In Moduli Stacks of Galois Representations we discussed the moduli stack of representations of the absolute Galois group when is a finite extension of , for representations valued in some -algebra, where is a prime number different from . When however, it turns out using the same strategy as for can lead to some moduli stacks that are difficult to study.

Instead, we are going to use the theory of -modules as an intermediary. Let be the residue field of , and let be the p-adic completion of the Laurent series field (where here denotes the ring of Witt vectors over ). For future reference, let us now also state that we will use to denote . An **etale -module** over is a finite -module equipped with commuting semilinear actions of (an endomorphism coming from the Frobenius of ) and (the subgroup of isomorphic to , see also Iwasawa theory, p-adic L-functions, and p-adic modular forms). One of the important facts about etale -modules is the following:

**The category of etale -modules is equivalent to the category of continuous -modules over finite -modules.**

This equivalence is given more explicit as follows. Let be the p-adic completion of the ring of integers of the maximal unramified extension of in . Then to obtain a -module from a -module , we take

and to obtain the -module from the – module , we take

where is adjoin all the -power roots of unity. Note that if we wanted bonafide Galois representations instead of -modules we can invert , and our Galois representations will be over . They will be equivalent to -modules over .

More generally we can consider etale -modules with coefficients in , where is some -algebra. This means they are finite -modules, where is the p-adic completion of , instead of -modules (again for future reference, we note that will be used for ). The category of etale -modules with coefficients in is equivalent to the category of continuous -representations over finite -modules.

We define the stack by letting be the groupoid of etale -modules with coefficients in for a p-adically complete -algebra and are projective of rank . That this is indeed a stack follows from the notion of a finitely generated projective module -module being local for the fpqc topology. But has more structure than just being a stack. It is an “ind-algebraic” stack, a notion which we shall explain in the next few paragraphs. As the approach we outline in this post was originally developed by Matthew Emerton and Toby Gee, the stack is also known as the **Emerton-Gee stack**.

### The ind-algebraicity of the moduli stack

As stated earlier, the moduli stack is an **ind-algebraic stack**. This means it can be written as the 2-colimit in the 2-category of stacks of 2-directed system of algebraic stacks . (Recall from Algebraic Spaces and Stacks that an algebraic stack is an fppf stack such that there exists a map from an affine scheme to and this map is representable by algebraic spaces, surjective, and smooth).

To understand why is an ind-algebraic stack, we need to understand it as the scheme-theoretic image of a certain map of certain moduli stacks. The idea is that the target stacks (which is going to be the moduli stack of -modules with a semilinear action of the “discretization” of , more on this later) is ind-algebraic and we can deduce the ind-algebraicity of from this.

First let us explain what “scheme-theoretic image” means. Let us suppose that we have a limit-preserving fppf sheaf whose diagonal is representable by algebraic spaces and a proper morphism where is an algebraic stack. If admits versal rings satisfying certain properties at all finite-type points then there exists an algebraic closed stack such that factors through and this map is scheme-theoretically dominant.

We will need to discuss moduli stacks of rank -modules, moduli stacks of rank -modules of -height at most , and moduli stacks of rank weak Wach modules of -height at most .

We define to be the stack over such that for any -algebra , is the groupoid of all -modules which are projective and of rank . We have that is also a stack over , and we can define , which is a stack over which we may think of as the moduli space of -modules which are projective and of rank .

Let be a polynomial in which is congruent to a power of modulo and let be a nonnegative integer. A **-module of -height at most over ** is a finitely generated -torsion free -module together with a -semilinear map such that the map is injective, and whose cokernel is annihilated by . We let be the stack such that is the groupoid of -module of -height at most over which are projective of rank .

In the special case that the polynomial is the minimal polynomial of the uniformizer of , a -module of -height at most over is also called a **Breuil-Kisin module of height at most **. We will encounter Breuil-Kisin modules again later.

We have the following important properties of the stacks and :

- The moduli stack is an algebraic stack of finite presentation over , with affine diagonal.
- The moduli stack is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
- The morphism is representable by algebraic spaces, proper, and of finite presentation.
- The diagonal morphism is representable by algebraic spaces, affine, and of finite presentation.

These properties were shown by Emerton and Gee following a strategy originally employed by George Pappas and Michael Rapoport involving relating these stacks to the affine Grassmannian. After taking limits over , we then have the following:

- The moduli stack is an p-adic formal algebraic stack of finite presentation over , with affine diagonal.
- The moduli stack is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
- The morphism is representable by algebraic spaces, proper, and of finite presentation.
- The diagonal morphism is representable by algebraic spaces, affine, and of finite presentation.

In the above, a **formal algebraic stack** is defined similarly to an algebraic stack except our atlas, instead of being a scheme, is a disjoint union of formal schemes (see also Formal Schemes), and we say that a formal algebraic stack over is a **p-adic formal algebraic stack** if it admits a morphism to that is representable by an algebraic stack.

