# 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

Moduli of Galois representations by Carl Wang-Erickson

# Trace Formulas

A trace formula is an equation that relates two kinds of data – “spectral” data related to representations (or eigenvalues of certain operators), and “geometric” data, related to integrals along “orbits” on some space.

The name “trace formula” comes from how this equation is obtained – by expanding the “trace” of a certain operator (let’s call it $R_{f}$. It will depend on a compactly supported “test function” $f(x)$ on a topological group $G$) on square-integrable functions on a compact quotient $\Gamma\backslash G$ of $G$ (which give a representation of $G$ by translation) by a discrete subgroup $\Gamma$.

The operator $R_{f}$ takes a function $\phi(x)$ on the group $G$, translates it by some element $y$ (recall for example that acting on functions by translation is how we defined the representation of the group $\mathbb{R}$ in Representation Theory and Fourier Analysis), multiplies it by the test function $f(x)$, then integrates over the group $G$ (the group $G$ must have a measure called “Haar measure” to do this) to obtain a new function $(R_{f}\phi)(y)$:

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(xy)f(x)dx$

We can also express this as

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(x)f(y^{-1}x)dx$

Let $\Gamma$ be a discrete subgroup of $G$, such that the quotient $\Gamma\backslash G$ is compact (this will turn out to be important later). Instead of integrating over all of $G$ we may instead integrate over the quotient $\Gamma\backslash G$ by re-expressing the integrand as follows:

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}\phi(x)\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)dx$

The sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$ is called the “kernel” of the operator $R_{f}$ and is denoted by $K(x,y)$. We have

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}K(x,y)\phi(x)dx$

So the operator $R_{f}$ looks like the integral of $K(x,y)\phi(x)dx$ over the quotient $\Gamma\backslash G$. Compare this with how a matrix with entries $A_{mn}$ acts on a finite dimensional vector $v_{n}$:

$\displaystyle v_{m}=\sum_{n}A_{mn}v_{n}$

Note that we think of integrals as analogous to sums for infinite dimensions, as functions are analogous to vectors in infinite dimensions. Now we can see that the kernel $K(x,y)$ is the analogue of the entries of some matrix!

The “trace” of a matrix is just the sum of its diagonal entries, i.e. the sum of $A_{nn}$ for all n. Therefore, the trace of the operator defined above is the integral of $K(x,x)$ (i.e. we set $x=y$) over $\Gamma\backslash G$.

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G} K(x,x)dx$

Now recall that the kernel $K(x,y)$ is given by the sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$. Therefore the trace will be given by

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}f(x^{-1}\gamma x)dx$.

Some analysis manipulations will allow us to re-express the trace as the sum

$\displaystyle \mathrm{tr}(R_{f})=\sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G} f(x^{-1}\gamma x)dx$

over representatives $\gamma$ of conjugacy classes in $\Gamma$ of the integrals of $f(x^{-1}\gamma x)$ over the quotient $G_{\gamma}\backslash G$ where $G_{\gamma}$ is the centralizer of $\gamma$ in $G$, multiplied by some factor called the “volume” of $\Gamma_{\gamma}\backslash G_{\gamma}$.

The integral of $f(x^{-1}\gamma x)$ over $G_{\gamma}\backslash G$ is called an “orbital integral“. This expansion of the trace is going to be the “geometric side” of the trace formula.

We consider another way to expand the trace. Recall that to define the operator $R_f$ we needed to act by translation. In this case that the quotient $\Gamma\backslash G$ is compact, as we stated earlier, this representation (let us call it $R$) by translation decomposes into a direct sum of irreducible representations $\pi$, with multiplicities $m(\pi,R)$. So we decompose first before getting the trace!

This other expansion is called the “spectral side“. Since we have now expanded the same thing, the trace, in two ways, we can equate the two expansions:

$\displaystyle \sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G}f(x^{-1}hx)dx=\sum_{\pi} m(\pi,R)\mathrm{tr}(\int_{G}f(x)\pi(x)dx)$

This equation is what is called the “trace formula”. Let us test it out for $G=\mathbb{R}$, $H=\mathbb{Z}$, like in Representation Theory and Fourier Analysis.

In the geometric side, $f(x^{-1}\gamma x)=f(\gamma)$, since $\mathbb{R}$ is abelian. $\mathbb{Z}$ is also abelian, so the conjugacy classes are just elements of $\mathbb{Z}$. We have $G_{\gamma}=G$ and $\Gamma_{\gamma}=\Gamma$. One can check that the volume is $1$ and the orbital integral is just $f(\gamma)$. Replacing $\gamma$ by $n$ for notational convenience, we see that the geometric side is just a sum of $f(n)$ over each integer $n$ in $\mathbb{Z}$.

Let us now look at the spectral side. Recall that the representation decomposes into irreducible representations, each with multiplicity $1$, which are given by multiplication by $e^{2 \pi i k x}$. We consider the operator $R_{f}$ now.

Recall that we let our representation act, then multiply it with the test function f, then integrate. We broke it up into irreducible representations, which act by multiplication by $e^{2 \pi i k x}$. What is multiplication of a function of the form $e^{2 \pi i k x}$ and integrating over $x$?

This is just the Fourier transform of the test function $f$! Since we have an irreducible representation for every integer $k$, we sum over those. So we have an equality between the sum of $f(n)$ where $n$ is an integer, and the corresponding sum of its Fourier transforms!

This is actually a classical result in Fourier analysis known as Poisson summation:

$\displaystyle \sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}} e^{2\pi i k x}f(x)dx$

Atle Selberg famously applied the trace formula to the representation of $G=\mathrm{SL}_{2}(\mathbb{R})$ on functions on a double quotient $H\backslash\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$. Note that the quotient $\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$ is the upper half-plane. H is chosen by Selberg so that the double quotient is a Riemann surface of genus $g\geq 2$.

Selberg used the trace formula to relate lengths of geodesics (given by orbital integrals) to eigenvalues of the 2D Laplacian. Note that the Laplacian already appears in our example of Poisson summation, because $e^{2 \pi i k x}$ is also an eigenfunction of the 1D Laplacian.

This may be why the spectral side is called “spectral”. The trace formula is fascinating on its own, but very commonly used with it is to study representations of certain groups via more familiar representations of other groups.

To do this, note that the spectral side contains information related to representations. If we could only somehow find a way to relate the geometric sides of trace formulas of two different representations, then we can relate their spectral sides!

This is an approach to the part of representation theory known as Langlands functoriality, which studies how representations are related given that the respective groups have “Langlands duals” that are related. Relating the geometric sides involves proving difficult theorems such as “smooth transfer” and the “fundamental lemma”.

Finally, it is worth noting that the spectral side is also used to study special values of L-functions. This is inspired by the work of Hecke expressing completed L-functions as Mellin transforms of modular forms. But that is for another time!

References:

Arthur-Selberg trace formula on Wikipedia

Poisson summation formula on Wikipedia

An introduction to the trace formula by James Arthur

Selberg’s trace formula: an introduction by Jens Marklof