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. 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

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

Automorphic Forms

An automorphic form is a kind of function of the adelic points (see also Adeles and Ideles) of a reductive group (see also Reductive Groups Part I: Over Algebraically Closed Fields), that can be used to investigate its representation theory. Choosing an important kind of function of a group that will be helpful in investigating its representation theory was also discussed in Representation Theory and Fourier Analysis, where we found the square-integrable functions on the circle to be useful in studying its representations (or that of the real line) since it decomposed into a direct sum of irreducible representations. In fact, the cuspidal automorphic forms we will introduce later on in this post will also have this property (called semisimple) of decomposing into a direct sum of irreducible representations.

Remark: We have briefly mentioned, in the unramified case, cuspidal automorphic forms as certain functions on \mathrm{Bun}_{G} in The Global Langlands Correspondence for Function Fields over a Finite Field. The function field (over a finite field) version of the cuspidal automorphic forms we define here are actually obtained as linear combinations of translates of such functions hence why it is enough to study them. In this post, we will discuss automorphic forms in more detail, beginning with the version over the field of rational numbers before generalizing to more general global fields.

Defining modular forms as functions on \mathrm{GL}_{2}(\mathbb{A})

In a way, automorphic forms can also generalize modular forms (see also Modular Forms), and this will give us a way to connect the two theories. We shall take this route first and recast modular forms in a new language – instead of functions on the upper half-plane, we shall now look at them as functions on the group \mathrm{GL}_{2}(\mathbb{A}) (here \mathbb{A} denotes the adeles of \mathbb{Q}).

Let K_{f} be a compact open subgroup of \mathrm{GL}_{2}(\mathbb{A}_{f}) whose elements all have determinants in \widehat{\mathbb{Z}}^{\times}. Here \mathbb{A}_{f} stands for the finite adeles, which are defined in the same way as the adeles, except we don’t include the infinite primes in the restricted product. We have that

\displaystyle \mathrm{GL_{2}}(\mathbb{A})=\mathrm{GL_{2}}(\mathbb{Q})\mathrm{GL}_{2}(\mathbb{R})^{+}K_{f}

where \mathrm{GL}_{2}(\mathbb{R})^{+} is the subgroup of \mathrm{GL}_{2}(\mathbb{R}) consisting of elements that have positive determinant. Now let us take the double quotient \mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}. By the above expression for \mathrm{GL}_{2}(\mathbb{A}) as a product, we have

\displaystyle \mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}\simeq \Gamma\backslash\mathrm{GL_{2}}(\mathbb{R})

where \Gamma is the subgroup of \mathrm{GL}_{2}(\mathbb{R}) given by projecting \mathrm{GL}_{2}(\mathbb{Q})\cap \mathrm{GL}_{2}(\mathbb{R})^{+}K_{f} into its archimedean component. Now suppose we are in the special case that K_{f} is given by \displaystyle \prod_{p} \mathrm{GL}_{2}(\mathbb{Z}_{p}). Then it turns out that \Gamma is just \mathrm{SL}_{2}(\mathbb{Z})! Using appropriate choices of K_{f}, we can also obtain congruence subgroups such as \Gamma_{0}(N) (see also Modular Forms).

The group \mathrm{GL}_{2}(\mathbb{R})^{+} acts on the upper half-plane by fractional linear transformations, i.e. if we have \displaystyle g_{\infty}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{R})^{+}, then g_{\infty} sends \tau in the upper half-plane to \displaystyle g_{\infty}(\tau)=\frac{az+b}{cz+d}. Let

\displaystyle j(g_{\infty},\tau)=\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d).

Now given a modular form f of weight m and level \Gamma_{0}(N), we may associate to it a function \phi_{\infty}(g_{\infty}) on \mathrm{GL}_{2}(\mathbb{R})^{+} as follows:

\displaystyle f\mapsto \phi_{\infty}(g_{\infty})=f(g_{\infty}(i))j(g_{\infty},i)^{-m}

We can also go the other way, recovering f from such a \phi_{\infty}:

\displaystyle \phi_{\infty}\mapsto f(g_{\infty}(i))=\phi_{\infty}(g_{\infty})j(g_{\infty},i)^{m}

for any g_{\infty} such that g_{\infty}(i)=\tau. Ultimately we want a function \phi on \mathrm{GL}_{2}(\mathbb{A}), and we achieve this by setting \phi(g)=\phi(\gamma g_{\infty} k_{f}) to just have the same value as \phi_{\infty}(g_{\infty}).

Translating properties of modular forms into properties of functions on \mathrm{GL}_{2}(\mathbb{A})

Invariance under \mathrm{GL}_{2}(\mathbb{Q} and K_{f}

Now we want to know what properties \phi must have, so that we can determine which functions on \mathrm{GL}_{2}(\mathbb{A}) come from modular forms. We have just seen that we must have

\displaystyle \phi(g)=\phi(\gamma g_{\infty} k_{f})=\phi_{\infty}(g).

The action of Z_{\infty}^{+} and K_{\infty}^{+}

Let us now consider the action of the center of \mathrm{GL}_{2}(\mathbb{R})^{+} (which we denote by Z_{\infty}) and the action of \mathrm{SO}(2), which is a maximal compact subgroup of \mathrm{GL}_{2}(\mathbb{R})^{+} (and therefore we shall also denote it by K_{\infty})^{+}. The center Z_{\infty} is composed of the matrices of the form z_{\infty} times the identity matrix, and it acts trivially on the upper half-plane. Therefore we will have

\displaystyle j(z_{\infty}g_{\infty},\tau)=\mathrm{sgn}(z_{\infty})\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d)

Now for the maximal compact subgroup K_{\infty}^{+}. As previously mentioned, this is the group \mathrm{SO}(2), and may be expressed as matrices of the form

\displaystyle k_{\theta}=\begin{pmatrix}\mathrm{cos}(\theta) & \mathrm{sin}(\theta)\\-\mathrm{sin}(\theta) & \mathrm{cos}(\theta)\end{pmatrix}.

Then in the action of \mathrm{GL}_{2}(\mathbb{R})^{+} on the upper half-plane, Z_{\infty}K_{\infty}^{+} is the stabilizer of i. We will also have

\displaystyle j(z_{\infty}k_{\theta},i)=\mathrm{sgn}(z_{\infty})e^{i\theta}

This leads us to the second property our function \phi must satisfy. First we consider \phi_{\infty}. For z_{\infty}k_{\theta}\in Z_{\infty}K_{\infty}^{+}, we must have

\displaystyle \phi_{\infty}(g_{\infty}z_{\infty}k_{\theta})=\phi_{\infty}(g_{\infty})\mathrm{sgn}(z)^{m}(e^{i\theta})^{m}.

Note the appearance of the weight m. Now when we extend this function \phi_{\infty} on \mathrm{GL}_{2}(\mathbb{R})^{+} to a function \phi on \mathrm{GL}_{2}(\mathbb{A}), we must replace Z_{\infty} by its connected component Z_{\infty}^{+}.

Holomorphicity and the action of the Lie algebra

Next we must translate the property that the modular form f is holomorphic into a property of \phi. For this we shall introduce certain “raising” and “lowering” operators.

Let \mathfrak{g}_{0} be the (real) Lie algebra of \mathrm{GL}_{2}(\mathbb{R})^{+}. An element X\in\mathfrak{g}_{0} acts on the space of smooth functions on \mathrm{GL}_{2}(\mathbb{R})^{+} as follows:

\displaystyle X\phi(g_{\infty})=\frac{d}{dt}\phi(g_{\infty}\mathrm{exp}(tX))\bigg\vert_{t=0}

We can extend this to an action of the complexified Lie algebra \mathfrak{g}, defined to be \mathfrak{g}\otimes_{\mathbb{R}} \mathbb{C}, by setting

\displaystyle (X+iY)\phi=X\phi+iY\phi

We now look at two special elements of \mathfrak{g}. They are

\displaystyle X_{+}=\frac{1}{2}\begin{pmatrix}1 & i\\i & -1\end{pmatrix}

and

\displaystyle X_{-}=\frac{1}{2}\begin{pmatrix}1 & -i\\-i & -1\end{pmatrix}.

Let us now look at how these special elements act on the smooth functions on \mathrm{GL}_{2}(\mathbb{R})^{+}. We have

\displaystyle X_{+}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m+2}

and

\displaystyle X_{-}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m-2}

In other words, the action of X_{+} raises the weight by 2, while the action of X_{-} lowers the weight by 2. Now it turns out that the condition that the function f on the upper half-plane is holomorphic is the same condition as the function \phi on \mathrm{GL}_{2}(\mathbb{R})^{+} satisfying X_{-}\phi=0!

Holomorphicity at the cusps

Now we have expressed the holomorphicity of our modular form f as a condition on our function \phi on \mathrm{GL}_{2}(\mathbb{A}). However not only do we want our modular forms to be holomorphic on the upper half-plane, we also want them to be “holomorphic at the cusps”, i.e. they do not go to infinity at the cusps. This is going to be accomplished by requiring the function g_{\infty}\mapsto \phi(g_{\infty}g_{f}) to be “slowly increasing” for all g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f}). This means that for all g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f}), we have

\displaystyle \vert \phi(g_{\infty}g_{f})\geq C\Vert g_{\infty}\Vert^{N}

where C and N are some positive constants and the norm on the right-hand side is given by, for g_{\infty}=\begin{pmatrix}a & b\\c & d\end{pmatrix},

\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=(a^{2}+b^{2}+c^{2}+d^{2})(1+\mathrm{det}(g_{\infty}^{-2}))=\mathrm{Tr}(g_{\infty}^{T}g_{\infty})+\mathrm{Tr}((g_{\infty}^{-1})^{T}g_{\infty}^{-1}).

Summary of the properties

Let us summarize now the properties we want our function \phi to have in order that it come from a modular form f:

  • For all \gamma\in\mathrm{GL}_{2}(\mathbb{Q}), we have \phi(\gamma g)=\phi(g).
  • For all k_{f}\in K_{f}, we have \phi(gk_{f})=\phi(g).
  • For all g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f}), the function g_{\infty}\mapsto \phi(g_{\infty}g_{f}) is smooth.
  • For all k_{\theta}\in K_{\infty} we have \phi(gk_{\theta})=\phi(g)e^{i\theta}.
  • The function \phi is invariant under Z_{\infty}^{+}.
  • We have \displaystyle X_{-}\phi=0.
  • The function given by g_{\infty}\mapsto\phi(g_{\infty}g_{f}) is slowly increasing.

Cuspidality

Now let us consider the case where f is a cusp form. We want to translate the cuspidality condition to a condition on \phi, and we do this by noting that this means that the Fourier expansion of f has no constant term. Given that Fourier coefficients can be expressed using Fourier transforms, we make use of the measure theory on the adeles to express this cuspidality condition as

\displaystyle \int_{\mathbb{Q}\setminus\mathbb{A}}\phi\left(\begin{pmatrix}1 & x\\0&1\end{pmatrix}\right)dx=0.

Automorphic forms

We have now defined modular forms as functions on \mathrm{GL}_{2}(\mathbb{A}), and enumerated some of their important properties. Modular forms, as functions on \mathrm{GL}_{2}(\mathbb{A}), turn out to be merely be specific examples of more general functions on \mathrm{GL}_{2}(\mathbb{A}) that satisfy similar, but more relaxed, properties. These are the automorphic forms.

The first few properties are the same, for instance for all \gamma\in\mathrm{GL}_{2}(\mathbb{Q}), we want \phi(\gamma g)=\phi(g), and for all k_{f}\in K_{f}, where K_{f} is a compact subgroup of \mathrm{GL}_{2}(\mathbb{A}_{f}), we want \phi(g k)=\phi(g). We will also want the function given by g_{\infty}\mapsto \phi(g_{\infty}g_{f}) to be smooth for all g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f}).

