# P-Divisible Groups Mini Course: Talk 4

This is the final post in my recent series on $p$-divisible groups and related topics. For the last talk, I want to discuss one place close to my own interests where this theory is used; namely, the Local Langlands program. In particular I want to discuss the case for the reductive group $GL_n$ where Lubin-Tate spaces are used to prove the Local Langlands correspondence for this group. The post is split into three parts. In the first, we review Lubin-Tate Theory as it appears in Local Class Field Theory. In the second part we discuss how this generalizes to Local Langlands for $GL_n$. Finally, in the third part we mention how the situation generalizes to other reductive groups and the connection to the theory of $p$-divisible groups.

The scope of this talk may sound impressive but first a disclaimer which I feel I should make out of honesty. A thorough description of the material in the second two sections requires greater knowledge of representation theory and the Langlands program than I currently possess. For this reason, I have had to be a bit vague in places and rely more heavily than normal on my resources. Moreover, though I have tried to be careful, I would not be surprised if I made some mistakes towards the end. As my understanding of the theory improves in the coming months, I’ll try to update this post.

## Local Class Field Theory Via Lubin Tate Formal Groups

In this first section we review the classical construction of a maximal Abelian extension of a local field $K$ over $\mathbb{Q}_p$ using Lubin-Tate formal groups. Let $\mathbb{F}_q$ be the residue field of $\mathcal{O}_K$, Let $L$ be the complete maximal unramified extension of $K$ and let $\mathcal{O}_L$ be its completed integer ring with residue field $\overline{\mathbb{F}_q}$.

In the Lubin-Tate setup, we will consider lifts of one dimensional formal groups over $\overline{\mathbb{F}_q}$. Take the $q$th power automorphism $\phi$ of $\overline{\mathbb{F}_q}$ which therefore fixes $\mathbb{F}_q$. If $\overline{G}=\mathrm{Spf} \overline{\mathbb{F}_q}[[X]]$ is a formal group over $\overline{\mathbb{F}_q}$ with group law $\overline{G}(X, Y)= X+Y+ \sum\limits{i,j} a_{i,j}X^iY^j$ then we can consider the $\overline{\phi}$-twist of $\overline{G}$ which we denote $\overline{G}^{\overline{\phi}}$ with group law $\overline{G}^{\overline{\phi}}(X,Y)= X+Y + \sum\limits_{i,j} \phi(a_{i,j})X^iY^j$. Put another way, $\overline{G}^{\overline{\phi}}$ is the relative Frobenius twist of $\overline{G}$ corresponding to the ‘Frobenius morphism’ $\overline{\phi}$. Thus we have a natural map $f: \overline{G} \to \overline{G}^{\overline{\phi}}$ defined over $\overline{\mathbb{F}_q}$ and given on rings by $X \mapsto X^q$  which we call the Frobenius morphism of $\overline{G}$.

Now we consider lifts of such formal groups to $\mathcal{O}_L$. Notice that $\overline{\phi}$ lifts uniquely to some automorphism $\phi$ of $L$ which fixes $K$.  Thus, given a lift $G$ of $\overline{G}$ we can also define its $\phi$-twist $G^{\phi}$. Furthermore, we are interested in lifts $G$ of $\overline{G}$ which are equipped with an $\mathcal{O}_K$-action such that the induced map of $\alpha \in \mathcal{O}_K$ on the tangent space of $G$ is given by multiplication by $\alpha$. We call such objects formal $\mathcal{O}_K$-modules.

Our analysis of the situation when $G$ has dimension one is greatly helped by the following fundamental theorem.

Theorem: Let $\pi$ be a uniformizer of $\mathcal{O}_L$, $f \in \mathcal{O}_L [[X]]$ such that $f \equiv \pi X \mathrm{mod} \mathrm{deg} 2 , f \equiv X^q \mathrm{mod} \pi$. Then there exists a unique formal group $G_f$ over $\mathcal{O}_L$ so that $f$ is the Frobenius map $f: G_f \to {G_f}^{\phi}$. Moreover, $G_f$ carries the structure of a formal $\mathcal{O}_K$ module. We call $G_f$ a Lubin-Tate formal group.

The proof of the above is actually quite simple and can be accomplished by determining $G_f$ inductively on rings of the form $\mathcal{O}_L/ (\pi)^n$.

