# The Theta Correspondence

In Siegel modular forms, we mentioned that one could construct Siegel modular forms from elliptic modular forms (see Modular Forms) via a process called “lifting”. In this post, we discuss a more general method that produces new automorphic forms (which generalize modular forms, and are also more properly a part of representation theory, see also Automorphic Forms) out of old ones. There is also a local version that deals with representations of p-adic Lie groups. Both of these form the theory of the (global and local) theta correspondence.

We begin with the local theory. Let $F$ be a nonarchimedean local field of characteristic zero (e.g. $\mathbb{Q}_{p}$ or a finite extension of $\mathbb{Q}_{p}$). Let $E$ be a quadratic etale $F$-algebra. Let $V$ be a vector space over $E$ equipped with a Hermitian form $\langle-,-\rangle_{V}$, and let $W$ be a vector space over $E$ equipped with a skew-Hermitian form $\langle -,-\rangle_{W}$. Their respective groups of isometries are the unitary groups $\mathrm{U}(V)$ and $\mathrm{U}(W)$. These two groups form an example of a reductive dual pair. The theory of the local theta correspondence relates representations of one of these groups to representations of the other.

Now the tensor product $V\otimes_{E} W$ can be viewed as a vector space over $F$ and we can equip it with a symplectic form $(-,-)=\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle -,-\rangle_{W})$. We have a map

$\displaystyle \mathrm{U}(V)\times\mathrm{U}(W)\to\mathrm{Sp}(V\otimes_{E} W)$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ has a cover called the metaplectic group; we will describe it in more detail shortly, but we say now that the importance of it for our purposes is that it has a special representation called the Weil representation, which as we shall shortly see will be useful in relating representations of $\mathrm{U}(V)$ to $\mathrm{U}(W)$, and vice-versa.

We first need to construct the Heisenberg group $H(V\otimes_{E} W)$. Its elements are given by $(V\otimes_{E} W)\oplus F$, and we give it the group structure

$\displaystyle (x_{1},t_{1})\cdot (x_{2},t_{2})=\left(x_{1}+x_{2},t_{1}+t_{2}+\frac{1}{2}(x_{1},x_{2})\right)$

The Stone-von Neumann theorem tells us that, for every nontrivial character $\psi:F\to\mathbb{C}^{\times}$ the Heisenberg group has a unique irreducible representation $\omega_{\psi}$ with central character $\psi$. Furthermore, the representation $\omega_{\psi}$ is unitary.

If $V\otimes_{E} W=X\oplus Y$ is a Lagrangian decomposition, we can realize the representation $\omega_{\psi}$ on the vector space of Schwarz functions on either $X$ or $Y$. Let us take it to be $Y$. In particular, we can express $\omega_{\psi}$ as follows. We first extend the character $\psi$ to $H(X)$ (defined to be the subgroup $X\oplus F$ of $H(V\otimes_{E}W)$) and then define $\omega_{\psi}$ as the induced representation (see also The Local Langlands Correspondence for General Linear Groups for another example of an induced representation)

$\displaystyle \omega_{\psi}=\mathrm{c-Ind}_{H(X)}^{H(V\otimes_{E}W)}\psi$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ acts on the Heisenberg group $H(V\otimes_{E}W)$ by $g\cdot (x,t)=(g\cdot x,t)$ for $g\in \mathrm{Sp}(V\otimes_{E} W)$ and $(x,t)\in H(V\otimes_{E}W)$. We can compose this action with the representation $\omega_{\psi}$ to get another representation ${}^{g}\omega_{\psi}=\omega_{\psi}\circ g^{-1}$ of $H(W)$. Now since the action of $\mathrm{Sp}(V\otimes_{E} W)$ on $H(V\otimes_{E}W)$ has trivial center, the central characters of ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ are the same. By the Stone-von Neumann theorem, ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ have to be isomorphic.

What this means now, is that for every $g\in \mathrm{Sp}(V\otimes_{E}W)$, we have a linear transformation $A_{\psi}(g)$ of the underlying vector space $\mathcal{S}$ of the representation $\omega_{\psi}$, so that

$\displaystyle A_{\psi}(g)\circ {}^{g}\omega_{\psi}=\omega_{\psi}\circ A_{\psi}(g)$

This action however is only defined up to a factor of $\mathbb{C}^{\times}$. Since $\omega_{\psi}$ is unitary, we can also require $A_{\psi}$ to be unitary, and so the action becomes well-defined up to $S^{1}$. All in all, this means that we have a representation

$\displaystyle A_{\psi}:\mathrm{Sp}(V\otimes W)\to \mathrm{GL}(\mathcal{S})/S^{1}$

Now if we pull back the map $\mathrm{GL}(\mathcal{S})\to\mathrm{GL}(\mathcal{S})/S^{1}$ by the map $A_{\psi}:\mathrm{Sp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})/S^{1}$, we get a map $\widetilde{A}_{\psi}:\mathrm{Mp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})$, where the group $\mathrm{Mp}(V\otimes_{E} W)$ is an $S^{1}$-cover of $\mathrm{Sp}(V\otimes_{E} W)$. This group $\mathrm{Mp}(V\otimes_{E} W)$ is the metaplectic group mentioned earlier.

Our construction allows us to extend the representation $\omega_{\psi}$ of $H(V\otimes_{E }W)$ to the semidirect product $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$. This representation of $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ is called the Heisenberg-Weil representation. The representation of $\mathrm{Mp}(V\otimes_{E} W)$ obtained by restriction is called the Weil representation.

Recall that we have a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Sp}(V\otimes_{E}W)$. If we could lift this to a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Mp}(V\otimes_{E}W)$, then we could obtain a representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by restricting the Weil representation $\omega_{\psi}$ from $\mathrm{Mp}(V\otimes_{E}W)$ to $\mathrm{U}(V)\times \mathrm{U}(W)$. It turns out such a lifting can be defined and is determined by a pair $(\chi_{V},\chi_{W})$ of characters of $E^{\times}$ satisfying certain conditions. Once we have this lifting, we denote the resulting representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by $\Omega$.

Now let $\pi$ be an irreducible representation of $V$. We consider the maximal $\pi$-isotypic quotient of $\Omega$, which is its quotient by the intersection of all the kernels of morphisms of representations of $U(V)$ from $\Omega$ to $\pi$. This quotient is of the form $\pi\otimes\theta(\pi)$, where $\Theta(\pi)$ is a representation of $U(W)$ called the big theta lift of $\pi$. The maximal semisimple quotient of $\Theta(\pi)$ is denoted $\theta(\pi)$, and is called the small theta lift of $\pi$.

Let us now look at the global picture. Let $k$ be a number field and let $k_{v}$ be the completion of $k$ at one of its places $v$. Let $E$ be a quadratic extension of $k$. Now we let $V$ and $W$ be vector spaces over $E$ equipped with Hermitian and skew-Hermitian forms $\langle-,-\rangle_{B}$ and $\langle--\rangle_{W}$, as in the local case, and consider the tensor product $V\otimes_{E} W$ as a vector space over $k$, and equip it with the symplectic form $\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle-,-\rangle_{W})$. We have localizations $(V\otimes_{E} W)_{v}$ for every $v$, and we have already seen that in this case we can construct the metaplectic group $\mathrm{Mp}((V\otimes_{E} W)_{v})$. We want to put each of these together for every $v$ to construct an “adelic” metaplectic group.

First we take the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$. “Restricted” means that all but finitely many of the factors in this product belong to the hyperspecial maximal compact subgroup $K_{v}$ of $\mathrm{Sp}((V\otimes_{E} W)_{v})$, which is also a compact open subgroup of $\mathrm{Mp}((V\otimes_{E} W)_{v})$. This restricted product contains $\bigoplus_{v}S^{1}$ as a central subgroup. Now if we quotient out the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ by the central subgroup $Z$ given by the set of all $(z_{v})\in\bigoplus_{v}S^{1}$ such that $\prod_{v}z_{v}=1$, the resulting quotient is the “adelic” metaplectic group $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ that we are looking for.

We have a representation $\bigotimes_{v}'\omega_{\psi_{v}}$ of $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ which acts trivially on the central subgroup $Z$ defined above and therefore gives us a representation $\omega_{\psi}$ of $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$.

What is the underlying vector space of the representation $\omega_{\psi}$? If $V\otimes_{E}W=X\oplus Y$ is a Lagrangian decomposition, we have seen that we can realize the local Weil representation $\omega_{\psi_{v}}$ on $\mathcal{S}(Y_{v})$, the vector space of Schwarz functions of $Y_{v}$ (the corresponding localization of $Y$). Likewise we can also realize the global Weil representation $\omega_{\psi}$ as functions on the vector space $\mathcal{S}(Y_{\mathbb{A}})$, defined to be the restricted product $\bigotimes'\mathcal{S}(Y_{v})$.