What we want to relax a little bit is the conditions on the actions of K_{\infty}, Z_{\infty}^{+}, and the Lie algebra \mathfrak{g}, in that we want the space we get by having them act on some function \phi to be finite-dimensional. Instead of looking at the action of the Lie algebra \mathfrak{g}, it is often convenient to instead look at the action of its universal enveloping algebra U(\mathfrak{g}). The universal enveloping algebra is an honest to goodness associative algebra that contains the Lie algebra (and is in fact generated by its elements) such that the commutator of the universal enveloping algebra gives the Lie bracket of the Lie algebra. We shall denote the center of U(\mathfrak{g}) by Z(\mathfrak{g}). Now it turns out that Z(\mathfrak{g}) is generated by the Lie algebra of Z_{\infty}^{+} and the Casimir operator \Delta, defined to be

\displaystyle \Delta=H^{2}+2X_{+}X_{-}+2X_{-}X_{+}

where H is the element given by \begin{pmatrix}0&-i\\i &0\end{pmatrix}. Therefore, the action of the center of the universal enveloping algebra encodes the action of Z_{\infty}^{+} and the Lie algebra \mathfrak{g} at the same time.

Let us now define automorphic forms in general. Even though the focus on this post is on \mathrm{GL}_{2} and over the rational numbers \mathbb{Q}, we can just give the most general definition of automorphic forms now, even for more general reductive groups and more general global fields. So let G be a reductive group and let F be a global field. The space of automorphic forms on G, denoted \mathcal{A}, is the space of functions \phi:G(\mathbb{A}_{F})\to\mathbb{C} satisfying the following properties:

  • For all \gamma\in G(F), we have \phi(\gamma g)=\phi(g).
  • For all k_{f}\in K_{f}, K_{f} a compact open subgroup of G(\mathbb{A}_{f}), we have \phi(gk_{f})=\phi(g).
  • For all g_{f}\in G(\mathbb{A}_{F,f}), the function g_{\infty}\mapsto \phi(g_{\infty}g_{f}) is smooth.
  • The function \phi is K_{\infty}-finite, i.e. the space \mathbb{C}[K_{\infty}]\cdot\phi is finite dimensional.
  • The function \phi is Z(\mathfrak{g})-finite, i.e. the space Z(\mathfrak{g})\cdot\phi is finite dimensional.
  • The function g_{\infty}\mapsto \phi(g_{\infty}g_{f}) is slowly increasing.

Here slowly increasing means that for all embeddings \iota:G_{\infty}\to\mathrm{GL}_{n}(\mathbb{R}) of the infinite part of G(\mathbb{A}_{F}), we have

\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=\mathrm{Tr}(\iota(g_{\infty})^{T}\iota(g_{\infty}))+\mathrm{Tr}((\iota(g_{\infty}^{-1})^{T}\iota(g_{\infty})^{-1}).

Furthermore, we say that the automorphic form \phi is cuspidal if, for all parabolic subgroups P\subseteq G, \phi satisfies the following additional condition:

\displaystyle \int_{N(\mathbb{F})\setminus N(\mathbb{A}_{F})}\phi(ng)dn=0

where N is the unipotent radical (the unipotent part of the maximal connected normal solvable subgroup) of the parabolic subgroup P.

These cuspidal automorphic forms, which we denote by \mathcal{A}_{0}, form a subspace of the automorphic forms \mathcal{A}.

Automorphic forms and representation theory

As stated earlier, automorphic forms give us a way of understanding the representation theory of G(\mathbb{A}_{F}) where G is a reductive group. Let us now discuss these representation-theoretic aspects.

We will actually look at automorphic forms not as representations of G(\mathbb{A}), but as (\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})-modules. This means they have actions of \mathfrak{g}, K_{\infty}, and G(\mathbb{A}_{F,f}) all satisfying certain compatibility conditions. A (\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})-module is called admissible if any irreducible representation K_{\infty}\times K_{f} shows up inside it with finite multiplicity, and irreducible if it has no proper subspaces fixed by \mathfrak{g}, K_{\infty}, and G(\mathbb{A}_{F,f}). Recall from The Local Langlands Correspondence for General Linear Groups, irreducible admissible representations of G(F_{v}), where F_{v} is some local field, are precisely the kinds of representations that show up in the automorphic side of the local Langlands correspondence.

In fact the global and the local picture are related by Flath’s theorem, which says that, for an irreducible admissible (\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})-module \pi, we have the following factorization

\displaystyle \pi=\bigotimes'_{v\not\vert\infty}\pi_{v}\otimes \pi_{\infty}

into a restricted tensor product (explained in the next paragraph) of irreducible admissible representations \pi_{v} of G(F_{v}), running over all places v of F. At the infinite place, \pi_{v} is an irreducible admissible (\mathfrak{g}, K_{\infty})-module.

The restricted tensor product is a direct limit over S of V_{S}=\bigotimes_{s\in S} \pi_{s} where for S\subset T we have the inclusion V_{S}\hookrightarrow V_{T} given by x_{S}\mapsto x_{S}\otimes\bigotimes_{v\in T\setminus S}\xi_{v}^{0}, where \xi_{v} is a vector fixed by a certain maximal compact open subgroup (called hyperspecial) K_{v} of G(F_{v}) (a representation of G(F_{v}) containing such a fixed vector is called unramified).

We have that \mathcal{A} and A_{0} are (\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{f})-modules. An automorphic representation of a reductive group G is an indecomposable (\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})-module that is isomorphic to a subquotient of \mathcal{A}. A cuspidal automorphic representation is an automorphic representation that is isomorphic to a subquotient of \mathcal{A}_{0}. It is a property of the space of cuspidal automorphic forms that it is semisimple, i.e. it decomposes into a direct product of cuspidal automorphic representations (a property that is not necessarily shared by the bigger space of automorphic forms!).

An automorphic form generates such an automorphic representation, and a theorem of Harish-Chandra states that such a representation is admissible. In fact, automorphic representations are always admissible. Again recalling that irreducible admissible representations of G(F_{v}) make up the automorphic side of the local Langlands correspondence, we therefore expect that automorphic representations of G will make up the automorphic side of the global Langlands correspondence.

However, to state the original global Langlands correspondence in general is still quite complicated, as it involves an as of-yet hypothetical object called the Langlands group, which plays a somewhat analogous role as the Weil group in the local Langlands correspondence. Instead there are certain variants of it that are considered easier to approach, for instance by imposing conditions on the representations such as being “algebraic at infinity“. These variants of the global Langlands correspondence will hopefully be discussed in future posts.

References:

Automorphic form on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

MSRI Summer School on Automorphic Forms and the Langlands Program by Kevin Buzzard

The Automorphic Project

An Introduction to the Langlands Program by Daniel Bump, James W. Cogdell, Ehud de Shalit, Dennis Gaitsgory, Emmanuel Kowalski, and Stephen S. Kudla (edited by Joseph Bernstein and Stephen Gelbart)

The Global Langlands Correspondence for Function Fields over a Finite Field

In The Local Langlands Correspondence for General Linear Groups, we introduced some ideas related to what is known as the Langlands program, and discussed in a little more detail the local Langlands correspondence, at least for general linear groups.

In this post, we will discuss the global Langlands correspondence, but we will focus on the case of function fields over a finite field. This will be somewhat easier to state than the case of number fields, and at the same time perhaps give us a bit more geometric intuition. Let us fix a smooth, projective, and irreducible curve X, defined over a finite field \mathbb{F}_{q}. We let F be its function field. For instance, if X is the projective line \mathbb{P}^{1} over \mathbb{F}_{q}, then F=\mathbb{F}(t).

The case of \mathrm{GL}_{1}: Global class field theory for function fields over a finite field

To motivate the global Langlands correspondence for function fields, let us first think of the \mathrm{GL}_{1} case, which is a restatement of (unramified) global class field theory for function fields. Recall that in Some Basics of Class Field Theory global class field theory tells us that for global field F, its maximal unramified abelian extension H, also called the Hilbert class field of F, has the property that \mathrm{Gal}(H/F) is isomorphic to the ideal class group.

We recall that there is an analogy between the absolute Galois group and the etale fundamental group in the case when there is no ramification. Therefore, in the case of function fields, the corresponding statement of unramified global class field theory may be stated as

\displaystyle \pi_{1}(X,\overline{\eta})^{\mathrm{ab}}\times_{\widehat{\mathbb{Z}}}\mathbb{Z}\xrightarrow{\sim} \mathrm{Pic}(\mathbb{F}_{q})

where \pi_{1}(X,\overline{\eta}) is the etale fundamental group of X, a profinite quotient of \mathrm{Gal}(\overline{F}/F) through which its action factors (\overline{\eta} here serves as the basepoint, which is needed to define the etale fundamental group). The Picard scheme \mathrm{Pic} is the scheme such that for any scheme S its S points \mathrm{Pic}(S) correspond to the isomorphism classes of line bundles on X\times S. This is analogous to the ideal class group. Taking the fiber product with \mathbb{Z} is analogous to taking the Weil group (see also Weil-Deligne Representations and The Local Langlands Correspondence for General Linear Groups).

The global Langlands correspondence, in the case of \mathrm{GL}_{1}, is a restatement of this in terms of maps from each side to some field (we will take this field to be \overline{\mathbb{Q}}_{\ell}). It states that there is a bijection between characters \sigma:\pi_{1}(X,\overline{\eta})\to \overline{\mathbb{Q}}_{\ell}^{\times}, and \chi:\mathrm{Pic}(\mathbb{F}_{q})/a^{\mathbb{Z}}\to \overline{\mathbb{Q}}_{\ell}^{\times} where a is any element of \mathrm{Pic}(\mathbb{F}_{q}) of nonzero degree. Again this is merely a restatement of unramified global class field theory, and nothing has changed in its content. However, this restatement points to us the way in which it may be generalized.

Generalizing to \mathrm{GL}_{n}, and then to more general reductive groups

To generalize this, we may take maps \sigma:\pi_{1}(X,\overline{\eta})\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell}) instead of maps \sigma:\pi_{1}(X,\overline{\eta})\to \overline{\mathbb{Q}}_{\ell}^{\times}, since \overline{\mathbb{Q}}_{\ell}^{\times} is just \mathrm{GL}_{1}(\overline{\mathbb{Q}}_{\ell}). To make it look more like the case of number fields, we may also define this same map as a map \sigma:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell}) which factors through \pi_{1}(U,\overline{\eta}) for some open dense subset U of X. This side we call the “Galois side” (as it involves the Galois group).

What about the other side (the “automorphic side”)? First we recall that \mathrm{Pic}(\mathbb{F}_{q}) classifies line bundles on X. We shall replace this by \mathrm{Bun}_{n}(\mathbb{F}_{q}), which classifies rank n vector bundles on X. It was figured out by Andre Weil a long time ago that \mathrm{Bun}_{n}(\mathbb{F}_{q}) may also be expressed as the double quotient \mathrm{GL}_{n}(F)\backslash\mathrm{GL}_{n}(\mathbb{A}_{F})/\mathrm{GL}_{n}(\prod_{v}\mathcal{O}_{F_{v}}) (this is known as the Weil parametrization). Now functions on this space will give representations of \mathrm{GL}_{n}(\mathbb{A}_{F}). We will be interested not in all functions on this space, but in particular certain kinds of functions called cuspidal automorphic forms, which gives a representation that decomposes into pieces that we then want to match up with the Galois representations.

In fact we can generalize even further and consider reductive groups (see also Reductive Groups Part I: Over Algebraically Closed Fields and Reductive Groups Part II: Over More General Fields) other than \mathrm{GL}_{n}! Let G be such a reductive group over F. Instead of \mathrm{Bun}_{n}(\mathbb{F}_{q}) we now consider \mathrm{Bun}_{G}(\mathbb{F}_{q}), the moduli stack (see also Algebraic Spaces and Stacks) of G-bundles on X. As above, we consider the space of cuspidal automorphic forms on \mathrm{Bun}_{G}(\mathbb{F}_{q}), which we shall denote by C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\overline{\mathbb{Q}}_{\ell}). Here \Xi is a subgroup of finite index in \mathrm{Bun}_{Z}(\mathbb{F}_{q}), where Z is the center of G.