In fact, the construction in the theorem shows that for uniformizers $\pi, \pi'$ and Frobenius lifts $f, f'$, the set ${\Theta^L}_{\pi, \pi'}=\{\theta \in \mathcal{O}_L: \frac{\theta^\phi}{\theta}=\frac{\pi'}{\pi} \}$ corresponds to morphisms $[\theta]_{f, f'}: G_f \to G_{f'}$ such that $[\theta]_{f,f'}$ is the unique morphism $G_f \to G_{f'}$ where the corresponding ring map satisfies $[\theta]_{f, f'} \equiv \theta X \mathrm{mod} \mathrm{deg} 2$ . Thus $\pi \in {\Theta^L}_{\pi, \phi(\pi)}$ and $f=[\pi]_{f, f^{\phi}}: G_f \to {G_f}^\phi$. Furthermore, we can take Frobenius twists of $\pi$ to define the morphisms $f_n=[\pi_n]: G_f \to {G_f}^{\phi^n}$ determined by $\pi_n= \prod\limits_{i=0}^{n-1} \phi^i(\pi)$ so that $f=f_1$. One can show that $f_n$ has distinct roots.

Key ExampleLet $K= \mathbb{Q}_p$ and consider the uniformizer $p$ and $f=(X+1)^p-1$. Then $G_f=\hat{G_m}$ with group law $G_f(X, Y)= X+ Y + XY$.

In fact the above example is crucial, because one can use the ${\Theta^L}_{\pi, \pi'}$  description of endomorphisms to construct an isomorphism between any two Lubin-Tate formal groups $G_f, G_{f'}$.

We now restrict to the case where $f \in \mathcal{O}_L [[ X]]$ is a monic polynomial. In particular, $f_n$ is of degree $q^n$.

Let $\mu_{f, n}$ be the collection of $q^n$ zeroes of the polynomial $f_n$ (in particular, for the example we gave above these are just the $q^n$th roots of unity. Clearly, $f_n | f_{n+1}$ and one can verify that $f_{n+1}/f_n$ is Eisenstein. Thus the splitting field of $f_{n+1}$ is a totally ramified extension ${L^n}_f$ of $L$.

The roots of $f_n$ have a natural $\mathcal{O}_K$-module structure induced from the group law and $\mathcal{O}_K$-action on $G_f$. One can concretely show that this module is isomorphic to $\mathcal{O}_K/ (\pi)^n$.  Now the key insight is that $\mu_{f,n}$ carries another action, namely that of $\mathrm{Gal}({L^n}_f/L)$. The power series involved in the $\mathcal{O}_L$ action have coefficients in $\mathcal{O}_L$, thus they commute with the Galois action and so we can concretely see that we have an embedding

$\mathrm{Gal}({L^n}_f/L) \hookrightarrow \mathrm{End}_{\mathcal{O}_K}( \mu_{f, n})$.

Since the latter is just $(\mathcal{O}_K/ (\pi)^n)^*$, one can compare orders and show that the splitting field of $\mu_{f,n}$ has order $(q-1)q^{n-1}$ over $L$ and that the Galois group is Abelian and isomorphic to $(\mathcal{O}_K/ (\pi)^n)^*$.

At this point, we’ve constructed a tower of totally ramified Abelien extensions of $L$. In fact the compositum of these will be a maximum Abelian ramified extension. Though this requires some work to prove, we can state that intersecting this with $K^{sep}$ gives the maximal Abelian extension of $K$. Moreover taking care to worry about compatibility issues,  the above identification of $\mathrm{Gal}({L^n}_f / L)$ and $(\mathcal{O}_K / (\pi)^n)^*$ allows us to construct the Artin map $\mathrm{Art}: K^* \to \mathrm{Gal}(K^{ab}/ K)$.

## Connection to Local Langlands

We continue with the notation as before. In this section, we’ll only consider formal groups of dimension $1$ but we will vary the height. All the formal groups we considered above were $p$-divisible and so everything we say can equally be translated into the language of $p$-divisible groups.

From the machinery we’ve built up from previous talks, we can easily show that there is a unique formal group over $\overline{\mathbb{F}_q}$ of dimension $1$ and height $n$. This is because the corresponding Newton polygon can have no slope zero part because of the assumption that we have a formal group and a single positive slope will increase the dimension to $1$. Thus the formal group has slope $\frac{1}{h}$. In fact, one can mess around with Dieudonne modules to show that there is a unique formal group of dimension one and slope $h$ up to isomorphism. If we restrict ourselves to dimension $1$ formal $\mathcal{O}_K$-modules over $\overline{\mathbb{F}_q}$ then there is a unique one of height $log_p(q)h$ which we denote $\overline{G}$.

