Serre-Tate Theory 2

I guess this will be the last post on this topic. I’ll explain a tiny bit about what goes into the proof of this theorem and then why anyone would care that such canonical lifts exist. On the first point, there are tons of details that go into the proof. For example, Nick Katz’s article, Serre-Tate Local Moduli, is 65 pages. It is quite good if you want to learn more about this. Also, Messing’s book The Crystals Associated to Barsotti-Tate Groups is essentially building the machinery for this proof which is then knocked off in an appendix. So this isn’t quick or easy by any means.

On the other hand, I think the idea of the proof is fairly straightforward. Let’s briefly recall last time. The situation is that we have an ordinary elliptic curve {E_0/k} over an algebraically closed field of characteristic {p>2}. We want to understand {Def_{E_0}}, but in particular whether or not there is some distinguished lift to characteristic {0} (this will be an element of {Def_{E_0}(W(k))}.

To make the problem more manageable we consider the {p}-divisible group {E_0[p^\infty]} attached to {E_0}. In the ordinary case this is the enlarged formal Picard group. It is of height {2} whose connected component is {\widehat{Pic}_{E_0}\simeq\mu_{p^\infty}}. There is a natural map {Def_{E_0}\rightarrow Def_{E_0[p^\infty]}} just by mapping {E/R \mapsto E[p^\infty]}. Last time we said the main theorem was that this map is an isomorphism. To tie this back to the flat topology stuff, {E_0[p^\infty]} is the group representing the functor {A\mapsto H^1_{fl}(E_0\otimes A, \mu_{p^\infty})}.

The first step in proving the main theorem is to note two things. In the (split) connected-etale sequence

\displaystyle 0\rightarrow \mu_{p^\infty}\rightarrow E_0[p^\infty]\rightarrow \mathbb{Q}_p/\mathbb{Z}_p\rightarrow 0

we have that {\mu_{p^\infty}} is height one and hence rigid. We have that {\mathbb{Q}_p/\mathbb{Z}_p} is etale and hence rigid. Thus given any deformation {G/R} of {E_0[p^\infty]} we can take the connected-etale sequence of this and see that {G^0} is the unique deformation of {\mu_{p^\infty}} over {R} and {G^{et}=\mathbb{Q}_p/\mathbb{Z}_p}. Thus the deformation functor can be redescribed in terms of extension classes of two rigid groups {R\mapsto Ext_R^1(\mathbb{Q}_p/\mathbb{Z}_p, \mu_{p^\infty})}.

Now we see what the canonical lift is. Supposing our isomorphism of deformation functors, it is the lift that corresponds to the split and hence trivial extension class. So how do we actually check that this is an isomorphism? Like I said, it is kind of long and tedious. Roughly speaking you note that both deformation functors are prorepresentable by formally smooth objects of the same dimension. So we need to check that the differential is an isomorphism on tangent spaces.

Here’s where some cleverness happens. You rewrite the differential as a composition of a whole bunch of maps that you know are isomorphisms. In particular, it is the following string of maps: The Kodaira-Spencer map {T\stackrel{\sim}{\rightarrow} H^1(E_0, \mathcal{T})} followed by Serre duality (recall the canonical is trivial on an elliptic curve) {H^1(E_0, \mathcal{T})\stackrel{\sim}{\rightarrow} Hom_k(H^1(E_0, \Omega^1), H^1(E_0, \mathcal{O}_{E_0}))}. The hardest one was briefly mentioned a few posts ago and is the dlog map which gives an isomorphism {H^2_{fl}(E_0, \mu_{p^\infty})\stackrel{\sim}{\rightarrow} H^1(E_0, \Omega^1)}.