As we are generalizing to more general reductive groups than just \mathrm{GL}_{n}, we need to modify the other side (the Galois side) as well. Instead of considering Galois representations, which are group homomorphisms \sigma: \mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell}), we must now consider L-parameters, which in this context are group homorphisms \sigma: \mathrm{Gal}(\overline{F}/F)\to \widehat{G}(\overline{\mathbb{Q}}_{\ell}), where \widehat{G} is the dual group of G (which as one may recall from Reductive Groups Part II: Over More General Fields, has the roots and coroots of G interchanged).

We may now state the “automorphic to Galois” direction of the global Langlands correspondence for function fields over a finite field \mathbb{F}_{q}, which has been proven by Vincent Lafforgue. It says that we have a decomposition

\displaystyle C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})=\bigoplus_{\sigma} \mathfrak{H}_{\sigma}

of the space C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) into subspaces \mathfrak{H}_{\sigma} indexed by L-parameters \sigma. It is perhaps instructive to compare this with the local Langlands correspondence as stated in Reductive Groups Part II: Over More General Fields, to which it should be related by what is known as local-global compatibility.

(The “Galois to automorphic direction” concerns whether an L-parameter is “cuspidal automorphic”, and we will briefly discuss some partial progress by Gebhard Böckle , Michael Harris, Chandrasekhar Khare, and Jack Thorne later at the end of this post.)

Furthermore the decomposition above must respect the action of Hecke operators (analogous to those discussed in Hecke Operators). Let us now discuss these Hecke operators.

Hecke operators

Let \mathcal{E},\mathcal{E}' be two G-bundles on X. Let x be a point of X, and let \phi:\mathcal{E}\to\mathcal{E}' be an isomorphism of G bundles over X\setminus x. We say that (\mathcal{E}',\phi) is a modification of \mathcal{E} at x. A modification can be bounded by a cocharacter, i.e. a homomorphism \lambda:\mathbb{G}_{m}\to G. This keeps track and bounds the vector bundles’ relative position.

To get an idea of this, we consider the case G=\mathrm{GL}_{n}. Consider the completion \mathcal{E}_{x}^{\wedge} of stalk of the vector bundle \mathcal{E} at x. It is a free module over the completion \mathcal{O}_{X,x}^{\wedge} of the structure sheaf at x, which happens to be isomorphic to \mathbb{F}_{q}[[t]]. Let (\mathcal{E}',\phi) be a modification of \mathcal{E} at x. There is a basis e_{1},\ldots,e_{n} of \mathcal{E}_{x}^{\wedge} such that t^{k_{1}}e_{1},\ldots,t^{k_{n}}e_{n} is a basis of \mathcal{E}_{x}^{'\wedge}, where k_{1}\geq\ldots\geq k_{n}. But the numbers k_{1},\ldots,k_{n} is the same as a cocharacter \lambda:\mathbb{G}_{m}\to\mathrm{GL}_{n}, given by \mu(t)=\mathrm{diag}(t^{k_{1}},\ldots,t^{k_{n}}).

The Hecke stack \mathrm{Hck}_{v,\lambda} is the stack whose points \mathrm{Hck}_{v,\lambda}(\mathbb{F}_{q}) correspond to modifications (\mathcal{E},\mathcal{E}',\phi) at v whose relative position is bounded by the cocharacter \lambda. It has two maps h^{\leftarrow} and h^{\rightarrow} to \mathrm{Bun}_{G}(\mathbb{F}_{q}), which send the modification (\mathcal{E},\mathcal{E}',\phi) to \mathcal{E} and \mathcal{E}' respectively. The Hecke operator T_{\lambda,v} is the composition h_{*}^{\rightarrow}\circ h^{\leftarrow *}. In essence what it does is it sends a function f in C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) to the function which sends a point in \mathrm{Bun}_{G}(\mathbb{F}_{q}) corresponding to the G-bundle \mathcal{E} to the sum of the values of f(\mathcal{E}') over all modifications of G-bundles \phi:\mathcal{E}'\to\mathcal{E} at v bounded by \lambda. In this last description one can see that it is in fact analogous to the description of Hecke operators for modular forms discussed in Hecke Operators.

More generally given a representation V of \widehat{G}, we can obtain a Hecke operator T_{V}, and these Hecke operators have the property that if V=V'\oplus V'', we must have T_{V,v}=T_{V',v}+T_{V'',v}, and if V=V'\otimes V'' , we must have T_{V,v}=T_{V',v}T_{V'',v}. If V is irreducible, then we can build T_{V,v} as a combination of T_{\lambda,v}, where the \lambda‘s are the weights of V.

Now let us go back to the decomposition

\displaystyle C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})=\bigoplus_{\sigma} \mathfrak{H}_{\sigma}.

The statement of the global Langlands correspondence for function fields over a finite field \mathbb{F}_{q} additionally requires that the Hecke operators preserve the subspaces \mathfrak{H}_{\sigma}; that is, they act on each of these subspaces, and do not send an element of such a subspace to another outside of it. Additionally, we require that the action of the Hecke operators are “compatible with the Satake isomorphism”. This means that the action of a Hecke operator T_{V,v} is given by multiplication by the scalar \mathrm{Tr}_{V}(\sigma(\mathrm{Frob}_{v})). This is somewhat analogous to the Eichler-Shimura relation relating the Hecke operators and the Frobenius briefly mentioned in Galois Representations Coming From Weight 2 Eigenforms.

Ideas related to the proof of the automorphic to Galois direction: Excursion operators and the cohomology of moduli stacks of shtukas

Let us now discuss some ideas related to Vincent Lafforgue’s proof of “automorphic to Galois direction” of the global Langlands correspondence for function fields over a finite field. An important part of these concerns the algebra of excursion operators, denoted by \mathcal{B}. These are certain endomorphisms of C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) which include the Hecke operators. The idea of the automorphic to Galois direction is that characters \nu:\mathcal{B}\to \overline{\mathbb{Q}}_{\ell}^{\times} correspond uniquely to some L-parameter \sigma.

To understand these excursion operators better, we will look at how they are constructed. The construction of the excursion operators involves the cohomology of moduli stacks of shtukas.

A shtuka is a very special kind of a modification of a vector bundle. Given an indexing set I, a shtuka over a scheme S over \mathbb{F}_{q} consists of the following data:

  • A set of points (x_{i})_{i\in I}:S\to X^{I} (the x_{i} are called the “legs” of the shtuka)
  • A G-bundle \mathcal{E} over X\times S
  • An isomorphism

\displaystyle \phi: \mathcal{E}\vert_{(X\times S)\setminus (\bigcup_{i\in I}\Gamma_{x_{i}}}\xrightarrow{\sim}(\mathrm{Id}\times \mathrm{Frob}_{S})^{*}\mathcal{E}\vert _{(X\times S)\setminus (\bigcup_{i\in I}\Gamma_{x_{i}})}

where \Gamma_{x_{i}} is the graph of the x_{i}‘s. Let us denote the moduli stack of such shtukas by \mathrm{Sht}_{I}. We take note of the important fact that the moduli stack of shtukas with no legs, \mathrm{Sht}_{\emptyset}, is a discrete set of points and is in fact the same as \mathrm{Bun}_{G}(\mathbb{F}_{q})!

We now want to define sheaves on \mathrm{Sht}_{I} which will serve as coefficients when we take its etale cohomology, and we want these sheaves to depend on representations W of \widehat{G}^{I}, for the eventual goal of having the cohomology (or appropriate subspaces of it that we want to consider) be functorial in W. This is to be accomplished by considering another moduli stack, the moduli stack of modifications over the formal neighborhood of the legs x_{i}. This parametrizes the following data:

  • The set of points (x_{i})_{i\in I}:S\to X^{I}
  • A pair of G-bundles \mathcal{E} and \mathcal{E}' on the formal completion \widehat{X\times S} of X\times S along the neighborhood of the union of the the graphs \Gamma_{x_{i}}
  • An isomorphism

\displaystyle \phi: \mathcal{E}\vert_{\widehat{X\times S}\setminus (\bigcup_{i\in I}\Gamma_{x_{i}}}\xrightarrow{\sim}\mathcal{E}'\vert _{\widehat{X\times S}\setminus (\bigcup_{i\in I}\Gamma_{x_{i}})}

We denote this moduli stack by \mathcal{M}_{I}. The virtue of this moduli stack \mathcal{M}_{I} is that a very important theorem called the geometric Satake equivalence associates to any representation W of \widehat{G}^{I} a certain object called a perverse sheaf on \mathcal{M}_{I}. Now there is a map from \mathrm{Sht}_{I} to \mathcal{M}_{I}, and pulling back this perverse sheaf associated to W we obtain a perverse sheaf \mathcal{F}_{I,W} on \mathrm{Sht}_{I}. Now we take the intersection cohomology (we just think of this for now as being somewhat similar to \ell-adic etale cohomology) with compact support of the fiber of \mathrm{Sht}_{I} over a geometric generic point of X^{I} with coefficients in \mathcal{F}_{I,W}. We cut down a “Hecke-finite” (this is a technical condition that we leave to the references for now) subspace of it, and call this subspace H_{I,W}. This subspace has an action of \mathrm{Gal}(\overline{F}/F)^{I}.

The above construction is functorial in W – that is, a map u:W\to W' gives rise to a map \mathcal{H}(u):H_{I,W}\to H_{I,W}. Furthermore, there is this very important phenomenon of fusion. Given a map of sets \zeta:I\to J this is an isomorphism H_{I,W}\xrightarrow{\sim}H_{J,W^{\zeta}}, where W^{\zeta} is a representation of \widehat{G}^{J} on the same underlying vector space as W, obtained by composing the map from \widehat{G}^{J} to \widehat{G}^{I} that sends (g_{j})_{j\in J} to (g_{\zeta(i)})_{i\in I} with W.

Now we can define the excursion operators. Let f be a function on \widehat{G}\backslash \widehat{G}^{I}/ \widehat{G}. We can then find a representation W of \widehat{G}^{I} and elements x\in W, \xi\in W^{*}, invariant under the diagonal action of \widehat{G}, such that

\displaystyle f((g_{i})_{i\in I})\langle \xi, (g_{i})_{i\in I}\cdot x\rangle.

Let (\gamma_{i})_{i\in I}\in \mathrm{Gal}(\overline{F}/F)^{I}. The excursion operator S_{I,f,(\gamma_{i})_{i\in I}} is defined to be

\displaystyle H_{\lbrace 0\rbrace,\mathbf{1}} \xrightarrow{\mathcal{H}(x)} H_{\lbrace 0\rbrace,W_{\mathrm{diag}}}\xrightarrow{\mathrm{fusion}} H_{I,W}\xrightarrow{(\gamma_{i})_{i\in I}} H_{I,W}\xrightarrow{\mathrm{fusion}} H_{\lbrace 0\rbrace,W_{\mathrm{diag}}}\xrightarrow{\mathcal{H}(\xi)} H_{ \lbrace 0\rbrace,\mathbf{1}}

where W_{\mathrm{diag}} is the diagonal representation of \widehat{G} on W, i.e. we compose the diagonal embedding \widehat{G}\hookrightarrow \widehat{G}^{I} given by g\mapsto (g,\ldots,g) with the representation W.

The excursion operators give endomorphisms of H_{\lbrace 0\rbrace,\mathbf{1}}. By fusion the subspace H_{\lbrace 0\rbrace,\mathbf{1}} is the same as H_{\emptyset,\mathbf{1}}, which, in turn, is the same as C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) (recall that the moduli stack of shtukas with no legs is the same as \mathrm{Bun}_{G}(\mathbb{F}_{q})). The algebra generated by these endomorphisms as I, f, and (\gamma_{i})_{i\in I} vary is called the algebra of excursion operators, and is denoted by \mathcal{B}. It is commutative and the different excursion operators satisfy certain natural relations amongst each other.