So now we have the global Weil representation $\omega_{\psi}$, which is a representation of the group $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ on the vector space $\mathcal{S}(Y_{\mathbb{A}})$. But suppose we want an automorphic representation, i.e. one realized on the vector space of automorphic forms for $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ (recall that one of our motivations in this post is to “lift” automorphic forms from one group to another). This is accomplished by the formation of theta functions $\theta(f)(g)$, so-called because it is a generalization of the Jacobi theta function discussed in Sums of squares and the Jacobi theta function. Let $f$ be a vector in the underlying vector space of the Weil representation. Then the theta function $\theta(f)(g)$ is obtained by summing the evaluations of the output of the action of Weil representation on $f$ over all rational points $y\in Y(k)$:

$\displaystyle \theta(f)(g)=\sum_{y\in Y(k)}(\omega_{\psi}(g)\cdot f)(y)$

Now suppose we have a pair of characters $\chi_{1},\chi_{2}$ of $E^{\times}$, so that we have a lifting of $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ to $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$. This lifting sends $U(V)(k)\times U(W)(k)$ to $\mathrm{Mp}(V\otimes_{E}W)(k)$, which means that we can consider $\theta(f)(g)$ as an automorphic form for $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$.

Now we can perform our lifting. Let $f$ be a cuspidal automorphic form for $U(V)$, and let $\varphi$ be a vector in the underlying vector space of the Weil representation. We can now obtain an automorphic form $\theta(\varphi,f)(g)$ on $U(W)$ as follows:

$\displaystyle \theta(\varphi,f)(g)=\int_{[\mathrm{U}(V)]}\theta(\varphi)(g,h)\cdot \overline{f(h)}dh$

The space generated in this way, for all vectors $f$ in a cuspidal automorphic representation $\pi$ of $U(V)$, and all vectors $\varphi$ in the in the underlying vector space of the Weil representation, is called the global theta lift of $\pi$, denoted $\Theta(\pi)$. It is an automorphic representation of $U(W)$.

There is also an analogue of all that we discussed for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ when $V$ and $W$ are vector space over some field, equipped with a quadratic form and symplectic form respectively.

Many cases of “lifting”, for instance the Saito-Kurokawa lift from elliptic modular forms to Siegel modular forms, can be considered special cases of the global theta lift (in particular for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$). The theory of theta lifting is itself part of the theory of Langlands functoriality (see also Trace Formulas). More aspects and examples of the theta correspondence will be discussed in future posts on this blog.

References:

Theta correspondence on Wikipedia

Heisenberg group on Wikipedia

Metaplectic group on Wikipedia

Saito-Kurokawa lift on Wikipedia

Automorphic forms and the theta correspondence by Wee Teck Gan

A brief survey of the theta correspondence by Dipendra Prasad

Non-tempered Arthur packets of G2 by Wee Teck Gan and Nadia Gurevich

A quaternionic Saito-Kurokawa lift and cusp forms on G2 by Aaron Pollack

# Sums of squares and the Jacobi theta function

Which numbers can be written as a sum of two squares (of integers)? To narrow the problem down a little bit, which prime numbers can be written as a sum of two squares? Notice that $2$ can be written as the sum of two squares, $2=1^{2}+1^{2}$. Meanwhile $3$ cannot be written as the sum of two squares; since squares are positive, we only need look at the numbers less than $3$ and we can exhaust all possibilities. Going to $5$, we can see that it can once again be written as the sum of two squares, $1^{2}+2^{2}$.

This problem was solved by Fermat, and the answer is that aside from $2$, which we have already resolved, it is precisely the prime numbers which are $1$ mod $4$ which can be written as the sum of two squares. More generally, for numbers which are not necessarily prime, such a number can be written as the sum of two squares if the numbers which are $3$ mod $4$ appear in its prime factorization with an even exponent. Fermat used the method of infinite descent to solve this problem, however, there are many other proofs, and this problem and its many variants have motivated many developments in mathematics. In this post, we will discuss a fascinating method due to Jacobi, which involves the theory of modular forms ( see also Modular Forms).

Before we start discussing the approach of Jacobi let us state another such variant of the problem. Which numbers can be written as the sum of four squares? This question was settled by Lagrange, and it turns out the answer is that all positive integers can be written as the sum of four squares! The approach of Jacobi that we will discuss turns out to solve this problem as well!

Furthermore, the method of Jacobi not only tells us whether a number is a sum of two squares or four squares, but it actually tells us how many ways such a number can be written in that form. For example, we have mentioned earlier that $5$ can be written as $1^{2}+2^{2}$. This is one way to write it as a sum of two squares – there are actually eight such ways:

$\displaystyle 1^{2}+2^{2}$

$\displaystyle (-1)^{2}+2^{2}$

$\displaystyle (1)^{2}+(-2)^{2}$

$\displaystyle (-1)^{2}+(-2)^{2}$

$\displaystyle (2)^{2}+1^{2}$

$\displaystyle (-2)^{2}+1^{2}$

$\displaystyle 2^{2}+(-1)^{2}$

$\displaystyle (-2)^{2}+(-1)^{2}$

In fact, this what Jacobi’s approach actually does – it gives us the number of ways $r_{k}(n)$ to write a number $n$ as the sum of $k$ squares (for the classical problems we mentioned $k=2$ or $k=4$). If the $r_{k}(n)$ is nonzero, then we know that $n$ can be written as a sum of $k$ squares.

Let us now discuss this method of Jacobi. We will streamline the discussion a bit using modern language that was probably not available to Jacobi. It hinges on a very special function $\theta(z)$ on the upper half-plane called the theta function, defined as follows:

$\displaystyle \theta(z)=\sum_{n=-\infty}^{\infty}e^{2\pi i n^{2}z}=\sum_{n=-\infty}^{\infty}q^{n^{2}}$

Here in the second equation we have just chosen to adopt the traditional notation $q=e^{2\pi i z}$. Re-indexing the summation we can also write the theta function as

$\displaystyle \theta(z)=1+\sum_{n=1}^{\infty}2q^{n^{2}}=1+2q+2q^{4}+2q^{9}+\ldots$

The square of the theta function is a modular form of weight $1$, level $\Gamma_{0}(4)$, and character $\chi_{-4}$ (see also Modular Forms). This means that $(\theta(z))^{2}$ is a holomorphic function on the upper half-plane, bounded as the imaginary part of $z$ goes to infinity, and satisfying the transformation law

$\displaystyle \left(\theta\left(\frac{az+b}{cz+d}\right)\right)^{2}=\chi_{-4}(a)(cz+d)(\theta(z))^{2}$

where $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is an element of $\Gamma_{0}(4)$, the group of $2\times 2$ integer matrices with determinant $1$ and which become upper triangular when the entries are reduced mod $4$ (i.e. $c$ is divisible by $4$), and $\chi_{-4}$ is a function which takes any integer $n$ and outputs $1$ if $n$ is $1$ mod $4$, outputs $-1$ if $n$ is $3$ mod $4$, and outputs $0$ if $n$ is even ($\chi_{-4}$ is an example of a Dirichlet character).

(In the literature the theta function $\theta(z)$ itself is referred to as a “modular form of weight $1/2$“, but we will avoid this terminology in this post to keep things less confusing.)

Now here is what relates the square of the Jacobi to sums of two squares. We can write

$\displaystyle (\theta(z))^{2}=\left(\sum_{a=-\infty}^{\infty}q^{a^{2}}\right)\left(\sum_{b=-\infty}^{\infty}q^{b^{2}}\right)$

Expanding the square of theta function as a Fourier series (again writing $q=e^{2 \pi i z}$) the above equation becomes

$\displaystyle (\theta(z))^{2}=\sum_{n=0}a_{n}q^{n}=\left(\sum_{a=-\infty}^{\infty}q^{a^{2}}\right)\left(\sum_{b=-\infty}^{\infty}q^{b^{2}}\right)$

Now the $n$-th term of this Fourier expansion will receive a contribution from each product of $q^{a^{2}}$ and $q^{b^{2}}$ such that $n=a^{2}+b^{2}$. In other words, the coefficient $a_{n}$ counts how many pairs $(a,b)$ there are such that $n=a^{2}+b^{2}$ – it counts the number of ways $n$ can be written as a sum of two squares! Therefore, the $n$-th Fourier coefficient of $(\theta(z))^{2}$ is just the function $r_{2}(n)$ we mentioned earlier that tells us how many ways there are to write $n$ as a sum of two squares.