Now noting that {H^2_{fl}(E_0, \mu_{p^\infty})=\mathbb{Q}_p/\mathbb{Z}_p} and that {T_0\mu_{p^\infty}\simeq H^1(E_0, \mathcal{O}_{E_0})} gives us enough compositions and isomorphisms that we get from the tangent space of the versal deformation of {E_0} to the tangent space of the versal deformation of {E_0[p^\infty]}. As you might guess, it is a pain to actually check that this is the differential of the natural map (and in fact involves further decomposing those maps into yet other ones). It turns out to be the case and hence {Def_{E_0}\rightarrow Def_{E_0[p^\infty]}} is an isomorphism and the canonical lift corresponds to the trivial extension.

But why should we care? It turns out the geometry of the canonical lift is very special. This may not be that impressive for elliptic curves, but this theory all goes through for any ordinary abelian variety or K3 surface where it is much more interesting. It turns out that you can choose a nice set of coordinates (“canonical coordinates”) on the base of the versal deformation and a basis of the de Rham cohomology of the family that is adapted to the Hodge filtration such that in these coordinates the Gauss-Manin connection has an explicit and nice form.

Also, the canonical lift admits a lift of the Frobenius which is also nice and compatible with how it acts on the above chosen basis on the de Rham cohomology. These coordinates are what give the base of the versal deformation the structure of a formal torus (product of {\widehat{\mathbb{G}_m}}‘s). One can then exploit all this nice structure to prove large open problems like the Tate conjecture in the special cases of the class of varieties that have these canonical lifts.


Serre-Tate Theory 1

Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and {p}-divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.

The idea is the following. Suppose you have an elliptic curve {E/k} where {k} is a perfect field of characteristic {p>2}. In most first courses on elliptic curves you learn how to attach a formal group to {E} (chapter IV of Silverman). It is suggestively notated {\widehat{E}}, because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.

Since an elliptic curve is isomorphic to it’s Jacobian {Pic_E^0} there is a conflation that happens. In general, if you have a variety {X/k} you can make the same formal group by completing this group scheme and it is called the formal Picard group of {X}. Although, in general you’ll want to do this with the Brauer group or higher analogues to guarantee existence and smoothness. Then you prove a remarkable fact that the elliptic curve is ordinary if and only if the formal group has height {1}. In particular, since the {p}-divisible group is connected and {1}-dimensional it must be isomorphic to {\mu_{p^\infty}}.

It might seem silly to think in these terms, but there is another “enlarged” {p}-divisible group attached to {E} which always has height {2}. This is the {p}-divisible group you get by taking the inductive limit of the finite group schemes that are the kernel of multiplication by {p^n}. It is important to note that these are non-trivial group schemes even if they are “geometrically trivial” (and is the reason I didn’t just call it the “{p^n}-torsion”). We’ll denote this in the usual way by {E[p^\infty]}.

I don’t really know anyone that studies elliptic curves that phrases it this way, but since this theory must be generalized in a certain way to work for other varieties like K3 surfaces I’ll point out why this should be thought of as an enlarged {p}-divisible group. It is another standard fact that {E} is ordinary if and only if {E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}. In fact, you can just read off the connected-etale decomposition:

\displaystyle 0\rightarrow \mu_{p^\infty}\rightarrow E[p^\infty] \rightarrow \mathbb{Q}_p/\mathbb{Z}_p\rightarrow 0

We already noted that {\widehat{E}\simeq \mu_{p^\infty}}, so the {p}-divisible group {E[p^\infty]} is a {1}-dimensional, height {2} formal group whose connected component is the first one we talked about, i.e. {E[p^\infty]} is an enlargement of {\widehat{E}}. For a general variety, this enlarged formal group can be defined, but it is a highly technical construction and would take a lot of work to check that it even exists and satisfies this property. Anyway, this enlarged group is the one we need to work with otherwise our deformation space will be too small to make the theory work.