As stated earlier, the Hecke operators are but particular cases of the excursion operators. Namely, the Hecke operator T_{V,v} is just the excursion operator S_{\lbrace 1,2\rbrace, f,(\mathrm{Frob}_{v},1)}, where f sends (g_{1},g_{2}) to \mathrm{Tr}_{V}(g_{1}g_{2}^{-1}).

Now the idea of the decomposition of C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) is as follows. The algebra of excursion operators \mathcal{B} partitions C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) into eigenspaces \mathfrak{H}_{\nu}, where it acts on each eigenspace as a character \nu:\mathcal{B}\to \overline{\mathbb{Q}}_{\ell}. Then as previously mentioned, every character \nu corresponds uniquely to an L-parameter \sigma, satisfying

\displaystyle \nu(S_{I,f,(\gamma_{i})_{i\in I}})=f(\sigma(\gamma_{i})_{i\in I}).

This says therefore that the decomposition of C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell}) is indexed by L-parameters. Everything we have discussed so far may also be applied with a level structure included, encoded in the form of a finite subscheme N of X. Then our L-parameters will be maps \pi_{1}(X\setminus N,\overline{\eta})\to \widehat{G}(\overline{\mathbb{Q}}_{\ell}) and we also replace \mathrm{Bun}_{G} by \mathrm{Bun}_{G,N}, which if G is split has a Weil parametrization given by G(F)\backslash G(\mathbb{A}_{F})/K, where K is the kernel of the map G( \prod_{v}\mathcal{O}_{F_{v}} )\to G(\mathcal{O}_{N}).

Other directions: The Galois to automorphic direction, and the geometric Langlands program

We have so far discussed the “automorphic to Galois direction” of the global Langlands correspondence for function fields over finite fields, and some ideas related to its proof by Vincent Lafforgue. We now briefly discuss the “Galois to automorphic direction” and related work by Gebhard Böckle, Michael Harris, Chandrasekhar Khare, and Jack A. Thorne. This concerns the question of whether a given L-parameter is “cuspidal automorphic”, i.e. it can be obtained from a character of the algebra of excursion operators as stated above.

What Böckle, Harris, Khare, and Thorne do not quite prove this “Galois to automorphic direction” in full. Instead what they prove is that given an everywhere unramified L-parameter \sigma:\mathrm{Gal}(\overline{F}/F)\to\widehat{G}(\mathbb{Q}_{\ell}) with dense Zariski image, then one can find an extension E of F such that the restriction \sigma\vert_{\mathrm{Gal}(\overline{E}/E)}:\mathrm{Gal}(\overline{E}/E)\to\widehat{G}(\mathbb{Q}_{\ell}) is cuspidal automorphic. We say that the L-parameter \sigma is potentially automorphic.

The way the above “potential automorphy” result is proved is by using techniques similar to that used in modularity (see also Galois Deformation Rings). We recall from our brief discussion in Galois Deformation Rings that the usual approach to modularity has two parts – residual modularity, and modularity lifting. The same is true in potential automorphy. The automorphy lifting part makes use of the same ideas as in the “R=T” theorems in modularity lifting, although in this context, they are called “R=B” theorems instead, since we are considering excursion operators instead of just the Hecke operators.

To obtain an analogue of the residual modularity part, Böckle, Harris, Khare, and Thorne make use of results of Alexander Braverman and Dennis Gaitsgory from what is known as the geometric Langlands correspondence (for function fields over a finite field). Although we will not discuss the work of Braverman and Gaitsgory here, we will end this post with a rough idea of what the geometric Langlands correspondence is about.

The geometric Langlands correspondence replaces the cuspidal automorphic forms (which as we recall are \overline{\mathbb{Q}}_{\ell}-valued functions on \mathrm{Bun}_{G}(\mathbb{F}_{q})) with \overline{\mathbb{Q}}_{\ell}-valued sheaves (actually a complex of \overline{\mathbb{Q}}_{\ell}-valued sheaves, or more precisely an object of the category D^{b}(\mathrm{Bun}_{G}) the “derived category of \overline{\mathbb{Q}}_{\ell}-valued sheaves with constructible cohomologies”) via Grothendieck’s sheaves to functions dictionary.

Suppose we have some scheme Y over \mathbb{F}_{q}. First let us suppose that Y=\mathrm{Spec}(\mathbb{F}_{q}). Then since Y is just a point, a complex \mathcal{F} of sheaves on Y is just a complex of vector spaces (we shall take the sheaves to be \overline{\mathbb{Q}}_{\ell}-valued, so this complex is a complex of \overline{\mathbb{Q}}_{\ell}-vector spaces). This complex has an action of \mathrm{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q}). Now we take the alternating sum of the traces of Frobenius acting on this complex, and this gives us an element of \overline{\mathbb{Q}}_{\ell}. For more general Y, for every point y:\mathrm{Spec}(\mathbb{F}_{q})\to Y we apply this same construction to the sheaf \mathcal{F}_{y} which is the pullback of the sheaf \mathcal{F} on Y to \mathrm{Spec}(\mathbb{F}_{q}) via the morphism y:\mathrm{Spec}(\mathbb{F}_{q})\to Y. This provides us with a \overline{\mathbb{Q}}_{\ell}-valued function on Y(\mathbb{F}_{q}).

One can also go in the other direction constructing complexes of sheaves given certain functions. Suppose we have a commutative connected algebraic group A and suppose we have a character \chi of A(\mathbb{F}_{q}). Then we can associate to this character an element of D^{b}(A) as follows. We have the Lang isogeny L:A(\mathbb{F}_{q})\to A(\mathbb{F}_{q}) given by \mathrm{Frob}(a)/a for some element a of A(\mathbb{F}_{q}). The Lang isogeny defines a covering map of A whose group of deck transformations is the group \mathrm{ker(L)}=A(\mathbb{F}_{q}). But because we have a character \chi (a \overline{\mathbb{Q}}_{\ell}^{\times}-valued function on A(\mathbb{F}_{q})), we can take the composition

\displaystyle \pi_{1}(Y,\overline{\eta})\to \mathrm{ker}(L)=A\xrightarrow{\chi}\overline{\mathbb{Q}}_{\ell}^{\times}

This gives us a 1-dimensional representation of \pi_{1}(Y,\overline{\eta}). This in turn gives us a 1-dimensional local system, which is known by the theory of constructible sheaves to be an object of D^{b}(A). This resulting sheaf is also called a character sheaf. In the case when A=\mathbb{G}_{m} it is called the Kummer sheaf, and when A=\mathbb{G}_{a} it is called the Artin-Schreier sheaf.

Grothendieck’s sheaves to functions dictionary is the inspiration for the geometric Langlands correspondence, which is stated entirely in terms of sheaves. We consider the same setting as before, but we now define a slightly modified version of the Hecke stack \mathrm{Hck} where aside from parametrizing modifications we also include in the data being parametrized the point being removed to make the modification. Let s:\mathrm{Hck}\to X be the map that gives us this point on X. Given a representation V of \widehat{G} we let \mathcal{S}_{V} be the perverse sheaf on D^{b}(\mathrm{Bun}_{G}) given by geometric Satake as discussed earlier, and we define the Hecke functor T_{V} that sends an object \mathfrak{F} of D^{b}(\mathrm{Bun}_{G}) to an object T(\mathfrak{F}) of D^{b}(X\times \mathrm{Bun}_{G}) follows

T_{V}(\mathfrak{F})=(s\times h_{\rightarrow})_{!}\circ((h^{\leftarrow})^{*}(\mathfrak{F})\otimes S_{V})

Then the geometric Langlands correspondence (for function fields over a finite field) states that given an L-parameter \sigma, one can find a Hecke eigensheaf, i.e. a sheaf \mathfrak{F}_{\sigma} such that applying the Hecke functor T to it we have T_{V}(\mathfrak{F}_{\sigma})=E_{V\circ\sigma})\boxtimes\mathfrak{F}_{\sigma} where E_{V\circ\sigma} is the local system associated to the representation V\circ\sigma.

A version of the geometric Langlands correspondence has also been formulated for function fields over \mathbb{C} instead of \mathbb{F}_{q}. Many things have to be modified, since in this case there is no Frobenius, and instead the theory of “D-modules” takes its place. This version of the geometric Langlands correspondence has found some fascinating connections to mathematical physics as well.

More recently, a very general and abstract formulation of the geometric Langlands correspondence has been formulated by replacing L-parameters by coherent sheaves on the moduli stack of L-parameters (a single L-parameter corresponding instead to a skyscraper sheaf on the corresponding point). This allows one to have the entire formulation be stated as an equivalence of categories between derived categories of constructible sheaves on \mathrm{Bun}_{G} on one side, and coherent sheaves on the moduli stack of L-parameters. This conjectural statement, appropriately modified to be made more precise (i.e. the moduli stack on the Galois side needs to be modified to parametrize “local systems with restricted variation” while the sheaves on both sides need to be ind-constructible, resp. ind-coherent, with nilpotent singular support), is also known as the categorical geometric Langlands correspondence.

We have given a rough overview of the ideas involved in the global Langlands correspondence for function fields over a finite field. Hopefully we will be able to dive deeper into the finer aspects of the theory, as well as discuss other closely related aspects of the Langlands program (for example the global Langlands correspondence for number fields) in future posts on this blog.

References:

Shtukas for reductive groups and Langlands correspondence for function fields by Vincent Lafforgue

Global Langlands parameterization and shtukas for reductive groups by Vincent Lafforgue (plenary lecture at the 2018 International Congress of Mathematicians)

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale by Vincent Lafforgue

Potential automorphy of \widehat{G}-local systems by Jack A. Thorne

Potential automorphy of \widehat{G}-local systems by Jack A. Thorne (invited lecture at the 2018 International Congress of Mathematicians)

\widehat{G}-local systems are potentially automorphic by Gebhard Böckle, Michael Harris, Chandrasekhar Khare, and Jack A. Thorne

Geometrization of the local Langlands program (notes by Tony Feng from a workshop at McGill University)

The geometric Langlands conjecture (notes from Oberwolfach Arbeitsgemeinschaft)

Recent progress in geometric Langlands theory by Dennis Gaitsgory

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support by Dima Arinkin, Dennis Gaitsgory, David Kazhdan, Sam Raskin, Nick Rozenblyum, and Yakov Varshavsky

An Introduction to the Langlands Program by Daniel Bump, James W. Cogdell, Ehud de Shalit, Dennis Gaitsgory, Emmanuel Kowalski, and Stephen S. Kudla (edited by Joseph Bernstein and Stephen Gelbart)

Reductive Groups Part II: Over More General Fields

In Reductive Groups Part I: Over Algebraically Closed Fields we learned about how reductive groups over algebraically closed fields are classified by their root datum, and how the based root datum helps us understand their automorphisms. In this post, we consider reductive groups over more general, not necessarily algebraically closed fields. We will discuss how they can be classified, and also define the Langlands dual of a reductive group, which will allow us to state the local Langlands correspondence (see The Local Langlands Correspondence for General Linear Groups) for certain groups other than just the general linear group.

Let F be a field. We will say an algebraic group G over F is a reductive group if G_{\overline{F}}, the base change of G to the algebraic closure \overline{F}, is a reductive group. Similarly, we say that G is a torus if the base change G_{\overline{F}} is a torus. This means that after base change to the algebraic closure it becomes isomorphic to the product of copies of the multiplicative group \mathbb{G}_{m}. However, if over F it is already isomorphic to the product of copies of \mathbb{G}_{m}, without the need for a base change, then we say that it is a split torus. If a reductive group G contains a maximal split torus, we say that G is split. We note that a “maximal split torus” is different from a “split maximal torus”!

The classification of split reductive groups is the same as that of reductive groups over algebraically closed fields – they are classified by their root datum. As such they will provide us with the first step towards classifying reductive groups over more general fields.

If G is a reductive group over F, a form of G is some other reductive group G' over F such that after base change to the algebraic closure \overline{F}, G_{\overline{F}} and G_{\overline{F}} are isomorphic. It happens that any reductive group is a form of a split reductive group. This follows from the fact that any abstract root datum is the root datum associated to some reductive group and some split maximal torus contained in it.