More generally, the same argument can be applied to other powers of the theta function. In particular, we can also look at $(\theta(z))^{4}$ and this will tell us about sums of four squares. More precisely, the $n$-th Fourier coefficient of $(\theta(z))^{4}$ is the function $r_{4}(n)$ that tells us how many ways there are to write $n$ as a sum of four squares.

Now we will use results from the theory of modular forms to give us proofs of the theorems of Fermat and Lagrange that we have mentioned earlier.

Modular forms of a certain weight and level form a complex vector space, and the dimension of this vector space can be computed via dimension formulas. In particular, the vector space of modular forms of weight $1$ and level $\Gamma_{0}(4)$ has dimension $1$, which means they are all just complex multiples of each other.

There is another modular form of weight $1$ and level $\Gamma_{0}(4)$ which is well-studied, called the Eisenstein series of weight $1$, level $\Gamma_{0}(4)$, and character $\chi_{-4}$. It is defined as follows:

$\displaystyle G_{1,\chi_{-4}}(z)=\frac{1}{4}+\sum_{n=1}^{\infty}\left(\sum_{d\vert n}\chi_{-4}(d)\right)q^{n}$

From the fact that modular forms of weight $1$ and level $\Gamma_{0}(4)$ form a vector space of dimension $1$, we know that the square of the theta function and this Eisenstein series are just multiples of each other. In fact, from a comparison of the leading terms, we can see that

$(\theta(z))^{2}=4G_{1,\chi_{-4}}(z)$

Therefore, comparing the Fourier expansions, we see that $r_{2}(n)=4(\sum_{d\vert n}\chi_{-4}(d))$. Specializing to when $n$ is a prime, the only divisors of $n$ are $1$ and $n$, and we have $r_{2}(n)=4(1+\chi_{-4}(n))$, which is $8$ when $n$ is $1$ mod $4$, and $0$ when $n$ is $3$ mod $4$, as follows from the definition of $\chi_{-4}$. Therefore this tells us that $n$ is a sum of two squares precisely when $n$ is $1$ mod $4$. With a little more effort, one can see that the formula $r_{2}(n)=4(\sum_{d\vert n}\chi_{-4}(d))$ also tells us that more generally $n$ (even when it is not prime) is a sum of two squares precisely when the prime divisors of $n$ which are $3$ mod $4$ have an even power in its prime factorization.

Let us now look at $(\theta(z))^{4}$ and the problem of writing a number as the sum of four squares. Now $(\theta(z))^{4}$ is actually a modular form of weight $2$ and level $\Gamma_{0}(4)$. This time the vector space of modular forms of weight $2$ and level $\Gamma_{0}(4)$ is a vector space of dimension $2$. So it is not quite as easy as the case of $(\theta(z))^{2}$ and sums of two squares, but we can still find two linearly independent modular forms of weight $2$ and level $\Gamma_{0}(4)$ which will form a convenient basis for us to express $(\theta(z))^{4}$ in terms of.

These modular forms are given by

$\displaystyle G_{2}(z)-2G_{2}(2z)$

and

$\displaystyle G_{2}(2z)-2G_{2}(4z)$

where

$\displaystyle G_{2}(z)=-\frac{1}{24}+\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n}$

is the Eisenstein series of weight $2$ and level $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ (here the symbol $\sigma_{1}(n)$ denotes the sum of the positive divisors of $n$ – note also that we are using a different normalization than in Modular Forms for convenience). It turns out that

$\displaystyle (\theta(z))^{4}=8(G_{2}(z)-2G_{2}(2z))+16(G_{2}(2z)-2G_{2}(4z))$

Similar to the earlier case for the sum of two squares, one can now expand both sides in a Fourier expansion and compare Fourier coefficients. It will turn out that $r_{4}(n)$ is equal to $8$ times the sum of the positive divisors of $n$ which are not divisible by $4$. Since there is always going to be such a divisor, this tells us that any positive integer can always be written as the sum of four squares.

We have seen, therefore, that the theory of modular forms can help us understand very classical problems in number theory. The theta function is in fact worthy of a whole entire theory itself – it is connected to many things in mathematics from representation theory to abelian varieties. We will discuss more of these aspects in future posts.

References:

Theta function on Wikipedia

Jacobi’s four-square theorem on Wikipedia

Sum of squares function on Wikipedia

Elliptic modular forms and their applications by Don Zagier

# The Geometrization of the Local Langlands Correspondence

In The Global Langlands Correspondence for Function Fields over a Finite Field, we introduced the global Langlands correspondence for function fields over a finite field, and Vincent Lafforgue’s work on the automorphic to Galois direction of the correspondence. In this post we will discuss the work of Laurent Fargues and Peter Scholze which uses similar ideas but applies it to the local Langlands correspondence (and this time it works not only for “equal characteristic” cases like Laurent series fields $\mathbb{F}_{q}((t))$ but also for “mixed characteristic” cases like finite extensions of $\mathbb{Q}_{p}$). Note that instead of having complex coefficients like in The Local Langlands Correspondence for General Linear Groups, here we will use $\ell$-adic coefficients.

### I. The Fargues-Fontaine Curve

Let us briefly discuss the idea of “geometrization” and what is meant by Fargues and Scholze making use of V. Lafforgue’s work. Recall that V. Lafforgue’s work concerns the global Langlands correspondence for function fields over a finite field $\mathbb{F}_{q}$, which on one side concerns the space of cuspidal automorphic forms, which are certain functions on $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which in turn parametrizes $G$-bundles on some curve $X$ over $\mathbb{F}_{q}$, and on the other side concerns representations (or more precisely L-parameters) of the etale fundamental group of $X$ (which can also be phrased in terms of the Galois group of its function field).

Perhaps the first question that comes to mind is, what is the analogue of the curve $X$ in the case of the local Langlands correspondence when the field is not a function field (or more correctly a power series field, since it has to be local) over $\mathbb{F}_{q}$, but some finite extension of $\mathbb{Q}_{p}$? Let $E$ be this finite extension of $\mathbb{Q}_{p}$. Since the absolute Galois group of $E$ is also the etale fundamental group of $\mathrm{Spec}(E)$, perhaps we should take $\mathrm{Spec}(E)$ to be our analogue of $X$.

However, in the traditional formulation of the local Langlands correspondence, it is the Weil group that appears instead of the absolute Galois group itself. Considering the theory of the Weil group in Weil-Deligne Representations, this means that we will actually want $\pi_{1}(\mathrm{Spec}(\breve{E})/\mathrm{Frob}^{\mathbb{Z}})$, where $\breve{E}$ is the maximal unramified extension of $E$ and $\mathrm{Frob}$ is the Frobenius, instead of $\pi_{1}(E)$.

Now, we want to “relativize” this. For instance, in The Global Langlands Correspondence for Function Fields over a Finite Field, we considered $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which parametrizes $G$-bundles on the curve $X$ over $\mathbb{F}_{q}$. But we may also want to consider say $\mathrm{Bun}_{G}(R)$, where $R$ is some $\mathbb{F}_{q}$-algebra; this would parametrize $G$-bundles on $X\times_{\mathrm{Spec}(\mathbb{F}_{q})}\mathrm{Spec}(R)$ instead. In fact, we need this “relativization” to properly define $\mathrm{Bun}_{G}$ as a stack (see also Algebraic Spaces and Stacks).

The problem with transporting this to the case of $E$ a finite extension of $\mathbb{Q}_{p}$ is that we do not have an “base” like $\mathbb{F}_{q}$ was for the function field case (unless perhaps if we have something like an appropriate version of the titular object in The Field with One Element, which is at the moment unavailable). The solution to this is provided by the theory of adic spaces and perfectoid spaces (see also Adic Spaces and Perfectoid Spaces).

For motivation, let us consider first the case where our field is $\mathbb{F}_{q}((t))$. Let $S=\mathrm{Spa}(R,R^{+})$ be a perfectoid space over $\overline{\mathbb{F}}_{q}$ with pseudouniformizer $\varpi$. Consider the product $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$. We may look at this as the punctured open unit disc over $S$. It sits inside $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$ as the locus where the pseudo-uniformizer $\pi$ of $R$ and the uniformizer $t$ of $\mathbb{F}_{q}[[t]]$ is invertible (or “nonzero”).

In the case where our field is $E$, a finite extension of $\mathbb{Q}_{p}$, as mentioned earlier we have no “base” like $\mathbb{F}_{q}$ was for $\mathbb{F}_{q}((t))$. So we cannot form the fiber products analogous to $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$ or $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$. However, notice that

$\displaystyle \mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])\cong \mathrm{Spa}(R^{+}[[t]])$.

