (indeed, flat) subgroup scheme .

Now we consider the quotient abelian scheme . This comes with a map , by construction, and the claim is that it also lifts ; in fact, we have a map

which is an *isomorphism*, as lifts the subgroup scheme . In particular, we find that the quotient is a deformation of with the appropriate -divisible group .

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