Here’s what Serre-Tate theory is all about. If you take a deformation of your elliptic curve {E} say to {E'}, then it turns out that {E'[p^\infty]} is a deformation of the {p}-divisible group {E[p^\infty]}. Thus we have a natural map {\gamma: Def_E \rightarrow Def_{E[p^\infty]}}. The point of the theory is that it turns out that this map is an isomorphism (I’m still assuming {E} is ordinary here). This is great news, because the deformation theory of {p}-divisible groups is well-understood. We know that the versal deformation of {E[p^\infty]} is just {Spf(W[[t]])}. The deformation problem is unobstructed and everything lives in a {1}-dimensional family.

Of course, let’s not be silly. I’m pointing all this out because of the way in which it generalizes. We already knew this was true for elliptic curves because for any smooth, projective curve the deformations are unobstructed since the obstruction lives in {H^2}. Moreover, the dimension of the space of deformations is given by the dimension of {H^1(E, \mathcal{T})}. But for an elliptic curve {\mathcal{T}\simeq \mathcal{O}_X}, so by Serre duality this is one-dimensional.

On the other hand, we do get some actual information from the Serre-Tate theory isomorphism because {Def_{E[p^\infty]}} carries a natural group structure. Thus an ordinary elliptic curve has a “canonical lift” to characteristic {0} which comes from the deformation corresponding to the identity.

Heights of Varieties

Now that we’ve defined the height of a {p}-divisible group we’ll define the height of a variety in positive characteristic. There are a few ways we can motivate this definition, but really it just works and turns out to be a very useful concept. We’ll mostly follow the paper of Artin and Mazur.

We could do this in more generality, but to keep things as simple as possible we’ll assume that we have a proper variety {X} over a perfect field {k} of characteristic {p}. The first motivation is that we can think about {\mathrm{Pic}(X)}. One way to get information about this group is to use deformation theory and look at the formal completion {\widehat{\mathrm{Pic}}(X)}.

The way to define this is to define the {S}-valued points ({S} an Artin local {k}-algebra with residue field {k}) to be the group fitting into the sequence {0\rightarrow \widehat{\mathrm{Pic}}(X)(S)\rightarrow H^1(X\times S, \mathbb{G}_m)\rightarrow H^1(X, \mathbb{G}_m)}.

So {\widehat{\mathrm{Pic}}(X)} is a functor which by Schlessinger’s criterion is prorepresentable by a formal group over {k}. Notice that {\widehat{\mathrm{Pic}}(X)(S)=\mathrm{ker}(\mathrm{Pic}(X\times S)\rightarrow \mathrm{Pic}(X))}, so there is a pretty concrete way to think about what is going on. We take our scheme and consider some nilpotent thickening. The line bundles on this thickening that are just extensions from the trivial line bundle are what is in this formal Picard group.

There is no reason to stop with just {H^1}. We could define {\Phi^r: Art_k\rightarrow Ab} by {\Phi^r(S)} is the kernel of the restriction map {H^r(X\times S, \mathbb{G}_m)\rightarrow H^r(X, \mathbb{G}_m)}. In the cases we care about, modulo some technical details, we can apply Schlessinger type arguments to this to get that if the dimension of {X} is {n}, then {\Phi^n} is not only pro-representable, but by formal Lie group of dimension {1}. We’ll call this {\Phi_X}.

When {n=2} this is just the well-known Brauer group, and so for instance the height of a K3 surface is the height of the Brauer group. We also have that if {\Phi_X} is not {\widehat{\mathbb{G}}_a} then it is a {p}-divisible group and amazingly the Dieudonne module of {\Phi_X} is related to the Witt sheaf cohomology via {D(\Phi_X)=H^n(X, \mathcal{W})}. Recall that {D(\Phi_X)} is a free {W(k)}-module of rank the height of {\Phi_X}, so in particular {H^n(X, \mathcal{W})} is a finite {W(k)}-module!

Remember that we computed an example where that wasn’t finitely generated. So non-finite generatedness of {H^n(X, \mathcal{W})} actually is related to the height in that if the variety is of finite height then {H^n(X, \mathcal{W})} is finitely generated. Since we call a variety of infinite height supersingular, we can rephrase this as saying that {H^n(X, \mathcal{W})} is not finitely generated if and only if {X} is supersingular.