This has an analogue in the mixed-characteristic, given by the theory of Witt vectors (compare, for instance $\mathbb{F}_{p}[[t]]$ and its “mixed-characteristic analogue” $\mathbb{Z}_{p}=W(\mathbb{F}_{p})$)! If $\kappa$ is the residue field of $\mathcal{O}_{E}$, we define the ramified Witt vectors $W_{\mathcal{O}_{E}}(R^{+})$ to be $W(R^{+})\otimes_{W(\kappa)}\mathcal{O}_{E})$. This is the analogue of $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$. Now all we have to do to find the analogue of $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$ that we are looking for is to define it as the locus in $W_{\mathcal{O}_{E}}(R^{+})$ where both the uniformizer $\varpi$ of $R^{+}$ and the uniformizer $\pi$ of $\mathcal{O}_{E}$ are invertible!

We denote this locus by $Y_{S}$. But recall again our discussion earlier, that due to the local Langlands correspondence being phrased in terms of the Weil group, we have to quotient out by the powers of Frobenius. Therefore we define the Fargues-Fontaine curve $X_{S}$ to be $Y_{S}/\mathrm{Frob}^{\mathbb{Z}}$.

Aside from our purpose of geometrizing the local Langlands correspondence, the Fargues-Fontaine curve $X_{S}$ is in itself a very interesting mathematical object. For instance, when $S$ is a complete algebraically closed nonarchimedean field over $\mathbb{F}_{q}$, the classical points of $X_{S}$ (i.e. maximal ideals of the rings $B$ such that $X_{S}$ is locally $\mathrm{Spa}(B,B^{+})$) correspond to untilts of $S$ (modulo the action of Frobenius)!

There is also a similar notion for more general $S$. To explain this we need the concept of diamonds, which will also be very important for the rest of the post. A diamond is a pro-etale sheaf on the category of perfectoid spaces over $\mathbb{F}_{p}$, which is the quotient of some perfectoid space $X$ over $\mathrm{Spa}(\mathbb{F}_{p})$ by a pro-etale equivalence relation $R \subset X\times X$ (we also say that the diamond is a coequalizer). An example of a diamond is given by $\mathrm{Spd}(\mathbb{Q}_{p})$. Note that $\mathbb{Q}_{p}$ is not perfectoid, but is the quotient of a perfectoid field we denoted $\mathbb{Q}_{p}^{\mathrm{cycl}}$ in Adic Spaces and Perfectoid Spaces by the action of $\mathbb{Z}_{p}^{\times}$. Now we can take the tilt $(\mathbb{Q}_{p}^{\mathrm{cycl}})^{\flat}$ and quotient out by $\underline{\mathbb{Z}_{p}}^{\times}$ (the underline notation will be explained later – for now we think of this as making the group $\mathbb{Z}_{p}^{\times}$ into a perfectoid space) – this is the diamond $\mathrm{Spd}(\mathbb{Q}_{p})$. More generally, if $X$ is an adic space over $\mathrm{Spa}(\mathbb{Z}_{p})$ satisfying certain conditions (“analytic”), we can define the diamond $X^{\diamond}$ to be such that $X^{\diamond}(S)$, for $S$ a perfectoid space over $\mathrm{Spa}(\mathbb{F}_{p})$, is the set of isomorphism classes of pairs $(S^{\#},S^{\#}\to X)$, $S^{\#}$ being the untilt of $S$. If $X=\mathrm{Spa}(R,R^{+})$, we also use $\mathrm{Spd}(R)$ to denote $X^{\diamond}$. Note that if $X$ is already perfectoid, $X^{\diamond}$ is just the same thing as the tilt $X^{\flat}$.

Now recall that $Y_{S}$ was defined to be the locus in $W_{\mathcal{O}_{E}}(S)$ where the uniformizer $\varpi$ of $S$ and the uniformizer $\pi$ of $E$ were invertible. We actually have that $Y_{S}^{\diamond}=S\times \mathrm{Spd}(E)$, and, for the Fargues-Fontaine curve $X_{S}$, we have that $X_{S}^{\diamond}=S\times \mathrm{Spd}(E)/(\mathrm{Frob}^{\mathbb{Z}}\times\mathrm{id})$.

Our generalization of the statement that the points of $X_{S}$ parametrize untilts of $S$ is now as follows. There exists a three-way bijection between sections of the map $Y^{\diamond}\to S$, maps $S\to\mathrm{Spd}(E)$, and untilts $S^{\#}$ over $E$ of $S$. Given such an untilt $S^{\#}$, this defines a closed Cartier divisor on $Y_{S}$, which in turn gives rise to a closed Cartier divisor on $X_{S}$. By the bijection mentioned earlier, these closed Cartier divisors on $X$ will be parametrized by maps $S\to \mathrm{Spd}(E)/\mathrm{Frob}^{\mathbb{Z}}$.

The closed Cartier divisors that arise in this way will be referred to as closed Cartier divisors of degree $1$. We have seen that they are parametrized by the following moduli space we denote by $\mathrm{Div}^{1}$ (this will also become important later on):

$\mathrm{Div}^{1}=\mathrm{Spd}(E)/\mathrm{Frob}^{\mathbb{Z}}$

Now that we have discussed the Fargues-Fontaine curve $X_{S}$ and some of its properties, we can define $\mathrm{Bun}_{G}$ as the stack that assigns to any perfectoid space $S$ over $\overline{\mathbb{F}}_{q}$ the groupoid of $G$-bundles on $X_{S}$.

When $G=\mathrm{GL}_{n}$, our $G$-bundles are just vector bundles. In this case we shall also denote $\mathrm{Bun}_{\mathrm{GL}_{n}}$ by $\mathrm{Bun}_{n}$.

### II. Vector Bundles on the Fargues-Fontaine Curve

Let us now try to understand a little bit more about vector bundles on the Fargues-Fontaine curve. They turn out to be related to another important thing in arithmetic geometry – isocrystals – and this will allow us to classify them completely.

Let $\breve{E}$ be the completion of the maximal unramified extension of $E$. Letting $\kappa$ denote the residue field of $\mathcal{O}_{E}$, $\breve{E}$ may also be expressed as the fraction field of $W(\kappa)$. It is equipped with a Frobenius lift $\mathrm{Frob}$. An isocrystal $V$ over $\breve{E}$ is defined to be a vector space over $\breve{E}$ equipped with a $\mathrm{Frob}$-semilinear automorphism.

Given an isocrystal $V$ over $\breve{E}$, we can obtain a vector bundle $\mathcal{E}$ on the Fargues-Fontaine curve $X_{S}$ by defining $\mathcal{E}=(V\times Y_{S})/\mathrm{Frob}^{\mathbb{Z}}$. It turns out all the vector bundles over $X_{S}$ can be obtained in this way!

Now the advantage of relating vector bundles on the Fargues-Fontaine curve to isocrystals is that isocrystals are completely classified via the Dieudonne-Manin classification. This says that the category of isocrystals over $\breve{E}$ is semi-simple (so every object is a direct sum of the simple objects), and the form of the simple objects are completely determined by two integers which are coprime, the rank (i.e. the dimension as an $\breve{F}$-vector space) $n$ which must be positive, and the degree (which determines the form of the $\mathrm{Frob}$-semilinear automorphism) $d$. Since these two integers are coprime and one is positive, there is really only one number that completely determines a simple $\breve{E}$-isocrystal – its slope, defined to be the rational number $d/r$. Therefore we shall also often denote a simple $\breve{E}$-isocrystal as $V(d/n)$. Since isocrystals over $\breve{E}$ and vector bundles on the Fargues-Fontaine curve $X_{S}$ are in bijection, if we have a simple $\breve{E}$-isocrystal $V(d/n)$ we shall denote the corresponding vector bundle by $\mathcal{E}(-d/n)$. More generally, an isocrystal is a direct sum of simple isocrystals and they can have different slopes. If an isocrystal only has one slope, we say that it is semistable (or basic). We use the same terminology for the corresponding vector bundle.

More generally, for more general reductive groups $G$, we have a notion of $G$-isocrystals; this can also be thought of functors from the category of representations of $G$ over $E$ to the category of isocrystals over $\breve{E}$. These are in correspondence with $G$-bundles over the Fargues-Fontaine curve. There is also a notion of semistable or basic for $G$-isocrystals, although its definition involves the Newton invariant (one of two important invariants of a $G$-isocrystal, the other being the Kottwitz invariant).

The set of $G$-isocrystals is denoted $B(G)$ and is also called the Kottwitz set. This set is in fact also in bijection with the equivalence classes in $G(\breve{E})$ under “Frobenius-twisted conjugacy”, i.e. the equivalence relation $g\sim \varphi(y)gy^{-1}$. Given an element $b$ of $B(G)$, we can define the algebraic group $G_{b}$ to be such that the elements of $G_{b}(F)$ are the elements $g$ of $G(\breve{F})$ satisfying the condition $\varphi(g)=bgb^{-1}$. If $b=1$, then $G_{b}=G$.