The forms of a split group are classified using Galois cohomology. Suppose we have an isomorphism f:G_{\overline{F}}\simeq G_{\overline{F}}. The Galois group \mathrm{Gal}(\overline{F}/F) (henceforth shortened to just \mathrm{Gal}_{F}) acts on the isomorphism f by conjugation, giving rise to another isomorphism ^{\sigma}f:G_{\overline{F}}\simeq G_{\overline{F}}'. Composing this with the inverse of f we get an automorphism f^{-1}\circ^{\sigma}f of G_{\overline{F}}. This automorphism is an example of a 1cocycle in Galois cohomology.

More generally, in Galois cohomology, for some group M with a Galois action (for instance in our case M=\mathrm{Aut}(G)_{\overline{F}})), a 1-cocycle is a homomorphism \varphi:\mathrm{Gal}_{F}\to M such that \varphi(\sigma\tau)=\varphi(\sigma)\cdot^{\sigma}\varphi(\tau). Two 1-cocycles \varphi, \psi are cohomologous if there is an element m\in M such that \psi(\sigma)=m^{-1}\varphi(\sigma)^{\sigma}m. The set of 1-cocycles, modulo those which are cohomologous, is denoted H^{1}(\mathrm{Gal}_{F},M).

By the above construction there is a map between the set of isomorphism classes of forms of G and the Galois cohomology group H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}})). This map actually happens to be a bijection!

Let BR be a based root datum corresponding to G together with a pinning. We have mentioned in Reductive Groups Part I: Over Algebraically Closed Fields the group of automorphisms of BR, the pinned automorphisms of G, and the outer automorphisms of G are all isomorphic to each other. We have the following exact sequence

\displaystyle 0\to\mathrm{Inn}(G_{\overline{F}})\to\mathrm{Aut}(G_{\overline{F}})\to\mathrm{Out}(G_{\overline{F}})\to 0

and when we are provided the additional data of a pinning this gives us a splitting of the exact sequence (i.e. a way to decompose the middle term into a semidirect product of the other two terms).

When the pinning is defined over F, we obtain a homomorphism

\displaystyle H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}}))\to H^{1}( \mathrm{Gal}_{F}, \mathrm{Out}(G_{\overline{F}}))

where H^{1}(\mathrm{Gal}_{F}, \mathrm{Out}(G_{\overline{F}})) is in bijection with the set of conjugacy classes of group homomorphisms \mu:\mathrm{Gal}_{F}\to \mathrm{Out}(G_{\overline{F}}). But we have said earlier that H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}})) is in bijection with the set of isomorphism classes of forms of G. Therefore, any form of G gives us such a homomorphism \mu:\mathrm{Gal}_{F}\to \mathrm{Out}(G_{\overline{F}}).

We say that a reductive group is quasi-split if it contains a Borel subgroup. Split reductive groups are automatically quasi-split.

An inner form of a reductive group G is another reductive group G' related by an isomorphism f:G_{\overline{F}}\simeq G_{\overline{F}}' such that the composition f^{-1}\circ^{\sigma}f is an inner automorphism of G_{\overline{F}}.

Once we have a split group G, and given the data of a pinning, we can now use any morphism \mu:\mathrm{Gal}_{F}\to\mathrm{Out}(G) together with the given pinning to obtain an element of H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}})), which in turn will give us a quasi-split form of G. Now it happens that any reductive group G has a unique quasi-split inner form!

Therefore, in summary, the classification of reductive groups over general fields proceeds in the following three steps:

  1. Classify the split reductive groups using the root datum.
  2. Classify the quasi-split forms using the homomorphisms \mathrm{Gal}_{F}\to\mathrm{Out}(G).
  3. Classify the inner forms of the quasi-split forms.

Let us now discuss the Langlands dual (also known as the L-group) of a reductive group. Since every abstract root datum corresponds to some reductive group G (say, over a field F), we can interchange the roots and coroots and get another reductive group \widehat{G}, which we refer to as the dual group of G.

The Langlands dual of G is the group (an honest to goodness group, not an algebraic group) given by the semidirect product \widehat{G}(\mathbb{C})\rtimes \mathrm{Gal}_{F}. In order to construct this semidirect product we need an action of \mathrm{Gal}_{F} on \widehat{G}(\mathbb{C}), and in this case this action is via its action on the based root datum of \widehat{G} together with a Borel subgroup B\subseteq G, which is the same as a pinned automorphism of \widehat{G}. We denote the Langlands dual of G by ^{L}G.

Let us recall that in The Local Langlands Correspondence for General Linear Groups we stated the local Langlands correspondence, in the case of \mathrm{GL}_{n}(F) where F is a local field, as a correspondence between the irreducible admissible representations of \mathrm{GL}_{n}(F) (over \mathbb{C}) and the F-semisimple Weil-Deligne representations of the Weil group W_{F} of F.

With the definition of the Langlands dual in hand, we can now state the local Langlands correspondence more generally, not just for \mathrm{GL}_{n}(F), and in this case it will not even be a one-to-one correspondence between irreducible admissible representations and Weil-Deligne representations anymore!

First, we will need the notion of a Langlands parameter, also called an L-parameter, which takes the place of the F-semisimple Weil-Deligne representation. It is defined to be a continuous homomorphism W_{F}\times \mathrm{SL}_{2}(\mathbb{C})\to ^{L}G such that, as a homomorphism from W_{F} to ^{L}G, it is semisimple, the composition W_{F}\to^{L}G\to\mathrm{Gal}_{F} is just the usual inclusion of W_{F} into \mathrm{Gal}_{F}, and as a function of \mathrm{SL}_{2}(\mathbb{C}) to \widehat{G}(\mathbb{C}) it comes from a morphism of algebraic groups from \mathrm{SL}_{2} to \widehat{G}.

And now for the statement: The local Langlands correspondence states that, for a reductive group G over a local field F, the irreducible admissible representations of G(F) are partitioned into a finite disjoint union of sets, called L-packets, labeled by (equivalence classes of, where the equivalence is given by conjugation by elements of \widehat{G}(\mathbb{C})) L-parameters. In other words, letting \mathrm{Irr}_{G} be the set of isomorphism classes of irreducible admissible representations of G, and letting \Phi be the set of equivalence classes of L-parameters, we have

\mathrm{Irr}_{G}=\coprod_{\phi\in\Phi}\Pi_{\phi}

where \Pi_{\phi} is the L-packet, a set of irreducible admissible representations of G(F). In the case that F is p-adic and G=\mathrm{GL}_{n}, each of these L-packets have only one element and this reduces to the one-to-one correspondence which we saw in The Local Langlands Correspondence for General Linear Groups.

The L-group and L-parameters are also expected to play a part in the global Langlands correspondence (in the case of function fields over a finite field, the construction of L-parameters was developed by Vincent Lafforgue using excursion operators). There is also much fascinating theory connecting the representations of the L-group to the geometry of a certain geometric object constructed from the original reductive group called the affine Grassmannian. We will discuss more of these topics in the future.

References:

Reductive group on Wikipedia

Root datum on Wikipedia

Inner form on Wikipedia

Langlands dual group on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Representation theory and number theory (notes by Chao Li from a course by Benedict Gross)

Algebraic groups by J. S. Milne

Formal Schemes

In Galois Deformation Rings we discussed the dual numbers as well as the concept of deformations. The dual numbers are numbers with an additional “tangent direction” or a “derivative” – we can further take into account higher order derivatives to consider deformations, which leads us to the concept of deformations.

In this post, we will consider spaces related to deformations, called formal schemes. Let us begin with a motivating example. Consider a field k. From algebraic geometry, we know that the underlying topological space of the scheme \mathrm{Spec}(k) is just a single point. What about the dual numbers, which is the ring k[x]/(x^{2}). What is the underlying topological space of the scheme \mathrm{Spec}(k[x]/(x^{2}))? It turns out it is also just the point!

So as far as the underlying topological spaces go, there is no difference between \mathrm{Spec}(k) and \mathrm{Spec}(k[x]/(x^{2})) – they are both just points. This is because they both have one prime ideal. For k this is the ideal (0), which is also its only ideal that is not itself. For k[x]/(x^{2}), its one prime ideal is the ideal (x); note that the ideal (0) in k[x]/(x^{2}) is not prime, which is related to this ring not being an integral domain. However a scheme is more than just its underlying topological space, one also has the data of its structure sheaf, i.e. its ring of regular functions, and in this regard \mathrm{Spec}(k) and \mathrm{Spec}(k[x]/(x^{2})) are different.

We sometimes consider \mathrm{Spec}(k[x]/(x^{2})) as a “thickening” of \mathrm{Spec}(k) – they are both just the point, but the functions on \mathrm{Spec}(k[x]/(x^{2})) have derivatives, as if there were tangent directions on the point that is the underlying space of \mathrm{Spec}(k[x]/(x^{2})) on which one can move “infinitesimally”, unlike on the point that is the underlying space of \mathrm{Spec}(k).

Just like in our discussion in Galois Deformation Rings, we may want to consider not only the “first-order derivatives” which appear in the dual numbers but also “higher-order derivatives”; we may even want to consider all of them together. This amounts to taking the inverse limit \varprojlim_{n}k[x]/(x^{n}), which is the formal power series ring k[[x]]. However, if we take \mathrm{Spec}(k[[x]]), we will see that it actually has two points, a “generic point” (corresponding to the ideal (0), which is prime because k[[x]] is an integral domain) and a “special point” (corresponding to the ideal (x), which is also prime and furthermore the lone maximal ideal), unlike \mathrm{Spec}(k) or \mathrm{Spec}(k[x]/(x^{2})) (or more generally \mathrm{Spec}(k[x])/(x^{n})), for any n, justified by similar reasons to the preceding argument).

This is where formal schemes come in – a formal scheme can express the “thickening” of some other scheme, with all the “higher-order derivatives”, where the underlying topological space is the same as that of the original scheme but the structure sheaf might be different, to reflect this “thickening”.

A topological ring is a ring A equipped with a topology such that the usual ring operations are continuous with respect to this topology. In this post we will mostly consider the I-adic topology, for some ideal I called the ideal of definition. This topology is generated by a basis consisting of sets of the form a+I^{n}, for a in A.

An example of a topological ring with the I-adic topology is the formal power series ring k[[x]] which we have discussed, together with the ideal of definition (x). Another example is the p-adic integers \mathbb{Z}_{p}, together with the ideal of definition (p). We note that all these examples are complete with respect to the I-adic topology.

More generally for higher dimension one can take, say, an affine variety cut out by some polynomial equation, say y^{2}=x^{3}, and consider the ring k[x,y]/(y^{2}-x^{3}). Note that \mathrm{Spec}(k[x,y]/(y^{2}-x^{3})) is an affine variety. Now we can form a topological ring complete with respect to the I-adic topology, by taking the completion with respect to the ideal I=(y^{2}-x^{3}), i.e. the inverse limit of the diagram

\displaystyle k[x,y]/(y^{2}-x^{3})\leftarrow  k[x,y]/(y^{2}-x^{3})^{2} \leftarrow  k[x,y]/(y^{2}-x^{3})^{3} \leftarrow\ldots

The formation of this ring is an important step in describing the “thickening” of the affine variety \mathrm{Spec}(k[x,y]/(y^{2}-x^{3})), but as said above, it cannot just be done by taking the “spectrum”. Therefore we introduce the idea of a formal spectrum.

Let A be a Noetherian topological ring and let I be an ideal of definition. In addition, let A be complete with respect to the I-adic topology. We define the formal spectrum of A, denoted \mathrm{Spf}(A), to be the pair (X,\mathcal{O}_{X}), where X is the underlying topological space of \mathrm{Spec}(A/I), and the structure sheaf \mathcal{O}_{X} is defined by setting \mathcal{O}_{X}(D_{f}) to be the I-adic completion of A[1/f], for D_{f} the distinguished open set corresponding to f. Applied to the examples earlier, this gives us what we want – a sort of “thickening” of some affine scheme, with an underlying topological space the same as that of the original scheme but with a structure sheaf of functions with “higher-order derivatives”.