Just as an example of what heights can be, an elliptic curve must have height {1} or {2} and a K3 surface can have height between {1} and {10} (inclusive). As of right now it seems that the higher dimensional analogue of if the finite height range of a Calabi-Yau threefold is bounded is still open. People have proved certain bounds in terms of hodge numbers. For instance {h(\Phi_X)\leq h^{1, 2}+1}. For a general CY {n}-fold we have {h\leq h^{1, n-1}+1}.

This is pretty fascinating because my interpretation of this (which could be completely wrong) is that since for K3 surfaces the moduli space is {20} dimensional, we get that (for non-supersingular) {h^{1,1}=20} since this is just the dimension of the tangent space of the deformations, which for a smooth moduli should match the dimension of the moduli space. Thus we get a uniform bound (not the one I mentioned earlier).

But for CY threefolds the moduli space is much less uniform. They aren’t all deformation equivalent. They lie on different components that have different dimensions (this is a guess, I haven’t actually seen this written anywhere). So this doesn’t allow us to say {h^{1,2}} is some number. It depends on the dimension of the component of the moduli that it is on (since {h^{1,2}=\dim H^2(X, \Omega)=\dim H^1(X, \mathcal{T})} using the CY conditions and Serre duality). So I think it is still an open problem for how big that can be. If it can get unreasonably large, then maybe we can arbitrarily large heights of CY threefolds.

Next time maybe we’ll prove some equivalent ways of computing heights for CY varieties and talk about how height has been used by Van der Geer and Katsura and others in a useful way for K3 surfaces.

Heights of p-divisible Groups

Let’s try to define a few words I’ve thrown around for a few weeks. What is a {p}-divisible group, and how do we know what its height is? I’m going to do two things that will either make this easier to understand or way more confusing. We will be working with group schemes. To keep from repeating everything twice with the word “formal” in front of everything I will rarely specify whether I mean formal group scheme or group scheme. Obviously some things are different in the formal case, but not so much. The other thing I’ll do is assume our group schemes are affine to simplify notation, but if you want to do this more generally you can.

The point of these posts should be to give an overview of how these things fit together. When trying to learn about this stuff, there are so many hundreds of terms and papers and details in the papers that it is really easy to forget what is going on. For instance, basically any reference on {p}-divisible groups will get very caught up in all the technical details of Dieudonne theory, and I want to massively downplay this aspect for the purpose of defining this invariant called the height of a variety.

On to the definition. A {p}-divisible group is just an inductive system {(G_\nu, i_\nu)} of group schemes satisfying two properties. First there must be an {h} so that the order of {G_\nu} is {p^{\nu h}}. Second, they must fit into an exact sequence {0\rightarrow G_\nu \stackrel{i_\nu}{\rightarrow} G_{\nu +1 } \stackrel{ p^\nu}{\rightarrow} G_{\nu +1}}. All this says is that if we look at the map that is multiplication by {p^{\nu}} on {G_{\nu +1}}, the kernel of this is the copy of {G_{\nu}} that sits inside {G_{\nu +1}} via {i_\nu}.

This might seem like a strange set of conditions at first, but really the two most natural examples of forming inductive systems of group schemes already satisfy both of these. The first one is to take an abelian variety {X} of dimension {g}. Then we have the isogenies multiplication by {p}, multiplication by {p^2}, etc. The kernels of these are all group schemes and it is well-known that they are isomorphic to {(\mathbb{Z}/p)^{2g}}, {(\mathbb{Z}/p^2)^{2g}}, etc. We just take the maps to be the inclusions and the {h=2g}.

The other main example is to do the same trick with {\mathbb{G}_m} by taking successive kernels of multiplication by {p}. We just get the inductive system {(\mu_{p^\nu}, i_{\nu})}. In this case the orders are just {p^\nu}, so {h=1}. We’ve suggestively labelled this number {h} which is the height of the {p}-divisible group. This one is usually denoted {\mathbb{G}_m(p)}.