The groups $G_{b}$ are inner forms of $G$ (see also Reductive Groups Part II: Over More General Fields). More precisely, the $G_{b}$ are the extended pure inner forms of $G$, which are all the inner forms of $G$ if the center of $G$ is connected. Groups which are inner forms of each other are in some way closely related under the local Langlands correspondence – for instance, they have the same Langlands dual group. It has been proposed that these inner forms should really be studied “together” in some way, and we shall see that the use of $\mathrm{Bun}_{G}$ to formulate the local Langlands correspondence provides a realization of this approach.

Let us mention one more important part of arithmetic geometry that vector bundles on the Fargues-Fontaine curve are related to, namely p-divisible groups. A p-divisible group (also known as a Barsotti-Tate group) $G$ is an direct limit of group schemes

$\displaystyle G=\varinjlim_{n} G_{n}=(G_{1}\to G_{2}\to\ldots)$

such that $G_{n}$ is a finite flat commutative group scheme which is $p^{n}$-torsion of order $p^{nh}$ and such that the inclusion $G_{n}\to G_{n+1}$ induces an isomorphism of $G_{n}$ with $G_{n+1}[p^{n}]$ (the kernel of the multiplication by $p^{n}$ map in $G_{n+1}$). The number $h$ is called the height of the p-divisible group.

An example of a p-divisible group is given by $\mu_{\infty}=\varinjlim_{n} \mu_{p^{n}}$. This is a p-divisible group of height $1$. Given an abelian variety of dimension $g$, we can also form a p-divisible group of height $2g$ by taking the direct limit of its $p$-torsion.

We can also obtain p-divisible groups from formal group laws (see also The Lubin-Tate Formal Group Law) by taking the direct limit of its $p^{n}$-torsion. In this case we can then define the dimension of such a p-divisible group to be the dimension of the formal group law it was obtained from. More generally, for any p-divisible group over a complete Noetherian local ring of residue characteristic $p$, the connected component of its identity always comes from a formal group law in this way, and so we can define the dimension of the p-divisible group to be the dimension of this connected component.

Now it turns out p-divisible groups can also be classified by a single number, the slope, defined to be the dimension divided by the height. If the terminology appears suggestive of the classification of isocrystals and vector bundles on the Fargues-Fontaine curve, that’s because it is! Isocrystals (and therefore vector bundles on the Fargues-Fontaine curve) and p-divisible groups are in bijection with each other, at least in the case where the slope is between $0$ and $1$. This is quite important because the cohomology of deformation spaces of p-divisible groups (such as that obtained from the Lubin-Tate group law) have been used to prove the local Langlands correspondence before the work of Fargues and Scholze! We will be revisiting this later.

### III. The Geometry of $\mathrm{Bun}_{G}$

Let us now discuss more about the geometry of $\mathrm{Bun}_{n}$. It happens that $\mathrm{Bun}_{G}$ is a small v-sheaf. A v-sheaf is a sheaf on the category of perfectoid spaces over $\overline{\mathbb{F}}_{q}$ equipped with the v-topology, where the covers of $X$ are any maps $X_{i}\to X$ such that for any quasicompact $U\subset X$ there are finitely many $U_{i}$ which cover $U$. A v-sheaf is small if it admits a surjective map from a perfectoid space. In particular being a small v-sheaf implies that $\mathrm{Bun}_{G}$ has an underlying topological space $\vert \mathrm{Bun}_{G}\vert$. The points of this topological space are going to be in bijection with the elements of the Kottwitz set $B(G)$.

If $G$ is a locally profinite topological group, we define $\underline{G}$ to be the functor from perfectoid spaces over $\mathbb{F}_{q}$ which sends a perfectoid space $S$ over $\mathbb{F}_{q}$ to the set $\mathrm{Hom}_{\mathrm{top}}(\vert S\vert,\vert G\vert)$. We let $[\ast/\underline{G}]$ be the classifying stack of $G$-bundles; this means that we can obtain any $\underline{G}$-bundle on any perfectoid space $S$ over $\mathbb{F}_{q}$ by pulling back a universal $\underline{G}$-bundle on $[\ast/\underline{G}]$.

We write $\vert \mathrm{Bun}_{G}^{\mathrm{ss}}\vert$ for the locus in $\vert \mathrm{Bun}_{G}\vert$ corresponding to the $G$-isocrystals that are semistable. We let $\mathrm{Bun}_{G}^{ss}$ the substack of $\mathrm{Bun}_{G}$ whose underlying topological space is $\vert\mathrm{Bun}_{G}^{\mathrm{ss}}\vert$. It turns out that we have a decomposition

$\displaystyle \mathrm{Bun}_{G}^{\mathrm{ss}}\cong\coprod_{b\in B(G)_{\mathrm{basic}}}[\ast/\underline{G_{b}(E)}]$

More generally, even is $b$ is not basic, we have an inclusion

$\displaystyle j:[\ast/\underline{G_{b}(E)}]\hookrightarrow \mathrm{Bun}_{G}$

Let us now look at some more of the properties of $\mathrm{Bun}_{G}$. In particular, $\mathrm{Bun}_{G}$ satisfies the conditions for an analogue of an Artin stack (see also Algebraic Spaces and Stacks) but with locally spatial diamonds instead of algebraic spaces and schemes.

A diamond $X$ is called a spatial diamond if it is quasicompact quasiseparated, and its underlying topological space $\vert X\vert$ is generated by $\vert U\vert$, where $U$ runs over all sub-diamonds of $X$ which are quasicompact. A diamond is called a locally spatial diamond if it admits an open cover by spatial diamonds.

Now we recall from Algebraic Spaces and Stacks that to be an Artin stack, a stack must have a diagonal that is representable in algebraic spaces, and it has charts which are representable by schemes. It turns out $\mathrm{Bun}_{G}$ satisfies analogous properties – its diagonal is representable in locally spatial diamonds, and it has charts which are representable by locally spatial diamonds.

We can now define a derived category (see also Perverse Sheaves and the Geometric Satake Equivalence) of sheaves on the v-site of $\mathrm{Bun}_{G}$ with coefficients in some $\mathbb{Z}_{\ell}$-algebra $\Lambda$. If $\Lambda$ is torsion (e.g. $\mathbb{F}_{\ell}$ or $\mathbb{Z}/\ell^{n}\mathbb{Z}$), this can be the category $D_{\mathrm{et}}(\mathrm{Bun}_{G},\Lambda)$, which is the subcategory of $D(\mathrm{Bun}_{G,v},\Lambda)$ whose pullback to any strictly disconnected perfectoid space $S$ lands in $D(S_{\mathrm{et}},\Lambda)$ (here the subscripts $v$ and $\mathrm{et}$ denote the v-site and the etale site respectively). If $\Lambda$ is not torsion (e.g. $\mathbb{Z}_{\ell}$ or $\mathbb{Q}_{\ell}$) one needs the notion of solid modules (which was further developed in the work of Clausen and Scholze on condensed mathematics) to construct the right derived category.

If $X$ is a spatial diamond and $j:U\to X$ is a pro-etale map expressible as a limit of etale maps $j_{i}U_{i}\to X$, we can construct the sheaf $\widehat{\mathbb{Z}}[U]$ as the limit $\varprojlim_{i}j_{i!}\widehat{\mathbb{Z}}$. We say that a sheaf $\mathcal{F}$ on $X$ is solid if $\mathcal{F}(U)$ is isomorphic to $\mathrm{Hom}(\widehat{\mathbb{Z}}[U],\mathcal{F})$. We can extend this to small v-stacks – if $X$ is a small v-stack and $\mathrm{F}$ is a v-sheaf on $X$, we say that $\mathcal{F}$ is solid if for every map from a spatial diamond $Y$ to $X$ the pullback of $\mathcal{F}$ to $Y$ coincides with the pullback of a solid sheaf from the quasi-pro-etale site of $Y$. We denote by $D_{\blacksquare}(X,\widehat{\mathbb{Z}})$ the subcategory of $D(X_{v},\widehat{\mathbb{Z}})$ whose objects have cohomology sheaves which are solid. Now if we have a solid $\widehat{\mathbb{Z}}$-algebra $\Lambda$, we can consider $D(X_{v},\Lambda)$ inside $D(X_{v},\widehat{\mathbb{Z}})$, and we denote by $D_{\blacksquare}(X,\Lambda)$ the subcategory of objects of $D(X_{v},\Lambda)$ whose image in $D(X_{v},\widehat{\mathbb{Z}})$ is solid.