More generally, to include the “non-affine” case a formal scheme is a topologically ringed space, i.e. a pair (X,\mathcal{O}_{X}) where X is a topological space and \mathcal{O}_{X} is a sheaf of topological rings, such that for any point x of X there is an open neighborhood of x which is isomorphic as a topologically ringed space to \mathrm{Spf}(A) for some Noetherian topological ring A.

Aside from being useful in deformation theory, formal schemes are also related to rigid analytic spaces (see also Rigid Analytic Spaces). For certain types of formal schemes (“locally formally of finite type over \mathrm{Spf}(\mathbb{Z}_{p})“) one can assign (functorially) a rigid analytic space. For example, this functor will assign to the formal scheme \mathrm{Spf}(\mathbb{Z}_{p}[[x]]) the open unit disc (the interior of the closed unit disc in Rigid Analytic Spaces). This functor is called the generic fiber functor, which is an odd name, because \mathrm{Spf}(\mathbb{Z}_{p}[[x]]) has no “generic points”! However, there is a way to make this name make more sense using the language of adic spaces, which also subsumes the theory of both formal schemes and rigid analytic spaces, and also provides a natural home for the perfectoid spaces we hinted at in Perfectoid Fields. The theory of adic spaces will hopefully be discussed in some future post on this blog.

References:

Formal Scheme on Wikipedia

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Modular Curves at Infinite Level by Jared Weinstein

Adic Spaces by Jared Weinstein

Algebraic Geometry by Robin Hartshorne

p-adic Hodge Theory: An Overview

In The Theory of Motives we discussed the notion of a Weil cohomology, and mentioned four “classical” examples, the singular (also known as Betti) cohomology, the de Rham cohomology, the \ell-adic cohomology, and the crystalline cohomology.

Cohomology theories may be thought of as a way to study geometric objects using linear algebra, by associating vector spaces (or more generally, modules or abelian groups) to such a geometric object. But the four Weil cohomology theories above actually give more than just a vector space:

  • The singular cohomology has an action of complex conjugation.
  • The de Rham cohomology has a Hodge filtration.
  • The \ell-adic cohomology has an action of the Galois group.
  • The crystalline cohomology has an action of Frobenius (and a Hodge filtration as well).

There are relations between these different cohomologies. For example, for a smooth projective variety X over the complex numbers \mathbb{C}, the singular cohomology of the corresponding complex analytic manifold X(\mathbb{C}), with complex coefficients (this can be obtained from singular cohomology with integral coefficients by tensoring with \mathbb{C}) and the de Rham cohomology are isomorphic:

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H_{\mathrm{dR}}^{k}(X)

The roots of this idea go back to de Rham’s work on complex manifolds, where chains in singular homology (which is dual to singular cohomology, see also Homology and Cohomology) can be paired with the differential forms of de Rham cohomology (see also Differential Forms), simply by integrating the differential forms along these chains. By the machinery developed by Alexander Grothendieck, this can be ported over into the world of algebraic geometry.

Again borrowing from the world of complex manifolds, the machinery of Hodge theory gives us the following Hodge decomposition (see also Shimura Varieties):

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{C}}^{j})

Now again for the case of smooth projective varieties over the complex numbers , \ell-adic cohomology also has such an isomorphism with singular cohomology – but this time if it has \ell-adic coefficients (i.e. in \mathbb{Q}_{\ell}).

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\simeq H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell})

Such isomorphisms are also known as comparison isomorphisms (or comparison theorems).

More generally, if we have a field B into which we can embed both \mathbb{Q}_{\ell} and \mathbb{C} (for instance \mathbb{C}), we obtain the following comparison theorem:

\displaystyle  H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q_{\ell}}} B\simeq  H_{\mathrm{dR}}^{k}(X) \otimes_{\mathbb{C}} B

Here is a very interesting thing that these comparison theorems can give us. Let X be a modular curve. Then the Hodge decomposition for the first cohomology gives us

\displaystyle H_{\mathrm{sing}}^{1}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H^{1}(X,\Omega_{X/\mathbb{C}}^{0})\oplus H^{0}(X,\Omega_{X/\mathbb{C}}^{1})

But the H^{0}(X,\Omega_{X/\mathbb{C}}^{1}) is the cusp forms of weight 2 as per the discussion in Modular Forms (see also Galois Representations Coming From Weight 2 Eigenforms). By the results of Hodge theory, the other summand H^{1}(X,\Omega_{X/\mathbb{C}}^{0}) is just the complex conjugate of H^{0}(X,\Omega_{X/\mathbb{C}}^{1}). But we now also have a comparison with etale cohomology, which has a Galois representation! For this the modular form must lie in the cohomology with \mathbb{Q} coefficients, which happens if it is a Hecke eigenform whose Hecke eigenvalues are in \mathbb{Q}. So one of the great things that these comparison theorems gives us is this way of relating modular forms and Galois representations.

The comparison isomorphisms above work for smooth projective varieties over the complex numbers, but let us now go to the p-adic world, and let us consider smooth projective varieties over the p-adic numbers.

It was observed by John Tate (and later explored by Gerd Faltings) that the p-adic cohomology (i.e. the etale cohomology of a smooth projective variety over \mathbb{Q}_{p}, or more generally some other p-adic field, with p-adic coefficients, distinguishing it from \ell-adic cohomology where another prime \ell different from p must be brought in) can have a decomposition akin to the Hodge decomposition, after tensoring it with the p-adic complex numbers (this is the completion of the algebraic closure of the p-adic numbers):

\displaystyle H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)

The p-adic complex numbers here play the role of the complex numbers in the singular cohomology case above or the \ell-adic numbers in the \ell-adic case.

The ideas conjectured by Tate, and later completed by Faltings, was but the prototype of what is now known as p-adic Hodge theory. In its modern form, p-adic Hodge theory concerns comparison isomorphisms between different Weil cohomology theories on smooth projective varieties over the p-adic numbers. However, the role played by the complex numbers, \ell-adic numbers (for the complex case), and p-adic complex numbers (for the p-adic case) must now be played by much more complicated objects called period rings, which were developed by Jean-Marc Fontaine. We will discuss the construction of the period rings at the end of this post, but first let us see how they work.

Let X be a smooth projective variety over \mathbb{Q}_{p} (or more generally some other p-adic field). Let H_{\mathrm{dR}}^{i}(X) and H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p}) be its de Rham cohomology and the p-adic etale cohomology of its base change to the algebraic closure \overline{\mathbb{Q}}_{p} respectively. The comparison isomorphism at the center of p-adic Hodge theory is the following:

\displaystyle H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}

The object denoted B_{\mathrm{dR}} here is the aforementioned period ring. It is equipped with both a Galois action and a filtration akin to the Hodge filtration. More than just that isomorphism above, we also have a way of obtaining the de Rham cohomology if we are given the p-adic etale cohomology, simply by taking the part that is invariant under the Galois action:

\displaystyle  \displaystyle H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

To go the other way, i.e. to recover the p-adic etale cohomology from the de Rham cohomology, we will need a different kind of period ring. This period ring is B_{\mathrm{cris}}, which aside from having a Galois action and a filtration also has an action of Frobenius. Aside from providing us the same isomorphism between de Rham and p-adic etale cohomology upon tensoring, it also provides us with a solution to our earlier problem (as long as X has a smooth proper integral model) as follows:

\displaystyle H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}

This idea can be further abstracted – since etale cohomology provides Galois representations, we can just take some p-adic Galois representation instead, without caring whether it comes from etale cohomology or not, and tensor it with a period ring, then take Galois invariants. For instance let V be some p-adic Galois representation. Then we can take the tensor product

V_{\mathrm{dR}}=(V\otimes B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

If the dimension of V_{\mathrm{dR}} is equal to the dimension of V, then we say that the Galois representation V is de Rham. Similarly we can tensor with B_{\mathrm{cris}}:

V_{\mathrm{cris}}=(V\otimes B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

If its V_{\mathrm{cris}} is equal to the dimension of V , we say that V is crystalline.

The idea of these “de Rham” and “crystalline” Galois representations is that if they come from the corresponding cohomologies then they will have these properties. But does the converse hold? If they are “de Rham” and “crystalline” does that mean that they come from the corresponding cohomologies (i.e. they “come from geometry”)? This is roughly the content of the Fontaine-Mazur conjecture.

Now let us say a few things about the construction of these period rings. These constructions make use of the concepts we discussed in Perfectoid Fields. We start with the ring A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}), which, as we recall from Perfectoid Fields, is the ring of Witt vectors of the tilt of \mathcal{O}_{\mathbb{C}_{p}}. By inverting p and taking the completion with respect to the canonical map \theta:  A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) \to\mathcal{O}_{\mathbb{C}_{p}}, we obtain a ring which we suggestively denote by B_{\mathrm{dR}}^{+}.

There is a special element t of B_{\mathrm{dR}}^{+} which we think of as the logarithm of the element (1, \zeta^{1/p},\zeta^{1/p},\ldots). Upon inverting this element t, we obtain the field B_{\mathrm{dR}}.

The field B_{\mathrm{dR}} is equipped with a Galois action, carried over from the fields involved in its construction, and a filtration, given by \mathrm{Fil}^{i}B_{\mathrm{dR}}=t^{i}B_{\mathrm{dR}}.

To construct B_{\mathrm{cris}}, we once again start with A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) and invert p. However, to have a Frobenius, instead of completing with respect to the kernel of the map \theta, we take a generator of this kernel (which we shall denote by \omega). Then we denote by B_{\mathrm{cris}}^{+} the ring formed by all the power series of the form \sum_{n=0}^{\infty} a_{n}\omega^{n}/n! where the a_{n}‘s are elements of A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p] which converge as n\to\infty, under the topology of A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p] (which is not the p-adic topology!). Once again there will be an element t like before; we invert t to obtain B_{\mathrm{cris}}.

There is yet another period ring called B_{\mathrm{st}}, where the subscript stands for semistable; in addition to a Galois action, filtration, and Frobenius, it has a monodromy operator. Since this is less extensively discussed in introductory literature, we follow this lead and leave this topic, and the many other wonderful topics related to p-adic Hodge theory, to future posts on this blog.

References:

p-adic Hodge theory on Wikipedia

de Rham cohomology on Wikipedia

Hodge theory on Wikipedia

An invitation to p-adic Hodge theory, or: How I learned to stop worrying and love Fontaine by Alex Youcis

Seminar on the Fargues-Fontaine curve (Lecture 1 – Overview) by Jacob Lurie

An introduction to the theory of p-adic representations by Laurent Berger

Reciprocity laws and Galois representations: recent breakthroughs Jared Weinstein

Reductive Groups Part I: Over Algebraically Closed Fields

In many posts on this blog we have talked about elliptic curves, which are examples of algebraic groups (which is itself a specific instance of a group scheme) – a variety whose points form a group. As varieties, elliptic curves (and abelian varieties in general) are projective. There are also algebraic groups which are affine, and in this post we will discuss a special class of affine algebraic groups, called reductive groups, which as we shall see are related to many familiar objects, and are well-studied. In particular, in the case when they are defined over algebraically closed fields we will discuss their classification in terms of root datum. We will also discuss how this root datum helps us understand the automorphisms of such a reductive group.

An example of a reductive group is \mathrm{GL}_{n}; let’s assume that this is a variety defined over some field F. If R is some F-algebra, then the R-valued points of \mathrm{GL}_{n} (in the “functor of points” point of view) is the group \mathrm{GL}_{n}(R) of n\times n matrices with nonzero determinant. Geometrically, we may think of the nonzero determinant condition as the polynomial equation that cuts out the variety \mathrm{GL}_{n}.

Linear algebraic groups are smooth closed algebraic subgroups of \mathrm{GL}_{n}, and they have their own “representation theory”, a “representation” in this context being a morphism from some linear algebraic group G to the algebraic group \mathrm{GL}(V), for some vector space V over some field E. The algebraic group \mathrm{GL}(V) is the algebraic group whose R-valued points give the group of linear transformations of the E-vector space R\otimes V.

