Moduli Stacks of Galois Representations

In Galois Deformation Rings we introduced the concept of Galois deformations and Galois deformation rings, which had the property that Galois deformations (which are certain equivalence classes of lifts of a fixed residual representation) correspond to maps from those Galois deformations to the one over the Galois deformation ring. In a way this allows us to consider all the deformations of this residual representation altogether.

In this post, we will consider not only the Galois representations that are lifts of some fixed residual representation, but consider Galois representations without the need to fix a residual representation. These Galois representations are going to be parametrized by the moduli stack of Galois representations, whose geometry we will study.

Before we consider Galois representations, let us first consider the simpler case of representations of a finitely presented group. Let $G$ be such a finitely presented group, with generators $g_{1},\ldots, g_{n}$ and relations $r_{1},\ldots r_{t}$ . Let us consider its $d$ -dimensional representations over some ring $A$ . The first thing we have to do is to give $d\times d$ matrices $M_{1},\ldots,M_{n}$ , with coefficients in $A$ , corresponding to the generators $g_{1},\ldots g_{n}$ . Then we have to quotient out by the relations $r_{1},\ldots,r_{t}$ , each viewed as a relation on the matrices $M_{1},\ldots,M_{n}$ . Then we may see the functor that assigns to a ring $A$ the set of $d$ -dimensional representations of $G$ over $A$ is representable by an affine scheme.

Now the theory of stacks (see also Algebraic Spaces and Stacks) comes in when we take into consideration that two representations that differ only by a change of basis may be considered to be “the same”. So we take the quotient of our affine scheme by this action of $\mathrm{GL}_{d}$ , and what we get is a stack.

Let us now go back to Galois representations. Note that the absolute Galois groups we will be interested in are not finitely presented, however, the idea is that we will have to find some clever way of relating these absolute Galois groups to some finitely presented groups so we can make use of what we have just learned.

Let us first discuss the local case, for $\ell\neq p$ , i.e. our representations will be on $A$ -modules, where $A$ is some $\mathbb{Z}_{\ell}$ -algebra. Consider $K$ , a finite extension of $\mathbb{Q}_{p}$ , and let $\kappa$ be its residue field. As a shorthand let us also denote $\mathrm{Gal}(\overline{K}/K)$ by $G_{K}$ . Let us recall (see also Splitting of Primes in Extensions and Weil-Deligne Representations) that we have the exact sequence

$\displaystyle 0\to I_{K}\to G_{K}\to\mathrm{Gal}(\overline{\kappa}/\kappa)\to 0$

Recall that $I_{K}$ is called the inertia group. An extension of $K$ is tamely ramified if its ramification index is prime to $p$ . Let $K^{\mathrm{tame}}$ be the maximal tamely ramified extension of $K$ and $K^{\mathrm{unr}}$ be the maximal unramified extension of $K$ . Let $G_{K}^{\mathrm{tame}}=\mathrm{Gal}(K^{\mathrm{tame}}/K)$ and let $G_{K}^{\mathrm{unr}}=\mathrm{Gal}(K^{\mathrm{unr}}/K)$ . We have an exact sequence

$\displaystyle 0\to I_{K}^{\mathrm{tame}}\to G_{K}^{\mathrm{tame}}\to G_{K}^{\mathrm{unr}}\to 0$

Where $I_{K}^{\mathrm{tame}}$ is called the tame inertia. It is a quotient of the inertia group $I_{K}$ by a subgroup $I_{K}^{\mathrm{wild}}$ , called the wild inertia. The tame inertia $I_{K}^{\mathrm{tame}}$ is of the form $\prod_{\ell\neq p}\mathbb{Z}_{\ell}(1)$ and is a pro-cyclic group.

Let $\tau$ be a generator of $I_{K}^{\mathrm{tame}}$ as a pro-cyclic group. Let $\sigma$ be a lift of Frobenius in $G_{K}^{\mathrm{tame}}$ . We consider the subgroup of $G_{K}^{\mathrm{tame}}$ given by

$\displaystyle \Gamma=\langle \tau,\sigma\vert\sigma\tau\sigma^{-1}=\tau^{q}\rangle$

where $q$ is the cardinality of the residue field $\kappa$ . This subgroup $\Gamma$ is is dense inside $G_{K}^{\mathrm{tame}}$ , and $G_{K}^{\mathrm{tame}}$ is its profinite completion.