This category $D_{\blacksquare}(X,\Lambda)$ is still too big for our purposes. Therefore we cut out a subcategory $D_{\mathrm{lis}}(X,\widehat{\mathbb{Z}})$ as follows. If we have a map of v-stacks $f:X\to Y$, we have a pullback map $f^{*}:D_{\blacksquare}(Y,\Lambda)\to D_{\blacksquare}(X,\Lambda)$. This pullback map has a left-adjoint $f_{\natural}:D_{\blacksquare}(X,\Lambda)\to D_{\blacksquare}(Y,\Lambda)$. We define $D_{\mathrm{lis}}(X,\Lambda)$ to be the smallest triangulated subcategory stable under direct sums that contain $f_{\natural}\Lambda$, for all $f:X\to Y$ which are separated, representable by locally spatial diamonds, and $\ell$-cohomologically smooth. If $\Lambda$ is torsion, then $D_{\mathrm{lis}}(X,\Lambda)$ coincides with $D_{\mathrm{et}}(X,\Lambda)$.

Let $D(G_{b}(E),\Lambda)$ be the derived category of smooth representations of the group $G_{b}(E)$ over $\Lambda$. We have

$\displaystyle D_{\mathrm{lis}}(\ast/\underline{G_{b}(E)},\Lambda)\cong D(G_{b}(E),\Lambda)$

Now taking the pushforward of this derived category of sheaves through the inclusion $j$, and using the isomorphism above, we get

$\displaystyle j_{!}:D(G_{b}(E),\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$

Now we can see that this derived category $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ of sheaves on $\mathrm{Bun}_{G}$ encodes the representation theory of $G$, which is one side of the local Langlands correspondence, but more than that, it encodes the representation theory of all the extended pure inner forms of $G$ altogether.

The properties of $\mathrm{Bun}_{G}$ mentioned earlier, in particular its charts which are representable by locally spatial diamonds, allow us to define properties of objects in $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ which translate into properties of interest in $D(G_{b}(E),\Lambda)$. For example, we have a notion of $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ being compactly generated, and this translates into a notion of compactness for $D(G_{b}(E),\Lambda)$. We also have a notion of Bernstein-Zelevinsky duality for $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$, which translates into Bernstein-Zelevinsky duality for $D(G_{b}(E),\Lambda)$, and finally, we have a notion of universal local acyclicity in $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$, which translates into being admissible for $D(G_{b}(E),\Lambda)$.

### IV. The Hecke Correspondence and Excursion Operators

Now let us look at how the strategy in The Global Langlands Correspondence for Function Fields over a Finite Field works for our setup. We will be working in the “geometric” setting (i.e. sheaves or complexes of sheaves instead of functions) mentioned at the end of that post, so there will be some differences from the work of Lafforgue that we discussed there, although the motivations and main ideas (e.g. excursion operators) will be somewhat similar.

Just like in The Global Langlands Correspondence for Function Fields over a Finite Field, we will have a Hecke stack $\mathrm{Hck}_{G}$ that parametrizes modifications of $G$-bundles over the Fargues-Fontaine curve. This means that $\mathrm{Hck}_{G}(S)$ is the groupoid of triples $(\mathcal{E},\mathcal{E}',\phi)$ where $\mathcal{E}$ and $\mathcal{E}'$ are $G$-bundles over $X_{S}$ and $\phi_{D_{S}}:\mathcal{E}\vert_{X_{S}\setminus D_{S}}\xrightarrow{\sim}\mathcal{E}'\vert_{X_{S}\setminus D_{S}}$ is an isomorphism of vector bundles meromorphic on some degree $1$ Cartier divisor $D_{S}$ on $X_{S}$ (which is part of the data of the modification). Note that we have maps $h^{\leftarrow}:\mathrm{Hck}_{G}\to\mathrm{Bun}_{G}$ and $h^{\rightarrow}:\mathrm{Hck}_{G}\to\mathrm{Bun}_{G}\times\mathrm{Div}^{1}$ which sends the triple $(\mathcal{E},\mathcal{E}'\phi_{D_{S}})$ to $\mathcal{E}$ and $(\mathcal{E}',D_{S})$ respectively.

Now we need to bound the relative position of the modification. Recall that this is encoded via (conjugacy classes of) cocharacters $\mu:\mathbb{G}_{m}\to G$. The way this is done in this case is via the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}$, which parametrizes modifications of $G$-bundles on the completion of $X_{S}$ along $D_{S}$ (compare the moduli stacks denoted $\mathcal{M}_{I}$ inThe Global Langlands Correspondence for Function Fields over a Finite Field). The local Hecke stack admits a stratification into Schubert cells labeled by conjugacy classes of cocharacters $\mu:\mathbb{G}_{m}\to G$. We can now pull back a Schubert cell $\mathcal{H}\mathrm{ck}_{G,\mu}$ to the global Hecke stack $\mathrm{Hck}_{G}$ to get a substack $\mathrm{Hck}_{G,\mu}$ with maps $h^{\leftarrow,\mu}$ and $h^{\rightarrow,\mu}$, and define a Hecke operator as

$\displaystyle Rh_{*}^{\rightarrow,\mu}h^{\leftarrow,\mu *}:D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G}\times\mathrm{Div}^{1},\Lambda)$

More generally, to consider compositions of Hecke operators we need to consider modifications at multiple points. For this we will need the geometric Satake equivalence.

Let $S$ be an affinoid perfectoid space over $\mathbb{F}_{q}$. For each $i$ in some indexing set $I$, we let $D_{i}$ be a Cartier divisor on $X_{S}$. Let $B^{+}(S)$ be the completion of $\mathcal{O}_{X_{S}}$ along the union of the $D_{i}$, and let $B(S)$ be the localization of $B$ obtained by inverting the $D_{i}$. For our reductive group $G$, we define the positive loop group $LG^{+}$ to be the functor which sends an affinoid perfectoid space $S$ to $G(B^{+}(S))$, and we define the loop group $LG$ to be the functor which sends $S$ to $G(B(S))$.

We define the Beilinson-Drinfeld Grassmannian $\mathrm{Gr}_{G}^{I}$ to be the quotient $LG^{+}/LG$. We further define the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}^{I}$ to be the quotient $LG\backslash\mathrm{Gr}_{G}^{I}$.

The geometric Satake equivalence tells us that the category $\mathrm{Sat}_{G}^{I}(\Lambda)$ of perverse sheaves on $\mathcal{H}\mathrm{ck}_{G}^{I}$ satisfying certain conditions (quasicompact over $\mathrm{Div}^{1})^{I}$, flat over $\Lambda$, universally locally acyclic) is equivalent to the category of representations of $(\widehat{G}\rtimes W_{E})^{I}$ on finite projective $\Lambda$-modules.

Let $V$ be such a representation of representations of $(\widehat{G}\rtimes W_{E})^{I}$. Let $\mathcal{S}_{V}$ be the corresponding object of $\mathrm{Sat}_{G}^{I}(\Lambda)$. The global Hecke stack $\mathrm{Hck}_{G}^{I}$ has a map $q$ to the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}^{I}$. It also has maps $h^{\leftarrow}$ to $h^{\rightarrow}$ to $\mathrm{Bun}_{G}$ and $\mathrm{Bun}_{G}\times(\mathrm{Div}^{1})^{I}$ respectively. We can now define the Hecke operator $T_{V}$ as follows:

$\displaystyle T_{V}=Rh_{*}^{\rightarrow}(h^{\leftarrow *}\otimes_{\Lambda}^{\mathbb{L}}q^{*}\mathcal{S}_{V}):D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G}\times(\mathrm{Div}^{1})^{I},\Lambda)$

Once we have the Hecke operators, we can then consider excursion operators and apply the strategy of Lafforgue discussed in The Global Langlands Correspondence for Function Fields over a Finite Field. We set $\Lambda$ to be $\overline{\mathbb{Q}}_{\ell}$. Let $(I,V,\alpha,\beta,(\gamma_{i})_{i\in I})$ be an excursion datum, i.e. $I$ is a finite set, $V$ is a representation of $(\widehat{G}\rtimes Q)^{I}$, $\alpha:1\to V$, $\beta:V \to 1$, and $\gamma_{i}\in W_{E}$ for all $i\in I$. An excursion operator is the following composition:

$\displaystyle A=T_{1}(A)\xrightarrow{\alpha} T_{V}(A)\xrightarrow{(\gamma_{i})_{i\in I}} T_{V}(A)\xrightarrow{\beta} T_{1}(V)=A$

Now this composition turns out to be the same as multiplication by the scalar determined by the following composition:

$\displaystyle \overline{\mathbb{Q}}_{\ell}\to V\xrightarrow{\varphi(\gamma_{i})_{i\in I}} V\to \overline{\mathbb{Q}}_{\ell}$

And the $\varphi$ that appears here is precisely the L-parameter that we are looking for. This therefore gives us the “automorphic to Galois” direction of the local Langlands correspondence.