There are lots of easy properties of {p}-divisible groups that can be verified mentally. For instance, you can put any two of {G_\nu} and {G_\alpha} into an exact sequence {0\rightarrow G_\nu \rightarrow G_{\nu + \alpha} \rightarrow G_\alpha \rightarrow 0}. A slightly harder property to check is that under mild base assumptions we have an equivalence of categories between {p}-divisible groups and divisible formal Lie groups. Under this equivalence we get that {\mathbb{G}_m(p)} corresponds to the one-dimensional Lie group with group law {F(X, Y)=X + Y + XY}, so our earlier notation of calling this {\widehat{\mathbb{G}_m}} makes sense because it comes from {\mathbb{G}_m(p)}.

Since ultimately we are concerned with computing heights, we should see if there is a way to figure out the height without computing orders. Let’s denote the Cartier dual of a group scheme by {G^D=Hom(G, \mathbb{G}_m)}. We can check that taking Cartier duals everywhere, we get another {p}-divisible group. I.e. {(G_\nu^D, i_\nu^D)} is an inductive system satisfying the properties of a {p}-divisible group. This is often called the Serre dual. If we denote the whole inductive system by {G}, then it is customary to write {G^t} for the Serre dual.

Note that duals can be quite different from the original group. In particular, the dimension can be different. In the case of {\mathbb{G}_m(p)} we get a dimension {1} etale group scheme, but it’s dual is {(\mathbb{Z}/p^\nu)} and hence a dimension {0} connected (with nilpotent structure) group scheme. If we add these two dimensions we get {1}, the height. This is true in general. We have the formula {h=\textrm{dim}(G)+\textrm{dim}(G^t)}. So we only need to know the dimensions of the group and its dual.

Another (much harder for calculating, but sometimes handy in theory) way to determine the height is to use the Dieudonne module we defined last time. If we take {D(G)} it is a {\mathbb{D}_k}-module, but also a free of finite rank {W(k)}-module. The rank of this module turns out to be the height of {G}. In a similar fashion, you can form another module out of {G} called the Tate module. By definition, multiplication by {p} is a map {G_{\nu +1}\rightarrow G_\nu}, and hence we get an inverse system. We define {T(G)=\lim G_{\nu}(\overline{k})} to be the Tate module. It is a free {\mathbb{Z}_p}-module. The rank of this is the height of {G}.

That is about the sketchiest crash course on {p}-divisible groups you can get, but I think it mentions enough to get to the next definition: the height of a variety in positive characteristic.

Formal Witt Vectors

Last time we checked that our explicit construction of the ring of Witt vectors was a ring, but in the proof we noted that {W} actually was a functor {\mathrm{Ring}\rightarrow\mathrm{Ring}}. In fact, since it exists and is the unique functor that has the three properties we listed, we could have just defined the ring of Witt vectors over {A} to be {W(A)}.

We also said that {W} was representable, and this is just because {W(A)=Hom(\mathbb{Z}[x_1, x_2, \ldots ], A)}. We can use our {\Sigma_i} to define a (co)commutative Hopf algebra structure on {\mathbb{Z}[x_1, x_2, \ldots]}.

For instance, define the comultiplication {\mathbb{Z}[x_1, x_2, \ldots ]\rightarrow \mathbb{Z}[x_1,x_2,\ldots]\otimes \mathbb{Z}[x_1,x_2,\ldots]} by {x_i\mapsto \Sigma_i(x_1\otimes 1, \ldots , x_i\otimes 1, 1\otimes x_1, \ldots 1\otimes x_i)}.

Since this is a Hopf algebra we get that {W=\mathrm{Spec}(\mathbb{Z}[x_1,x_2,\ldots])} is an affine group scheme. The {A}-valued points on this group scheme are by construction the elements of {W(A)}. In some sense we have this “universal” group scheme keeping track of all of the rings of Witt vectors.