Now let be a topological generator of , and let , so that . Let

be the moduli stack of projective etale -modules of rank together with a semilinear action of (in the above is the diagonal and is the graph of ). The stack is an ind-algebraic stack, which follows from the properties stated earlier. Now the stack maps into , however since it may not be a closed substack this is not yet enough to prove the ind-algebraicity of . So we need to exhibit it as the scheme-theoretic image of an appropriate map into , and this is where the weak Wach modules come in.

A **rank projective weak Wach module of -height at most and level at most over ** is a rank projective -module of -height at most over , such that has a semilinear action of satsifying .

Let be the moduli stack of rank projective weak Wach modules, of -height at most , and level at most . This is a p-adic formal algebraic stack of finite presentation over . To show this we make the following steps.

We consider the fiber product . This is the moduli stack of rank projective -modules over of -height at most , equipped with a semilinear action of on . It is a p-adic formal algebraic stack of finite presentation over .

Now consider , the moduli stack of rank projective weak Wach modules of height at most . We have an isomorphism , and has a closed immersion of finite presentation into the fiber product .

We let . This is a closed substack of . We define to be the scheme-theoretic image of the composition . The stack is a closed substack of , and in fact we will see that is isomorphic to .

Let us explain very briefly how the last statement works. The existence of a morphism from to (which factors through basically comes down to being able to extend the action of to a continuous action of .

Now to show that the morphism , we have to show that for any -algebra any morphism must factor through , for some and some . It is in fact enough to show this for such that there is a scheme-theoretically dominant map and such that if is the -module corresponding to , then is free. The freeness of allows us to find a -invariant lattice inside it which corresponds to a weak Wach module over . Associating to gives us a map . Recalling that the scheme-theoretic image of in is , we see that our map factors through and thus is an isomorphism. The existence of satisfying such properties is guaranteed by the work of Emerton and Gee.

### Crystalline moduli stacks

We briefly mentioned in p-adic Hodge Theory: An Overview that Galois representations that come from the etale cohomology of some scheme are expected to have certain properties related to **p-adic Hodge theory** (this is part of the **Fontaine-Mazur conjecture**), It will therefore be interesting to us to have a moduli space of Galois representations that satisfy such p-adic Hodge-theoretic properties. Namely, we can investigate the moduli space of crystalline and semistable representations, and there are going to be corresponding substacks and of .

Let denote , where denotes the tilt of the ring of integers of the p-adic complex numbers (see also Perfectoid Fields) and is an element of the maximal ideal of whose image in is nonzero. A **Breuil-Kisin-Fargues module of height at most with -coefficients** is a finitely generated -module together with a -semilinear map such that the map is injective, and whose cokernel is annihilated by , where is the minimal polynomial of the uniformizer of . A **Breuil-Kisin-Fargues -module of height at most ** is a Breuil-Kisin-Fargues module of height at most together with a semilinear action that commutes with .

Let us note that given a Breuil-Kisin module , we can obtain a Breuil-Kisin-Fargues module by taking . To be able to take the tensor product we need a map from to , which in this case is provided by sending the element in to a compatible system of p-power roots of the uniformizer in (we also say that we are in the “**Kummer case**“, as opposed to the “**cyclotomic case**” where p-power roots of unity are used; in the literature the symbol is also used in place of , which is reserved for the cyclotomic case; note also that will be replaced by in this case, being adjoin all p-power roots of ).

There is a notion of a Breuil-Kisin-Fargues -module of height at most **admitting all descents**. This means that, for every a uniformizer of and every the p-power roots of in , we can find a Breuil-Kisin module inside the part of the Breuil-Kisin-Fargues module fixed by the absolute Galois group of the field obtained by adjoining all p-power roots of to (satisfying some conditions related to certain submodules being independent of the choice of and ). If is a Breuil-Kisin-Fargues -module and is a finite extension of , we say that admits all descents over if the Breuil-Kisin Fargues -module obtained by restricting the action to admits all descents.

Let be a Breuil-Kisin-Fargues -module of height at most admitting all descents. We say that is **crystalline** if, for all and for any choice of and we have

.

As the name implies, the importance of the crystalline condition is that it gives rise to crystalline Galois representations (see p-adic Hodge Theory: An Overview). To obtain a Galois representation from a Breuil-Kisin-Fargues -module of height at most admitting all descents, first we take . Then is a -module. Then we can take to get a -module, and finally we can tensor with to get a Galois representation, which we shall denote by . As hinted at earlier, the Galois representation will be crystalline if and only if is crystalline. Furthermore will have Hodge-Tate weights in the range if and only if has height at most .