Now we want to consider the space of deformations of $\overline{G}$. In particular, we want to consider the functor $M_0$ which assigns to each complete local Noetherian  $\mathcal{O}_L$-algebra $A$ with residue field $\overline{\mathbb{F}_q}$ the set of pairs $(G, i)$ where $G$ is a one dimensional formal group over $A$ and $i: \overline{G} \to G \times_{\mathcal{O}_L} \overline{\mathbb{F}_q}$ is an isomorphism.

Theorem (Lubin-Tate): The functor $M_0$ is representable by a formal scheme non-canonically isomorphic to $\mathrm{Spf} \mathcal{O}_L [[ X_1, ..., X_{h-1}]]$.

We define the Lubin-Tate deformation space of level $0$, $\mathcal{M}_0$, to be the rigid generic fiber (the rigid open ball of radius $1$) of $M_0$. We should think of $\mathcal{M}_0$ as the local analog of a Shimura variety. In particular one would like to attach level structure to $\mathcal{M}_0$. This was done by Drinfeld, who constructed an adic space $\mathcal{M}_n$ for each is which is etale over $\mathcal{M}_0$ with Galois group $GL_h(\mathcal{O}_K/ \pi^n \mathcal{O}_K)$. The $L$ points of $\mathcal{M}_n$ correspond to triples $(G, i , \alpha)$ where $G$ and $i$ are as before and $\alpha: (\mathcal{O}_K/ \pi^n \mathcal{O}_K)^h \to G[{\pi}^n]$ is an isomorphism of $\mathcal{O}_K/ \pi^n \mathcal{O}_K$ modules.

The system of spaces $\mathcal{M}_n$ has natural actions by lots of interesting groups. $g \in GL_h(\mathcal{O}_K)$ acts by $(G, i, \alpha) \mapsto (G , i ,\alpha \circ g)$. The endomorphism ring of $\overline{G}$ is isomorphic to the maximal order $\mathcal{O}(B)$ of the division ring $B$ with invariant $\frac{1}{h}$ over $K$ and this acts by $(G, i, \alpha) \mapsto (G, i \circ b, \alpha)$. Finally we have an action of the Weil group of $K$ which we denote $W_K$, though this takes a bit to explain. Let $\sigma \in W_K$ and consider the valuation map defined so that $v(\sigma)$ is the power of Frobenius that $\sigma$ corresponds to restricted to $K^{unr}$. Let $v(\sigma)=n$. Then $\sigma$ takes $G$ to $G^{\phi^n}$, $i$ to $i^{\sigma} \circ F^n$ where $i^{\sigma}: \overline{G}^{(q^n)} \to G^{\phi^n} \times_{\mathcal{O}_L} \overline{\mathbb{F}_q}$ and $F^n$ is the Frobenius twist $\overline{G} \to \overline{G}^{(q^n)}$, and $\alpha$ to $\alpha^{\sigma}$ which we get by twisting $\alpha$ by $\phi^n$.

This got really complicated all of a sudden so lets take a step back and see what all this implies for the case when $h=1$. These formal groups are the Lubin-Tate formal groups we constructed earlier. $\mathcal{M}_0$ is just $\mathrm{Spec} O_L$ with fiber $\mathrm{Spec} L$.

Let’s think about what $\mathcal{M}_n$ should be in this case. As a scheme it corresponds to the constant $\mathcal{O}_K$-module $\mathcal{O}_K/ \pi^n \mathcal{O}_K$. We think of $\mathcal{M}_n$ as being $\mu_{f, n}$ thought of as a constant group scheme with its $\mathcal{O}_K$-module structure.  We have $GL_1(\mathcal{O}_K)={\mathcal{O}_K}^*, \mathcal{O}(B)^*={\mathcal{O}_K}^*$. The relevant actions are that $GL_1$ acts by module automorphisms, $b \in \mathcal{O}(B)$ acts by multiplying the module by $b^{-1}$ and the inertia group in $W_K$ acts as ${\mathcal{O}_K}^*$ via the Artin map and Frobenius acts by twisting.

Now we briefly discuss the relation to Local Langlands. The point is that by taking rigid $l$-adic etale cohomology, we can convert the tower $\mathcal{M}_n$ of rigid spaces to a direct system of $\overline{\mathbb{Q}_l}$-vector spaces. The group actions on the rigid spaces translate to actions on the cohomology and thus turn the cohomology into an interesting representation of $GL_h(\mathcal{O}_K) \times \mathcal{O}(B)^* \times W_K$. The key insight is that decomposing this representation into a sum of irreducible representations gives relationships between the irreducible representations of these groups, concretely realizing Local Langlands correspondences.