Another thing we could notice is that {\Sigma_1(X,Y)}, {\Sigma_2(X,Y)}, {\ldots} are polynomials and hence power series. If we go through the tedious (yet straightfoward since it is just Witt addition) details of checking, we will find that they satisfy all the axioms of being an infinite-dimensional formal group law. We will write this formal group law as {\widehat{W}(X,Y)} and {\widehat{W}} as the associated formal group.

Next time we’ll start thinking about the length {n} formal group law of Witt vectors (truncated Witt vectors).

Formal Groups 4

Let’s back up from all these definitions for now and see one situation in which all of these things just pop out. It also will show us where these names came from. Suppose {G} is a smooth algebraic affine group scheme over {k}. Then {G} is represented by a finitely generated Hopf algebra, say {A}. If {I} is the augmentation ideal (i.e. {I=\text{ker}(\epsilon^*: A\rightarrow k)}), then by smoothness we get that {I/I^2} is free on the generators {x_1, \ldots, x_r} where {r} is the number of generators of {A} over {k}.

Likewise {I^n/I^{n+1}} is free and generated by monomials {x_1^{m_1}\cdots x_r^{m_r}}, thus if we take the completion with respect to the {I}-adic topology, we get {\widehat{A}=\lim (A/I^n)\simeq k[[x_1, \ldots x_r]]}. Now notice that {\Delta (I)\subset I\otimes A+A\otimes I}, and this is just the maximal ideal defining {(e,e)} in the product. Thus the maps pass to the successive quotients, and we get an induced map on completions.

This is just {\widehat{\Delta}:k[[x_1, \ldots , x_r]] \rightarrow k[[x_1', \ldots, x_r', x_1'', \ldots , x_r'']]}. But any such map is completely described by where each of the {x_i} get sent to. But they are sent to power series in {2r} variables! So we get {r} power series in {2r} variables, namely {\widehat{\Delta}(x_i)=F_i(X,Y)}. Notice these came from maps of Hopf algebras (not merely ring maps, otherwise this wouldn’t work), so tracing the coassociativity axiom we get precisely the associativity we need to say that these power series form a formal group law of dimension {r}.

Even more importantly, we could have chosen different generators originally, and this would have changed the construction and given a different formal group law, but they are isomorphic. The isomorphism is just given by a change of variables. So given any smooth affine algebraic group scheme over {k} we get a unique formal group up to isomorphism, and moreover (given appropriate other technical conditions we won’t discuss) there is actually an (anti)equivalence of categories between formal groups and Hopf algebras.

This is exactly how the formal group laws {\widehat{\mathbb{G}}_a} and {\widehat{\mathbb{G}}_m} were formed. We take the affine algebraic group scheme {\mathbb{G}_a} which is represented by {k[z]}, then {\Delta(z)=z\otimes 1+1\otimes z}. The completion is clearly {k[[z]]\rightarrow k[[x,y]]} and the map {\widehat{\Delta}(z)=x+y}. Thus our one-dimensional additive formal group (scheme) is appropriately named, and it is just as easy to go through and check the multiplicative one gives the right law as well.

Now we see that these definitions of formal groups aren’t just arbitrary isolated things. They actually arise in practice. We will also see another important way in which they appear next time (or the time after) in defining new rings with particularly nice properties.

Lastly, we need to do one more thing that doesn’t really fit anywhere nicely. We started with one-dimensional formal groups, and then moved to arbitrary finite dimension, but really we can just keep going and define infinite dimensional in exactly the same way. If we have an index set {I}, then take a collection of indeterminates {(x_i)} indexed by {i}. Define the formal power series ring {A[[x_i]]} to be all formal sums {\sum c_nx^n} where {n} runs through functions {n:I\rightarrow \mathbb{N}\cup \{0\}} with finite support, and {x^n=\prod x_i^{n(i)}}.