Let be the limit-preserving category of groupoids over such that , for a finite type -algebra, is the groupoid of Breuil-Kisin-Fargues -modules with -coefficients of height at most , admitting all descents, and crystalline. We let .

There is a map from to given by sending a Breuil-Kisin-Fargues -module to the -module . We now let be the maximal substack of which is flat over , and define to be the scheme-theoretic image of under the map from to as described above.

Let us now introduce the notion of Hodge types. A **Hodge type** is a set of tuples of nonnegative integers such that for all and all . A Hodge type is **regular** if for all and all .

We also have the notion of an **inertial type**, which is defined to be a -representation of the inertia group which extends to a representation of the Weil group with open kernel (which implies that it has finite image).

We can associate to a Breuil-Kisin-Fargues -module a Hodge type and an inertial type as we now discuss. We let be a finite extension of large enough so that it contains all the embeddings of into . Let be a p-adically complete flat -algebra topologically of finite type over , and let . Let be a Breuil-Kisin-Fargues module with coefficients admitting all descents over . We write for the associated Breuil-Kisin module, and define . We write

and

We have the decomposition . We have idempotents corresponding to each factor of this decomposition, and we have the decomposition

of the -module into -modules .

Now let be a Hodge type. We say that a Breuil-Kisin-Fargues -module has Hodge type if has constant rank equal to .

Now on to inertial types. Let be a Breuil-Kisin-Fargues -module admitting all descents over and let be the associated Breuil-Kisin module. Consider , a submodule of . Let be the residue field of and let . We have a -semilinear action of on induced from the action of on , which in turn induces an action of on the -module .

Fix an embedding . As before we have a corresponding idempotent . Now let be an inertial type. Given a Breuil-Kisin-Fargues -module we say that it has inertial type if as an -module, is isomorphic to the base change of to .

We now define to be the moduli stacks of Breuil-Kisin-Fargues -modules of rank , height at most , and admitting all descents to , that give rise to Galois representations which become crystalline over and with associated Hodge type and inertial type . We define to be the scheme-theoretic image of in .

It is known, via what we know about the corresponding versal rings , that the moduli stacks are equidimensional of dimension equal to the quantity

In particular, if is a regular Hodge type, then this quantity is equal to . This plays a role in the formulation of the geometric Breuil-Mezard conjecture as we shall see later.

### The reduced substack

Let us now consider the reduced substack . This is an algebraic stack of finite presentation over , equidimensional of dimension , and its irreducible components are labeled by Serre weights.

To see more explicitly the geometry of let us focus on the case and .

For we are looking at characters . These are of the form . In the picture of -modules, these are obtained from the trivial -module over by twisting by and twisting by . For each the representations are therefore parametrized by , but we also have automorphisms parametrized by .

For , the irreducible representations are of the form . These form a -dimensional substack inside . The reducible ones which are of the form

will belong to the irreducible component of labeled by the Serre weight (this is unambiguous except in the case where or , in which case the component labeled by is one where the representations with are dense, and the component labeled by is one where the representations with are dense). Such a representation will correspond to a closed point if it is semisimple.

More generally, given a family of Galois representations, Emerton and Gee outline a way to construct extensions of this family by some irreducible Galois representation.

Suppose we have a family of -dimensional Galois representations parametrized by a reduced finite scheme (this family corresponds to a map ). Let be a fixed Galois representation of dimension .

The theory of the Herr complex then allows us to find a bounded complex of finite rank locally free -modules

whose cohomology computes (the finite type points of correspond to the usual Galois cohomology). If is locally free of some rank , then we have a bounded complex of finite rank locally free -modules

where is defined to be the kernel of the map . There is a surjection .

Let be the vector bundle over corresponding to , and let be the pullback of to . Then we can use the surjection to construct a “universal extension” that fits into the following exact sequence:

This universal extension is a family of Galois representations parametrized by , i.e. a map . Being able to construct families of higher-dimensional Galois representations as extensions of lower-dimensional ones helps us study the moduli stacks of Galois representations for any dimension, and is used for instance, to prove the earlier stated facts about the dimension and irreducible components of these moduli stacks.

### The “coarse moduli space” and the Bernstein center

Let us now look at a “coarse moduli space” associated to . This coarse moduli space is a moduli space of pseudorepresentations. The associated reduced space should be a chain of projective lines, as we shall shortly explain.

A map from Galois representations to pseudorepresentations should factor through semisimplification. If a reducible mod p Galois representation is semisimple then it must be of the form

and from our earlier discussion we can associate to it the Serre weight . But we can also see this as

and now the associated Serre weight is . Therefore there are two Serre weights that we can associate to this reducible mod p Galois representation! Now if we fix the determinant of our Galois representation to be, say , then besides the two Serre weights our Galois representation only depends on the parameter (because in this case we must have ).