### V. Relation to Local Class Field Theory

It is interesting to look at how this all works in the case $G=\mathrm{GL}_{1}$, i.e. local class field theory. There is historical precedent for this in the work of Pierre Deligne for what we might now call the $\mathrm{GL}_{1}$ case of the (geometric) global Langlands correspondence for function fields over a finite field, but which might also be called geometric class field theory.

Let us go back to the setting in The Global Langlands Correspondence for Function Fields over a Finite Field, where we are working over a function field of some curve $X$ over the finite field $\mathbb{F}_{q}$. Since we are considering $G=\mathrm{GL}_{1}$, our $\mathrm{Bun}_{G}$ in this case will be the Picard group $\mathrm{Pic}_{X}$, which parametrizes line bundles on $X$. The statement of the geometric Langlands correspondence in this case is that there is an equivalence of character sheaves on $\mathrm{Pic}_{X}$ (see the discussion of Grothendieck’s sheaves to functions dictionary at the end of The Global Langlands Correspondence for Function Fields over a Finite Field) and $\overline{\mathbb{Z}}_{\ell}$-local systems of rank $1$ on $X$ (these are the same as one-dimensional representations of $\pi_{1}(X)$).

We have an Abel-Jacobi map $\mathrm{AJ}: X\to \mathrm{Pic}_{X}$, sending a point $x$ of $X$ to the corresponding divisor $x$ in $\mathrm{Pic}_{X}$. More generally we can define $\mathrm{AJ}^{d}:X^{(d)}\to\mathrm{Pic}_{X}^{d}$, where $X^{(d)}$ is the quotient of $X^{d}$ by the symmetric group on its factors, and $\mathrm{Pic}_{X}^{d}$ is the degree $d$ part of $\mathrm{Pic}_{X}$.

Now suppose we have a rank $1$ $\overline{\mathbb{Z}}_{\ell}$-local system on $X$, which we shall denote by $\mathcal{F}$. We can form a local system $\mathcal{F}^{\boxtimes d}$ on $X^{d}$. We can push this forward to $X^{(d)}$ and get a sheaf $\mathcal{F}^{(d)}$ on $X^{(d)}$. What we hope for is that this sheaf $\mathcal{F}^{(d)}$ is the pullback of the character sheaf on $\mathrm{Pic}_{X}^{d}$ that we are looking for via $\mathrm{AJ}^{(d)}$. This is in fact what happens, and what makes this possible is that the fibers of $\mathrm{AJ}^{(d)}$ are simply connected for $d>2g-2$, by the Riemann-Roch theorem. So for this $d$, by taking fundamental groups of the fiber sequence, we have that $\pi_{1}(X^{(d)})\cong\pi_{1}(\mathrm{Pic}_{X}^{d})$. So representations of $\pi_{1}(X^{(d)})$ give rise to representations of $\pi_{1}(\mathrm{Pic}_{X}^{d})$, and since representations of the fundamental group are the same as local systems, we see that there must be a local system on $\mathrm{Pic}_{X}^{d}$, and furthermore the sheaf $\mathcal{F}^{(d)}$ is the pullback of this local system. There is then an inductive method to extend this to $d\leq 2g-2$, and we can check that the local system is a character sheaf.

Now let us go back to our case of interest, the local Langlands correspondence. Instead of the curve $X$ we will use $\mathrm{Div}^{1}$, the moduli of degree $1$ Cartier divisors. It will be useful to have an alternate description of $\mathrm{Div}^{1}$ in terms of Banach-Colmez spaces.

For any perfectoid space $T$ over $S$ and any vector bundle $\mathcal{E}$ over $X_{S}$, the Banach-Colmez space $\mathcal{BC}(\mathcal{E})$ is the locally spatial diamond such that $\mathcal{BC}(\mathcal{E})(S)=H^{0}(X_{T},\mathcal{E}\vert_{X_{T}})$. We define $\mathcal{BC}(\mathcal{E})\setminus \lbrace 0\rbrace$ to be such that $\mathcal{BC}(\mathcal{E})\setminus \lbrace 0\rbrace (S)$ are the sections in $H^{0}(X_{T},\mathcal{E}\vert_{X_{T}})$ which are nonzero fiberwise on $S$.

There is a map from $\mathcal{BC}(\mathcal{O}(1))\setminus \lbrace 0\rbrace$ to $\mathrm{Div}^{1}$ which sends a section $f$ to $V(f)$, which in turn induces an isomorphism $(\mathcal{BC}(\mathcal{O}(1))\setminus \lbrace 0\rbrace)/\underline{E^{\times}}\cong \mathrm{Div}^{1}$. A more explicit description of this map is given by Lubin-Tate theory (see also The Lubin-Tate Formal Group Law). After choosing a coordinate, the Lubin-Tate formal group law $\mathcal{G}$ with an action of $\mathcal{O}_{E}$, over $\mathcal{O}_{E}$, is isomorphic to $\mathrm{Spf}(\mathcal{O}_{E}[[x]])$. We can form the universal cover $\widetilde{\mathcal{G}}$ which is isomorphic to $\mathrm{Spf}(\mathcal{O}_{E}[[x^{1/q^{\infty}}]])$. Now let $S=\mathrm{Spa}(R,R^{+})$ be a perfectoid space with tilt $S^{\#}=\mathrm{Spa}(R^{\#},R^{\#+})$. We have $\widetilde{\mathcal{G}}(R^{\#+})=R^{\circ\circ}$, where $R^{\circ\circ}$ is the set of topologically nilpotent elements in $R$, and the map which sends a topologically nilpotent element $x$ to the power series $\sum_{i}\pi^{i}[x^{q^{-i}}]$ gives a map to $H^{0}(Y_{S},\mathcal{O}(1))$, which upon quotienting out by the action of Frobenius gives an isomorphism between $\widetilde{\mathcal{G}}(R^{\#+})$ and $H^{0}(X_{S},\mathcal{O}(1))$.

What this tells us is that $H^{0}(X_{S},\mathcal{O}(1))\cong \mathrm{Spd}(\mathbb{F}[[x^{1/p^{\infty}}]])$. Defining $E_{\infty}$ to be the completion of the union over all $n$ of the $\pi^{n}$-torsion points of $\mathcal{G}$ in $\overline{E}$, we have that $H^{0}(X_{S},\mathcal{O}(1))\cong \mathrm{Spd}(E_{\infty})$. This is an $\underline{\mathcal{O}_{E}^{\times}}$-torsor over $\mathrm{Spd}(E)$, and then quotienting out by the action of Frobenius we obtain our map to $\mathrm{Div}^{1}$.

More generally, we have an isomorphism $(\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace)/\underline{E^{\times}}\cong \mathrm{Div}^{d}$, where $\mathrm{Div}^{d}$ parametrized degree $d$ relative Cartier divisors on $X_{S,E}$.

Now that we have our description of $\mathrm{Div}^{1}$ (and more generally $\mathrm{Div}^{d}$) in terms of Banach-Colmez spaces, let us now see how we can translate the strategy of Deligne to the local case. Once again we have an Abel-Jacobi map

$\displaystyle \mathrm{AJ}^{d}:\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace\to\mathrm{Pic}^{d}$

Given a local system on $\mathcal{BC}(\mathcal{O}(d))$, we want to have a character sheaf on $\mathrm{Pic}^{d}$ whose pullback to $\mathcal{BC}(\mathcal{O}(d))$ is precisely this local system. Again what our strategy hinges will be whether $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace$ will be simply connected. And in fact this is true for $d\geq 3$, and by a result called Drinfeld’s lemma for diamonds this will actually be enough to prove the local Langlands correspondence for $\mathrm{GL}_{1}$ (i.e. it is not needed for $d<3$ – in fact this is false for $d=1$!). The fact that $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace$ is simply connected for $d\geq 3$ is a result of Fargues, and, at least for the characteristic $p$ case, follows from expressing $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace=\mathrm{Spa}(\mathbb{F}_{q}[[x_{1}^{1/p^{\infty}},\ldots,x_{d}^{1/p^{\infty}}]])\setminus V(x_{1},\ldots x_{d})$, whose category of etale covers is the same as that of $\mathrm{Spa}(\mathbb{F}_{q}[[x_{1},\ldots,x_{d}]])\setminus V(x_{1},\ldots x_{d})$. Then Zariski-Nagata purity allows one to reduce this to showing that $\mathrm{Spa}(\mathbb{F}_{q}[[x_{1},\ldots,x_{d}]])$ is simply connected, which it is by Hensel’s lemma.