Now an infinite dimensional formal group law is a collection of elements {F_i(X, Y)\in A[[X_i, Y_i]]_{i\in I}} which we write {\displaystyle F_i(X,Y)=\sum_{m,n} c_{m,n}(i)X^mY^n}, and the usual conditions {F_i(X,Y)=X_i+Y_i+}(higher degree) and {F_i(F(X,Y),Z)=F_i(X, F(Y,Z))} and we need one extra finiteness condition that {c_{m,n}(i)\neq 0} for only finitely many {i}.

This may seem implausible to appear in practice, but there is a really simple example where you get this occuring. Take any ring {A}, then we can form a new ring {A[X_i, Y_i]} an infinite-dimensional polynomial algebra over {A}. Then take just a single variable power series over this to get {A[X_i, Y_i][[t]]}. Taking any two arbitrary elements with constant coefficient {1}, we can multiply them: {(1+\sum X_it^i)(1+\sum Y_it^i)=1+\sum F_i(X,Y)t^i}. We know the product has this form, and the {F_i(X,Y)} form an infinite dimensional formal group law.

Reversing this process, we can define a multiplication given an infinite dimensional group law, and this is how we’ll define our new rings, including the Witt vectors.

Formal Groups 3

Today we move on to higher dimensional formal group laws over a ring {A}. Side note: later on we’ll care about the formal group attached to a Calabi-Yau variety in positive characteristic which is always one-dimensional, but to talk about Witt vectors we’ll need the higher dimensional ones.
An {n}-dimensional formal group law over {A} is just an {n}-tuple of power series, each of {2n} variables and no constant term, satisfying certain relations. We’ll write {F(X,Y)=(F_1(X,Y), F_2(X,Y), \ldots , F_n(X,Y))} where {X=(x_1, \ldots , x_n)} and {Y=(y_1, \ldots , y_n)} to simplify notation.

There are a few natural guesses for the conditions, but the ones we actually use are that {F_i(X,Y)=x_i+y_i+}(higher degree) and for all {i} {F_i(F(X,Y),Z)=F_i(X, F(Y,Z))}. We call the group law commutative if {F_i(X,Y)=F_i(Y,X)} for all {i}.

We still have our old examples that are fairly trivial {\widehat{\mathbb{G}}_a^n(X,Y)=X+Y}, meaning the i-th one is just {x_i+y_i}. For a slightly less trivial example, let’s explicitly write a four-dimensional one





It would beastly to fully check even one of those associative conditions. I should probably bring this to your attention, but the condition {F_i(F(X,Y), Z)} has you input as the first four variables those four equations, so in this case checking the first condition amounts to {F_1(F_1(X,Y),F_2(X,Y),F_3(X,Y),F_4(X,Y),z_1,z_2,z_3,z_4)}

{=F_1(x_1,x_2,x_3,x_4, F_1(Y,Z),F_2(Y,Z),F_3(Y,Z),F_4(Y,Z))}.

But on the other hand, the fact that {\widehat{\mathbb{G}}_a^n} satisfies it is trivial since you are only adding everywhere which is associative.

Now we can do basically everything we did with the one-dimensional case now. For one thing we can form the set {\mathcal{C}(F)} of {n}-tuples of power series in one indeterminant and no constant term. This set has an honest group structure on it given by {\gamma_1(t)+_F \gamma_2(t)=F(\gamma_1(t), \gamma_2(t))}.

We can define a homomorphism between an {n}-dimensional group law {F} and {m}-dimensional group law {G} to be an {m}-tuple of power series in {n}-indeterminants {\alpha(X)} with no constant term and satisfying {\alpha(F(X,Y))=G(\alpha(X), \alpha(Y))}.

From here we still have the same inductively defined endomorphisms for any {n} (not just the dimension) from the one-dimensional case {[0]_F(X)=0}, {[1]_F(X)=X} and {[n]_F(X)=F(X, [n-1]_F(X))}. That’s a lot of information to absorb, so we’ll end here for today.