We have the following exact sequence:

$\displaystyle 0\to I_{K}^{\mathrm{wild}}\to G_{K}\to G_{K}^{\mathrm{tame}}\to 0$

Inside $G_{K}^{\mathrm{tame}}$ we have the subgroup $\Gamma$ , and we have another exact sequence as follows:

$\displaystyle 0\to I_{K}^{\mathrm{wild}}\to\mathrm{WD}_{K}\to\Gamma\to 0$

The middle term $\mathrm{WD}_{K}$ is defined to be the limit $\varprojlim_{Q}\mathrm{WD}_{K}/Q$ , where $Q$ is an open subgroup of $I_{K}^{\mathrm{wild}}$ which is normal in $G_{K}$ , and $\mathrm{WD}_{K}/Q$ is in turn defined to be the extension of the finitely presented group $\Gamma$ by the finite group $I_{K}^{\mathrm{wild}}/Q$ , i.e. $\mathrm{WD}_{K}/Q$ is the middle term in the exact sequence

$\displaystyle 0\to I_{K}^{\mathrm{wild}}/Q\to\mathrm{WD}_{K}/Q\to\Gamma\to 0$

Now the idea is that $\mathrm{WD}_{K}/Q$ , being an extension of a finitely presented group by a finite group, is finitely presented, and we can use what we have learned about moduli stacks of finitely presented groups at the beginning of this post. At the same time, $\mathrm{WD}_{K}/Q$ is dense inside $G_{K}/Q$ , and we have $G_{K}=\varprojlim_{Q} G_{K}/Q$ .

Therefore, we let $V_{Q}$ be the moduli stack of representations of the finitely presented group $\mathrm{WD}_{K}/Q$ , and our moduli stack of Galois representations will be given by the direct limit $V=\varinjlim V_{Q}$ .

Now all of what we just discussed applies to the $\ell\neq p$ case, but the $\ell=p$ case is much more subtle. To properly construct the moduli stack of Galois representations for the $\ell=p$ case we will need the theory of $(\varphi,\Gamma)$ -modules, which will not discuss in this post, though hopefully we will be able to in some future post.

Let us now discuss briefly the global case. Let $K$ be a number field, and let $S$ be a finite set of places of $S$ . Let $G_{K,S}$ denote the Galois group of the maximal Galois extension of $K$ unramified outside $S$ . We want to consider $d$ -dimensional representations of $G_{K,S}$ over a $\mathbb{Z}_{p}/p^{a}\mathbb{Z}_{p}$ -algebra $A$ , for some $a$ . The functor that assigns to such an $A$ this set of representations gives us a stack $\mathfrak{X}$ over the formal scheme $\mathrm{Spf}(\mathbb{Z}_{p})$ (see also Formal Schemes).

Not only can we consider representations, but we can also consider pseudo-representations, which are sort of generalizations of the concept of the trace of a representation. These pseudo-representations also have a corresponding moduli space, which is a formal scheme, denoted by $X$ , also over $\mathrm{Spf}(\mathbb{Z}_{p})$ . Since we can associate a pseudo-representation to a representation, we have a map $\mathfrak{X}\to X$ .

It is a theorem of Chenevier that $X$ is a disjoint union of components $X_{\overline{\rho}}$ indexed by residual pseudo-representations (semi-simple pseudo-representations over a finite field). Similarly, $\mathfrak{X}$ will be a disjoint union of components $\mathfrak{X}_{\overline{\rho}}$ , each with a map to the corresponding $X_{\overline{\rho}}$ . In the case that $\overline{\rho}$ is irreducible, $X_{\overline{\rho}}$ will be $\mathrm{Spf}(R_{\overline{\rho}})$ , while $\mathfrak{X}_{\rho}$ will be $\mathrm{Spf}(R_{\overline{\rho}})/\widehat{\mathbb{G}}_{m}$ , where $R_{\overline{\rho}}$ is the universal deformation ring, and $\widehat{\mathbb{G}}_{m}$ is some formal completion of $\widehat{\mathbb{G}}_{m}$ .

We end this post by mentioning a conjecture related to the conjectural categorical geometric Langlands correspondence mentioned at the end of The Global Langlands Correspondence for Function Fields over a Finite Field. This is currently part of ongoing work by Matthew Emerton and Xinwen Zhu. There is a “restriction” map

$\displaystyle f:\mathfrak{X}\to\prod_{v\in S}\mathfrak{X}_{v}$

from the global moduli stack $\mathfrak{X}$ to the product of local moduli stacks $\mathfrak{X}_{v}$ , for all $v$ in the set $S$ (defined at the start of the discussion of the global case). It is then conjectured that there are coherent sheaves $\mathfrak{A}_{v}$ on each $\mathfrak{X}_{v}$ , which come from representations of $\mathrm{GL}_{n}(K_{v})$ . We can form the product of these sheaves and pull back to get a sheaf $\mathfrak{A}$ on the global stack $\mathfrak{X}$ , and after tensoring with the universal Galois representation on $\mathfrak{X}$ , it is conjectured that this gives the compactly supported cohomology of Shimura varieties.

One can also form, more generally, moduli stacks not just of Galois representations but of Langlands parameters. More on these, as well as more in-depth details on these moduli stacks and the conjectures regarding coherent sheaves on these moduli stacks, will hopefully be discussed in future posts.

References:

Moduli stacks of Galois representations by Matthew Emerton on YouTube

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

Moduli of Langlands parameters by Jan-Francois Dat, David Helm, Robert Kurinczuk, and Gilbert Moss

Coherent sheaves on the stack of Langlands parameters by Xinwen Zhu

Moduli of Galois representations by Carl Wang-Erickson

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Moduli Stacks of Galois Representations

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