## Generalizing to Reductive Groups

Lets end by discussing how one might generalize the above. The key to the Lubin-Tate construction was building a tower of geometric spaces (in this case the $\mathcal{M}_n$) which carry actions by the relevant groups. In the $GL_n$ case we saw that this was accomplished by constructing a deformation space of formal groups of a particular height and dimension. In this case, the formal groups correspond to $p$-divisible groups. Thus, we might hope to build geometric spaces out of moduli of $p$-divisible groups. In fact, this is exactly what is done.

Lets analogize with the case PEL type Shimura varieties. Such a variety is naturally a moduli space for Abelian varieties with extra structure such as polarization and endomorphism data. This data is controlled by a reductive group $G$ which in turn acts on the Shimura variety. Similary, one can consider deformation spaces  of $p$-divisible groups with additional structures which will admit actions by reductive groups.  Just as in the case of Shimura varieties, we can add in level structure data to get a tower of spaces. These spaces are called Rapoport-Zink spaces.

Of course many cases of the Local Langlands correspondence are still open. Rapoport-Zink spaces have only been constructed for certain reductive groups and for cases where the correct space is known, the cohomology representations can prove very difficult to analyze.

## References

Carayol has an article in Volume II of Automorphic Forms, Shimura Varieties, and L-functions which sketches a proof of Local Langlands for $GL_2$ using Lubin-Tate theory.

Yoshida has written a very readable article proving local class field theory using Lubin-Tate formal groups. Yoshida

Mantovan Is a good source for reading about the connections between Lubin-Tate theory and Rapoport-Zink Spaces.

Weinstein has a very readable introduction to many of the ideas surrounding Lubin-Tate theory.

Rapoport and Viehmann have written a survey paper on Rapoport-Zink spaces and connections to Local Langlands titled Towards a Theory of Local Shimura Varieties

## One thought on “P-Divisible Groups Mini Course: Talk 4”

1. My old friend Bertie,

1) I think you have a typo in the second paragraph of ‘Connections to Langlands’. Namely, you are missing ‘up to isogeny’ in your first claim about connected $p$-divisible groups of dimension $1$ and height $n$ being unique.

2) Maybe it’s worth mentioning what this formal group is. So, for height $1$ it’s $\mu_{p^\infty]$, for height $2$ it’s $E[p^\infty]$ with $E/\overline{\mathbb{F}_p}$ supersingular, and in general it’s one of the finite Witt schemes that, if we wanted to, we could figure out by slope considerations. OK, let’s do this. In your langue from the second talk we have that $G_{m,n}$ has height $m+n$ and dimension $m$. Thus, it has to be $G_{1,h-1}$. This is, not shockingly, related to the fact that the Shimura variety coming into play for Taylor-Harris’s proof is $\mathrm{GU}(1,n-1)$.

3) One makes precise the fact that $\mathcal{M}_0$ is like the local analogue of a Shimura variety, I believe, by the statement that it’s the rigid generic fiber of the completion of the local ring of the $\mathrm{GU}(1,n-1)$ Shimura variety at the associated point.

4) I think it’s more natural, and less scary if you don’t like adic spaces, to think about these $\mathcal{M}_n$‘s as being rigid generic fibers of formal schemes $M_n$ which are, literally, just deformation rings of formal $\mathcal{O}$-modules with level structures.

5) One usually wants to get the whole action of $B^\times$ (I think you meant $\mathcal{O}(B)^\times$ by the way) on the cohomology of our space to get Jacquet-Langlands. One does this by considering the moduli problem of deformations but only requiring that the special fiber have a quasi-isogeny to our fixed formal group.

6) I think you mean to say “$M_0$ is just $\mathrm{Spf}(\mathcal{O}_L)$ with formal fiber $\mathrm{Spa}(L,\mathcal{O}_L)$“.

7) Pedantic point is that $G$ doesn’t act on the Shimura variety. In fact, $G(\mathbb{A}_f)$ (or $G(\mathbb{A}_f^p)$) doesn’t even act on the Shimura variety. It acts by ‘correspondences’ which, in turn, gives an *honest* action on *cohomology*.

8) As mentioned in the lecture, one needs to be careful about the direct analogy to higher $n$. Namely, for $n=1$ and $n=2$ one can use $\text{GL}_1$ and $\text{GL}_2$ as the global Shimura varieties. Unfortunately, there is no $\text{GL}_n$ Shimura variety for $n\geqslant 3$. Thus, one uses something like $\mathrm{GU}(1,n-1)$.

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