We can consider our two Serre weights now to be the and of a projective line (these points will also correspond to irreducible representations) and the points of the projective line in between these gives us the values of the parameter which parametrize the reducible representations. But a “” Serre weight could also be considered as the “” Serre weight associated to another family of Galois representations. Therefore we have a chain of projective lines parametrizing our semisimple Galois representations. This is the reduced space of our “coarse moduli space” .

One interesting application of these ideas, which is currently part of ongoing work by Andrea Dotto, Matthew Emerton, and Toby Gee, is that the category of mod p representations of (which we shall denote by ) forms a stack over the Zariski site of !

That is, to every Zariski open set of , we can associate a category and these categories glue together well and form a stack over the Zariski site of . To define these categories we need to use the theory of “blocks” developed by Vytautas Paskunas. Namely, Paskunas showed that the category of locally admissible representations of decomposes into “blocks” labeled by semisimple Galois representations .

We can now construct the category as follows. Let be a closed subset of . Then we define to be the full sub category of consisting of all representations of whose irreducible subquotients live in blocks labeled by the -points of (since these correspond to semisimple Galois representations, which in turn label the blocks). Then for an open subset of , we define to be the Serre quotient , where .

If is an additive category, the **Bernstein center** of , denoted , is defined to be the ring of endomorphisms of the identity functor .

It is expected that , the category of mod p representations of , forms a sheaf over , and the Bernstein center coincides with the structure sheaf of .

### Relation to p-adic local Langlands and modularity

Let us now discuss how the moduli stack is related to the p-adic local Langlands correspondence and questions of modularity.

We want there to be a sheaf on which realizes the p-adic local Langlands correspondence. In the case and , we can apply a construction of Colmez to the universal -module on and obtain a quasi-coherent sheaf of -representations on it. Being a quasi-coherent sheaf it has an action of the structure sheaf , which is expected to be the same as the action of the Bernstein center of the category of smooth -representations on -modules which are locally -power torsion.

In the case when either or , there is so far no known satisfactory analogue of Colmez’ construction. However, it is believed that if there is such a sheaf it must coincide with a certain patched module construction , which is a module over the deformation ring , after pulling back over the map .

The sheaf is also expected to play a role in the geometric version of the **Breuil-Mezard conjecture**, which in its original form concerns the geometry of Galois deformation rings and has applications in modularity and the Fontaine-Mazur conjecture.

Let be a Hodge type and let be an inertial type. Let , where . Let be the algebraic -representation of with highest weight , and let .

Now to the inertial type , there is an “inertial local Langlands correspondence” that associates to a smooth admissible representation of over . Let be a -stable -lattice in , and let . Finally, we let be the semisimplification of . We may now view this as an representation of , where is the residue field of . Now let be the irreducible representation of associated to the tuple (these are higher-dimensional versions of the Serre weights discussed in The mod p local Langlands correspondence for GL_2(Q_p)). We have the decomposition

Let . Let be the support of on . It is expected that .

Let and let be the support of on . The **geometric Breuil-Mezard conjecture** states that

.

The Breuil-Mezard conjecture is expected to have applications in modularity, i.e. knowing when a Galois representation comes from a modular form. Some progress towards the conjecture has recently been obtained by Daniel Le, Bao Viet Le Hung, Brandon Levin, and Stefano Morra by the use of local models, which are geometric objects of a more group-theoretic origin (related to affine Grassmannians and flag varieties) which can make them easier to study. Their work also has applications to a generalization of the weight part of Serre’s conjecture. We leave this work and other related topics to future posts.

References:

Moduli stacks of (phi, Gamma)-modules: a survey by Matthew Emerton and Toby Gee

Moduli stacks of etale (phi, Gamma)-modules and the existence of crystalline lifts by Matthew Emerton and Toby Gee

Moduli stacks of (phi, Gamma)-modules by Toby Gee (recording of a talk at the Serre weight conjectures and geometry of Shimura varieties workshop at Centre de Recherches Mathematique)

Moduli of Galois representations by David Savitt (recording of a talk at the 2020 Connecticut Summer School in Number Theory)

“Scheme-theoretic images” of morphisms of stacks by Matthew Emerton and Toby Gee

Phi-modules and coefficient spaces by George Pappas and Michael Rapoport

Mod p Bernstein centers of p-adic groups by Andrea Dotto ( recording of a talk at the Serre weight conjectures and geometry of Shimura varieties workshop at Centre de Recherches Mathematiques)

Localizing GL_2(Q_p) representations by Matthew Emerton (recording of a talk at the INdaM program on Serre conjectures and the p-adic Langlands program)

Local models for Galois deformation rings and applications by Daniel Le, Bao Viet Le Hung, Brandon Levin, Stefano Morra