A linear algebraic group is a reductive group if it is geometrically connected and every representation is semisimple (a direct product of irreducible representations).

We also denote the reductive group \mathrm{GL}_{1} by \mathbb{G}_{m}. A torus is a reductive group which is isomorphic to a product of copies of \mathbb{G}_m. A torus contained in a reductive group G is called maximal if it is not contained in some strictly larger torus contained in G.

Let T be a maximal torus of the reductive group G. The Weyl group (G,T) is the quotient N(T)/Z(T) where N(T) is the normalizer of T in G (the subgroup consisting of all elements g in G such that for any element t in T gtg^{-1} is an element of T) and Z(T) is the centralizer of T in G (the subgroup consisting of all elements in G that commute with all the elements of T).

Now let us discuss the classification of reductive groups, for which we will need the concept of roots and root datum.

For a maximal torus T in a reductive group G, the characters (homomorphisms from T to \mathbb{G}_{m}) and the cocharacters (homomorphisms from T to \mathbb{G}_{m}) will play an important role in this classification. Let us denote the characters of T by X^{*}(T), and the cocharacters of T by X_{*}(T).

Just like Lie groups, reductive groups have a Lie algebra (the tangent space to the identity), on which it acts (therefore giving a representation of the reductive group, called the adjoint representation). We may restrict to a maximal torus T contained in the reductive group G, so that the Lie algebra \mathfrak{g} of G gives a representation of T. This gives us a decomposition of \mathfrak{g} as follows:

\displaystyle \mathfrak{g}=\mathfrak{g}_{0}\oplus \bigoplus_{\alpha}\mathfrak{g}_{\alpha}

Here \mathfrak{g}_{\alpha} is the subspace of \mathfrak{g} on which T acts as a character \alpha:T\to\mathbb{G}_{m}. The nonzero characters \alpha for which \mathfrak{g}_{\alpha} is nonzero are called roots. We denote the set of roots by \Phi.

For a character \alpha, let T_{\alpha} be the connected component of the kernel of \alpha. Let G_{\alpha} be the centralizer of T_{\alpha} in G. Then the Weyl group W(G_{\alpha},T) will only have two elements, the identity and one other element, which we shall denote by s_{\alpha}. There will be a unique cocharacter \alpha^{\vee} satisfying the equation

s_{\alpha}(x)=x-\langle \alpha^{\vee},x\rangle\alpha

for all characters x:T\to\mathbb{G}_{m}. This cocharacter is called a coroot. We denote the set of coroots by \Phi^{\vee}.

The datum (\Phi, X^{*}(T), \Phi^{\vee}, X_{*}(T)) is called the root datum associated to G. This root datum is actually independent of the chosen maximal torus, which follows from all maximal tori being contained in a unique conjugacy class in G.

There is also a concept of an “abstract” root datum, a priori having seemingly nothing to do with reductive groups, just some datum (M, \Psi, M^{\vee}, \Psi^{\vee}) where M and M^{\vee} are finitely generated abelian groups, \Psi is a subset of M\setminus \lbrace 0\rbrace, and \Psi^{\vee} is a subset of M^{\vee}\setminus \lbrace 0\rbrace, and they satisfy the following axioms:

  • There is a perfect pairing \langle,\rangle:M\times M^{\vee}\to\mathbb{Z}.
  • There is a bijection between \Psi and \Psi^{\vee}.
  • For any \alpha\in \Psi, and \alpha^{\vee} its image in \Psi under the aforementioned bijection, we have \langle \alpha,\alpha^{\vee}\rangle=2.
  • For any \alpha\in \Psi, the automorphism of M given by x\mapsto \alpha-\langle x,\alpha^{\vee}\rangle\alpha preserves \alpha.
  • The subgroup of \mathrm{Aut}(M) generated by x\mapsto \alpha-\langle x,\alpha^{\vee}\rangle\alpha is finite.

Again, a priori, such a datum of finitely generated abelian groups and their subsets, satisfying these axioms, seems to have nothing to do with reductive groups. However, we have the following amazing theorem:

Any abstract root datum is the root datum associated to some reductive group.

For reductive groups over an algebraically closed field, the root datum classifies reductive groups:

Two reductive groups over an algebraically closed field have the same root datum if and only if they are isomorphic.

Let us now discuss how root datum helps us understand the automorphisms of a reductive group. For this we need to expand the information contained in the root datum.

A root basis is a subset of the roots such that any root can be expressed as a unique linear combination of the roots, where the integer coefficients are either all positive or all negative. A based root datum is given by (\Phi, X^{*}(T), S, \Phi^{\vee}, X_{*}(T), S^{\vee}), i.e. the usual root datum together with the additional datum of a root basis S.

The root datum already determines the reductive group G. What does the additional data of a root basis mean? The root basis corresponds to a Borel subgroup of G that contains our chosen maximal torus T. A Borel subgroup of G is a maximal connected solvable Zariski closed algebraic subgroup of G.

A pinning is the datum (G,T,B,\lbrace x_{\alpha}\rbrace_{\alpha\in S}) where T is a maximal torus, B is a Borel subgroup containing T, and \lbrace x_{\alpha}\rbrace_{\alpha\in S} is a basis element of \mathfrak{g}_{\alpha}. Given a pinning, a pinned automorphism of G is an automorphism of G that preserves the pinning.

An inner automorphism of a group G is one that comes from conjugation by some element; in a way they are the automorphisms that are easier to understand. The inner automorphisms form a normal subgroup \mathrm{Inn}(G) of the group of automorphisms \mathrm{Aut}(G), and the quotient \mathrm{Aut}(G)/\mathrm{Inn}(G) is called \mathrm{Out}(G). We have similar notions for algebraic groups.

Now a pinned automorphism is an automorphism, therefore has a map to \mathrm{Out}(G). A pinned automorphism also has a map to the automorphisms of the corresponding based root datum. Both of these maps are actually isomorphisms! Therefore we have a description of \mathrm{Aut}(G) as follows:

The automorphisms of G as an algebraic group are given by the semidirect product of the inner automorphisms and the automorphisms of the based root datum.

In this post we have only focused on the case of reductive groups over algebraically closed fields. Over more general fields the theory of reductive groups, for instance the classification, is more complicated. This will hopefully be tackled in future posts on this blog.

References:

Algebraic group on Wikipedia

Linear algebraic group on Wikipedia

Reductive group on Wikipedia

Root datum on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Representation theory and number theory (notes by Chao Li from a course by Benedict Gross)

Lectures on the geometry and modular representation theory of algebraic groups by Geordie Williamson and Joshua Ciappara

Algebraic groups by J. S. Milne

Rigid Analytic Spaces

This blog post is inspired by and follows closely an amazing talk given by Ashwin Iyengar at the “What is a…seminar?” online seminar. I hope I can do the talk, and this wonderful topic, some justice in this blog post.

One of the most fascinating and powerful things about algebraic geometry is how closely tied it is to complex analysis, despite what the word “algebraic” might lead one to think. To state one of the more simple and common examples, we have that smooth projective curves over the complex numbers \mathbb{C} are the same thing as compact Riemann surfaces. In higher dimensions we also have Chow’s theorem, which tells us that an analytic subspace of complex projective space which is topologically closed is an algebraic subvariety.

This is all encapsulated in what is known as “GAGA“, named after the foundational work “Géometrie Algébrique et Géométrie Analytique” by Jean-Pierre Serre. We refer to the references at the end of this post for the more precise statement, but for now let us think of GAGA as giving us a fully faithful functor from proper algebraic varieties over \mathbb{C} to complex analytic spaces, which gives us an equivalence of categories between their coherent sheaves.

One may now ask if we can do something similar with the p-adic numbers \mathbb{Q}_{p} (or more generally an extension K of \mathbb{Q}_{p} that is complete with respect to a valuation that extends the one on \mathbb{Q}_{p}) instead of \mathbb{C}. This leads us to the theory of rigid analytic spaces, which was originally developed by John Tate to study a p-adic version of the idea (also called “uniformization”) that elliptic curves over \mathbb{C} can be described as lattices on \mathbb{C}.

Let us start defining these rigid analytic spaces. If we simply naively try to mimic the definition of complex analytic manifolds by having these rigid analytic spaces be locally isomorphic to \mathbb{Q}_{p}^{m}, with analytic transition maps, we will run into trouble because of the peculiar geometric properties of the p-adic numbers – in particular, as a topological space, the p-adic numbers are totally disconnected!

To fix this, we cannot just use the naive way because the notion of “local” would just be too “small”, in a way. We will take a cue from algebraic geometry so that we can use the notion of a Grothendieck topology to fix what would be issues if we were to just use the topology that comes from the p-adic numbers.

The Tate algebra \mathbb{Q}_{p}\langle T_{1},\ldots,T_{n}\rangle is the algebra formed by power series in n variables that converge on the n-dimensional unit polydisc D^{n}, which is the set of all n-tuples (c_{1},\ldots,c_{n}) of elements of \mathbb{Q}_{p} that have p-adic absolute value less than or equal to 1 for all i from 1 to n.

There is another way to define the Tate algebra, using the property of power series with coefficients in p-adic numbers that it converges on the unit polydisc D^{n} if and only if its coefficients go to zero (this is not true for real numbers!). More precisely if we have a power series

\displaystyle f(T_{1},\ldots,T_{n})=\sum_{a} c_{a}T_{1}^{a_{1}}\ldots T_{n}^{a_{n}}

where c_{a}\in \mathbb{Q}_{p} and a=a_{1}+\ldots+a_{n} runs over all n-tuples of natural numbers, then f converges on the unit polydisc D^{n} if and only if \lim_{a\to 0}c_{a}=0.

The Tate algebra has many important properties, for example it is a Banach space (see also Metric, Norm, and Inner Product) with the norm of an element given by taking the biggest p-adic absolute value among its coefficients. Another property that will be very important in this post is that it satisfies a Nullstellensatz – orbits of the absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}) in D^{n} correspond to maximal ideals of the Tate algebra. Explicitly, this correspondence is given by the evaluation map x\in D^{n}\mapsto \lbrace f\vert f(x)=0\rbrace.

A quotient of the Tate algebra by some ideal is referred to as an affinoid algebra. The maximal ideals of the underlying set of an affinoid algebra A will be denoted \mathrm{Max}(A), and this should be reminiscent of how we obtain the closed points of a scheme.

Once we have the underlying set \mathrm{Max}(A) , we need two other things to be able to define “affinoid” rigid analytic spaces, which we shall later “glue” to form more general rigid analytic spaces – a topology, and a structure sheaf (again this should be reminiscent of how schemes are defined).

Again taking a cue from how schemes are defined, we define “rational domains”, which are analogous to the distinguished Zariski open sets of (the underlying topological space of) a scheme. Given elements f_{1},\ldots,f_{r},g of the affinoid algebra A, the rational domain \displaystyle A\left(\frac{f}{g}\right) is the set of all x\in\mathrm{Max}(A) such that f_{i}(x)\leq g(x) for all 1\leq i\leq r.

The rational domains generate a topology, however this is still just the p-adic topology, and so it still does not solve the problems that we originally ran into when we tried to define rigid analytic spaces by mimicking the definition of a complex analytic manifold. The trick will be to make use of something that is more general than just a topology – a Grothendieck topology (see also More Category Theory: The Grothendieck Topos).

Let us now define the particular Grothendieck topology that we will use. Unlike other examples of a Grothendieck toplogy, the covers will involve only subsets of the space being covered (it is also referred to as a mild Grothendieck topology). Let X=\mathrm{Max}(A). A subset U of X is called an admissible open if it can be covered by rational domains \lbrace U_{i}\rbrace_{i\in I} such that for any map Y\to X where Y=\mathrm{Max}(B) for some affinoid algebra B, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover.

If U is an admissible open covered by admissible opens \lbrace U_{i}\rbrace_{i\in I}, then this covering is called admissible if for any map Y\to X whose image is contained in U, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover. These admissible coverings satisfy the axioms for a Grothendieck topology, which we denote G_{X}.

If A is an affinoid algebra, and f_{1},\ldots,f_{k},g are functions, we let \displaystyle A\left\langle \frac{f}{g}\right\rangle denote the ring A\langle T_{1},\ldots T_{k}\rangle/(gT_{i}-f_{i}). By associating to a rational domain \displaystyle A\left(\frac{f}{g}\right) this ring \displaystyle A\left\langle\frac{f}{g}\right\rangle, we can define a structure sheaf \cal{O}_{X} on this Grothendieck topology.

The data consisting of the set X=\mathrm{Max}(A), the Grothendieck topology G_{X}, and the structure sheaf \mathcal{O}_{X} is what makes up an affinoid rigid analytic space. Finally, just as with schemes, we define a more general rigid analytic space as the data consisting of some set X, a Grothendieck topology G_{X} and a sheaf \mathcal{O}_{X} such that locally, with respect to the Grothendieck topology G_{X}, it is isomorphic to an affinoid rigid analytic space.

Under this definition, we have in fact a version of GAGA that holds for rigid analytic spaces – a fully faithful functor from proper schemes of finite type over \mathbb{Q}_{p} to rigid analytic spaces over \mathbb{Q}_{p} that gives an equivalence of categories between their coherent sheaves.

Finally let us now look at an example. Consider the affinoid rigid analytic space obtained from the affinoid algebra \mathbb{Q}_{p}\langle T\rangle. By the Nullstellensatz the underlying set is the unit disc D. The “boundary” of this is the rational subdomain (and therefore an admissible open) \displaystyle D\left(\frac{1}{T}\right), and its complement, the “interior” is covered by rational subdomains \displaystyle D\left(\frac{T^{n}}{p}\right). With this covering the interior may also be shown to be an admissible open.

While it appears that, since we found two complementary admissible opens, we can disconnect the unit disc, we cannot actually do this in the Grothendieck topology, because the set consisting of the boundary and the interior is not an admissible open! And so in this way we see that the Grothendieck topology is the difference maker that allows us to overcome the obstacles posed by the peculiar geometry of the p-adic numbers.

Since Tate’s innovation, the idea of a p-adic or nonarchimedean geometry has blossomed with many kinds of “spaces” other than the rigid analytic spaces of Tate, for example adic spaces, a special class of which generalize perfectoid fields (see also Perfectoid Fields) to spaces, or Berkovich spaces, which are honest to goodness topological spaces instead of relying on a Grothendieck topology like rigid analytic spaces do. Such spaces will be discussed on this blog in the future.

References:

Rigid analytic space on Wikipedia

Algebraic geometry and analytic geometry on Wikipedia

Several approaches to non-archimedean geometry by Brian Conrad

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

p-adic families of modular forms [after Coleman, Hida, and Mazur] by Matthew Emerton

The Local Langlands Correspondence for General Linear Groups

The Langlands program is, roughly, a part of math that bridges the representation theory of reductive groups and Galois representations. It consists of many interrelated conjectures, among which are the global and local Langlands correspondences, which may be seen as higher-dimensional analogues of global and local class field theory (see also Some Basics of Class Field Theory) in some way. Just as in class field theory, the words “global” and “local” refer to the fact that these correspondences involve global and local fields, respectively, and their absolute Galois groups.

In this post we will discuss the local correspondence (the global correspondence requires the notion of automorphic forms, which involves a considerable amount of setup, hence is postponed to future posts). We will also only discuss the case of the general linear group \mathrm{GL}_{n}(F), for F a local field, and in particular \mathrm{GL}_{1}(F) and \mathrm{GL}_{2}(F) where we can be more concrete. More general reductive groups will bring in the more complicated notions of L-packets and L-parameters, and therefore also postponed to future posts.

Let us give the statement first, and then we shall unravel what the words in the statement mean.

The local Langlands correspondence for general linear groups states that there is a one-to-one correspondence between irreducible admissible representations of \mathrm{GL}_{n}(F) (over \mathbb{C}) and F-semisimple Weil-Deligne representations of the Weil group W_{F} (also over \mathbb{C}).

Let us start with “irreducible admissible representations of \mathrm{GL}_{n}(F)“. These representations are similar to what we discussed in Representation Theory and Fourier Analysis (in fact as we shall see, many of these representations are also infinite-dimensional, and constructed somewhat similarly). Just to recall, irreducible means that the only subspaces held fixed by \mathrm{GL}_{n}(F) are 0 and the entire subspace.

Admissible means that, if we equip \mathrm{GL}_{n}(F) with the topology that comes from the p-adic topology of the field F, for any open U subgroup of \mathrm{GL}_{n}(F) the fixed vectors form a finite-dimensional subspace.

Now we look at the other side of the correspondence. We already defined what a Weil-Deligne representation is in Weil-Deligne Representations. A Weil-Deligne representation (\rho_{0},N) is F-semisimple if the representation \rho_{0} is the direct sum of irreducible representations.

In the case of \mathrm{GL}_{1}(F), the local Langlands correspondence is a restatement of local class field theory. We have that \mathrm{GL}_1(F)=F^{\times}, and the only irreducible admissible representations of \mathrm{GL}_1(F) are continuous group homomorphisms \chi:F^\times\to\mathbb{C}^{\times}.

On the other side of the correspondence we have the one-dimensional Weil-Deligne representations (\rho_{0},N) of W_F, which must have monodromy operator N=0 and must factor through the abelianization W_F^{\mathrm{ab}}.

Recall from Weil-Deligne Representations that we local class field theory gives us an isomorphism \mathrm{rec}:F^{\times}\xrightarrow{\sim}W_{F}^{\mathrm{ab}}, also known as the Artin reciprocity map. We can now describe the local Langlands correspondence explicitly. It sends \chi to the Weil-Deligne representation (\rho_{0},0), where \rho_{0} is the composition W_{F}\to W_{F}^{\mathrm{ab}}\xrightarrow{\mathrm{rec}^{-1}}F^{\times}\xrightarrow{\chi}\mathbb{C}^{\times}.

Now let us consider the case of \mathrm{GL}_{2}(F). If the residue field of F is not of characteristic 2, then the irreducible admissible representations of \mathrm{GL}_{2}(F) may be enumerated, and they fall into four types: principal series, special, one-dimensional, and supercuspidal.

Let \chi_{1},\chi_{2}:F^{\times}\to\mathbb{C}^{\times} be continuous admissible characters and let I(\chi_{1},\chi_{2}) be the vector space of functions \phi:\mathrm{GL}_{2}(F)\to\mathbb{C} such that

\displaystyle \phi \left(\begin{pmatrix}a&b\\0&d\end{pmatrix}g\right)=\chi_{1}(a)\chi_{2}(d)\Vert a/d\Vert^{1/2}\phi(g)

The group \mathrm{GL}_{2}(F) acts on the functions \phi, just as in Representation Theory and Fourier Analysis. Therefore it gives us a representation of \mathrm{GL}_{2}(F) on the vector space I(\chi_{1},\chi_{2}), which we say is in the principal series.

Now the representation I(\chi_{1},\chi_{2}) might be irreducible, in which case it is one of the things that go into our correspondence, or it might be reducible. This is decided by the Bernstein-Zelevinsky theorem, which says that I(\chi_{1},\chi_{2}) is irreducible precisely if the ration of the characters \chi_{1} and \chi_{2} is not equal to plus or minus 1.

In the case that \chi_{1}/\chi_{2}=1, then we have an exact sequence

\displaystyle 0\to\rho\to I(\chi_{1},\chi_{2})\to S(\chi_{1},\chi_{2})\to 0

where the representations S(\chi_{1},\chi_{2}) and \rho are both irreducible representations of \mathrm{GL}_{2}(F). The representation S(\chi_{1},\chi_{2}) is infinite-dimensional and is known as the special representation. The representation \rho is the one-dimensional representation and is given by \chi_{1}\Vert\cdot\Vert^{1/2}\det.

If \chi_{1}/\chi_{2}=-1 instead, then we have a “dual” exact sequence

\displaystyle 0\to S(\chi_{1},\chi_{2}) \to I(\chi_{1},\chi_{2})\to \rho\to 0

So far the irreducible admissible representations of \mathrm{GL}_{2}(F) that we have seen all arise as subquotients of I(\chi_{1},\chi_{2}). Since characters such as \chi_{1} and \chi_{2} are the irreducible admissible representations of \mathrm{GL}_{1}(F), we may consider the principal series, special, and one-dimensional representations as being built out of these more basic building blocks.

However there exist irreducible admissible representations that do not arise via this process, and they are called supercuspidal representations. For \mathrm{GL}_{2}(F) there is one kind of supercuspidal representation denoted \mathrm{BC}_{E}^{F}(\psi) for E a quadratic extension of F and \psi an admissible character \psi:E\to\mathbb{C}^{\times}.

Now we know what the irreducible admissible representations of \mathrm{GL}_{2}(F) are. The local Langlands correspondence says that they will correspond to F-semisimple Weil-Deligne representations. We can actually describe explicitly which Weil-Deligne representation each irreducible admissible representation of \mathrm{GL}_{2}(F) gets sent to!

Let \chi_{1},\chi_{2}:F^{\times}\to \mathbb{C}^{\times} be the same continuous admissible characters used to construct the irreducible representations as above, and let \rho_{1},\rho_{2} :W_{F}\to \mathbb{C}^{\times} be the corresponding representation of the Weil group given by the local Langlands correspondence for \mathrm{GL}_{1}, as discussed earlier. Then to each irreducible admissible representation of \mathrm{GL}_2(F) we associate a 2-dimensional Weil-Deligne representation as follows:

To the principal series representation I(\chi_{1},\chi_{2}) we associate the Weil-Deligne representation (\rho_{1}\oplus\rho_{2},0).

To the special representation S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert), we associate the Weil-Deligne representation \left(\begin{pmatrix}\Vert\cdot\Vert\rho_{1} & 0\\0 & \rho_{1}\end{pmatrix},\begin{pmatrix} 0 & 1\\0 & 0\end{pmatrix}\right).

To the one-dimensional representation \chi_{1}\circ\det, we associate the Weil-Deligne representation \left(\begin{pmatrix}\rho_{1}\times\Vert\cdot\Vert^{1/2} & 0\\0 & \rho_{1}\times\Vert\cdot\Vert^{-1/2}\end{pmatrix},0\right).

Finally, to the supercuspidal representation \mathrm{BC}_{E}^{F}(\psi) we associate the Weil-Deligne representation (\mathrm{Ind}_{W_{E}}^{W_{F}}\sigma,0), where \sigma is the unique nontrivial element of \mathrm{Gal}(E/F).

We have been able to describe the local Langlands correspondence for \mathrm{GL}_{1}(F) and \mathrm{GL}_{2}(F) explicitly (in the latter case as long as the characteristic of the residue field of F is not 2). The local Langlands correspondence for \mathrm{GL}_{n}(F), for more general n on the other hand, was proven via geometric means – namely using the geometry of certain Shimura varieties (see also Shimura Varieties) as well as their local counterpart, the Lubin-Tate tower, which parametrizes deformations of Lubin-Tate formal group laws (see also The Lubin-Tate Formal Group Law) together with level structure.

There has been much recent work regarding the local Langlands program for groups other than \mathrm{GL}_{n}(F). For instance there is work on the local Langlands correspondence for certain symplectic groups making use of “theta lifts”, by Wee Teck Gan and Shuichiro Takeda. Very recently, there has also been work by Laurent Fargues and Peter Scholze that makes use of ideas from the geometric Langlands program. These, and more, will hopefully be discussed more here in the future.

References:

Langlands program on Wikipedia

Local Langlands conjectures on Wikipedia

Local Langlands conjecture on the nLab

MSRI Summer School on Automorphic Forms and the Langlands Program by Kevin Buzzard

Langlands Correspondence and Bezrukavnikov Equivalence by Anna Romanov and Geordie Williamson

The Local Langlands Conjecture for GL(2) by Colin Bushnell and Guy Henniart