### VI. The Cohomology of Local Shimura Varieties

Many years before the work of Fargues and Scholze, the $\mathrm{GL}_{n}$ case of the local Langlands correspondence (see also The Local Langlands Correspondence for General Linear Groups) was originally proven using the cohomology of the Lubin-Tate tower (which we shall denote by $\mathcal{M}_{\infty}$) which parametrizes deformations of the Lubin-Tate formal group law (see also The Lubin-Tate Formal Group Law) with level structure, together with the cohomology of Shimura varieties. Let us now investigate how the cohomology of the Lubin-Tate tower can be related to what we have just discussed.

It turns out that because of the relationship between Lubin-Tate formal group laws, p-divisible groups, and vector bundles on the Fargues-Fontaine curve, the Lubin-Tate tower is also a moduli space of modifications of vector bundles on the Fargues-Fontaine curve, but of a very specific kind! Namely, it parametrizes modifications where we fix the two vector bundles, and furthermore one has to be the trivial bundle $\mathcal{O}^{n}$ and the other a degree $1$ bundle $\mathcal{O}(1/n)$, and so the only thing that varies is the isomorphism between them (as opposed to the Hecke stack, where the vector bundles can also vary) away from a point. So we see that the Lubin-Tate tower is a part of the Hecke stack (we can think of it as the fiber of the Hecke stack above $(\mathcal{E}_{1},\mathcal{E}_{b})\in \mathrm{Bun}_{G}\times\mathrm{Bun}_{G}$).

More generally, the Lubin-Tate tower is a special case of a local Shimura variety at infinite level, which is itself related to a special case of a moduli stack of local shtukas. These parametrize modifications of $G$-bundles $\mathcal{E}_{1}$ and $\mathcal{E}_{b}$, which are bounded by some cocharacter $\mu:\mathbb{G}_{m}\to G(E)$. This moduli stack of local shtukas, denoted $\mathrm{Sht}_{G,b,\mu,\infty}$, is an inverse limit of locally spatial diamonds $\mathrm{Sht}_{G,b,\mu,K}$ with “level structure” given by some compact open subgroup $K$ of $G(E)$. In the case where the cocharacter $\mu$ is miniscule, the data $(G,b,\mu)$ is called a local Shimura datum, and we define the local Shimura variety at infinite level, denoted $\mathcal{M}_{G,b,\mu,\infty}$, to be such that $\mathrm{Sht}_{G,b,\mu,\infty}=\mathcal{M}_{G,b,\mu,\infty}^{\diamond}$. It is similarly a limit of local Shimura varieties at finite level $K$, denoted $\mathcal{M}_{G,b,\mu,K}$, and for each $K$ we have $\mathrm{Sht}_{G,b,\mu,K}=\mathcal{M}_{G,b,\mu,K}^{\diamond}$.

Let us now see how the cohomology of the moduli stack of local shtukas is related to our setup. We will consider the case of finite level, i.e. $\mathrm{Sht}_{G,b,\mu,K}$, since the cohomology at infinite level may be obtained as a limit. Consider the inclusion $j_{1}:[\ast/\underline{G(E)}]\hookrightarrow \mathrm{Bun}_{G}$. Now consider the object $A=j_{1!}\mathrm{c-ind}_{K}^{G(E)}\mathbb{Z}_{\ell}$ of $D_{\mathrm{lis}}(\mathrm{Bun_{G}},\mathbb{Z}_{\ell})$. Now for our cocharacter $\mu:\mathbb{G}_{m}\to G(E)$, we have a Hecke operator $T_{\mu}$, and we apply this Hecke operator to obtain $T_{\mu}(A)$. Now we pull this back through the inclusion $j_{b}:[\ast/\underline{G_{b}(E)}]\hookrightarrow \mathrm{Bun}_{G}$, to get an object $j_{b}^{*}T_{\mu}(A)$ of $D_{\mathrm{lis}}(\ast/\underline{G_{b}(E)},\mathbb{Z}_{\ell})$. We can think of all this happening not on the entire Hecke stack, but only on $\mathrm{Sht}_{G,b,\mu,K}$, since we are specifically only considering this very special kind of modification parametrized by $\mathrm{Sht}_{G,b,\mu,K}$. But the derived pushforward from $D_{\mathrm{lis}}(\mathrm{Sht}_{G,b,\mu,K})$ to a point gives $R\Gamma(\mathrm{Sht}_{G,b,\mu,K},\mathbb{Z}_{\ell})$ (from which we can compute the cohomology).

This relationship between the cohomology of the moduli stack of local shtukas and sheaves on $\mathrm{Bun}_{G}$, as we have just discussed, has been used to obtain new results. For instance, David Hansen, Tasho Kaletha, and Jared Weinstein used this formulation together with the concept of the categorical trace to prove the Kottwitz conjecture.

Let $\rho$ be a smooth irreducible representation of $G_{b}(E)$ over $\overline{\mathbb{Q}}_{\ell}$. We define

$\displaystyle R\Gamma(G,b,\mu)[\rho]=\varinjlim_{K\subset G(E)}R\mathrm{Hom}(R\Gamma_{c}(\mathrm{Sht}_{G,b,\mu,K},\mathcal{S}_{\mu}),\rho)$

Let $S_{\varphi}$ be the centralizer of $\varphi$ in $\widehat{G}$. Given a representation $\pi$ in the L-packet $\Pi_{\varphi}(G)$ and a representation $\rho$ in the L-packet $\Pi_{\varphi}(G_{b})$, the refined local Langlands correspondence gives us a representation $\delta_{\pi,\rho}$ of $S_{\varphi}$. We let $r_{\mu}$ be the extension of the highest-weight representation of $\widehat{G}$ to ${}^{L}G$. The Kottwitz conjecture states that

$\displaystyle R\Gamma(G,b,\mu)[\rho]=\sum_{\pi\in\Pi_{\varphi}(G)}\pi\boxtimes\mathrm{Hom}_{S_{\varphi}}(\delta_{\pi,\rho},r_{\mu}\circ \varphi)$

The approach of Hansen, Kaletha, and Weinstein involve first using a generalized Jacquet-Langlands transfer operator $T_{b,\mu}^{G\to G_{b}}$. We define the regular semisimple elements in $G$ to be the semisimple elements whose connected centralizer is a maximal torus, and we define the strongly regular semisimple elements to be the regular semisimple elements whose centralizer is connected. We denote their corresponding open subvarieties in $G$ by $G_{\mathrm{rs}}$ and $G_{\mathrm{rs}}$ respectively. The generalized Jacquet-Langlands transfer operator $T_{b,\mu}^{G\to G_{b}}: C(G(E)_{\mathrm{sr}}\sslash G(E))\to C(G_{b}(E)_{\mathrm{sr}}\sslash G(E))$ is defined to be

$\displaystyle [T_{b,\mu}^{G\to G_{b}}f](g')=\sum_{(g,g',\lambda)\in\mathrm{Rel}_{b}}f(g)\dim r_{\mu}[\lambda]$

Here the set $\mathrm{Rel_{b}}$ is the set of all triples $(g,g',\lambda)$ where $g\in G(E)$, $g'\in G_{b}(E)$, and $\lambda$ is a certain specially defined element of $X_{*}(T)$ ($T$ being the centralizer of $g$ in $G$) that depends on $g$ and $g'$. When applied to the Harish-Chandra character $\Theta_{\rho}$, we have

$\displaystyle [T_{b,\mu}^{G\to G_{b}}\Theta_{\rho}](g)=\sum_{\pi\in\Pi_{\varphi}(G)}\dim \mathrm{Hom}_{S_{\varphi}}(\delta_{\pi,\rho},r_{\mu})\Theta_{\pi}(g)$

Next we have to relate this to the cohomology of the moduli stack of local shtukas. We first need the language of distributions. We define

$\mathrm{Dist}(G(E),\Lambda)^{G(E)}:=\mathrm{Hom}_{G(F)}(C_{c}(G(E),\Lambda)\otimes \mathrm{Haar}(G,\Lambda),\Lambda$

To any object $A$ of $D(G(E),\Lambda)$, we can associate an object $\mathrm{tr.dist}(A)$ of $\mathrm{Dist}(G(E),\Lambda)^{G(E)}$. We also have “elliptic” versions of these constructions, i.e. an object $\mathrm{tr.dist}_{\mathrm{ell}}(A)$ of the category $\mathrm{Dist}_{\mathrm{ell}}(G(E),\Lambda)^{G(E)}$. Now we can define the action of the generalized Jacquet-Langlands transfer operator on $\mathrm{Dist}_{\mathrm{ell}}(G(E),\Lambda)^{G(E)}$. The hope will be that we will have the following equality:

$\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho=\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]$

Proving this equality is where the geometry of $\mathrm{Bun}_{G}$ (and the Hecke stack) and the trace formula come into play. The action of the generalized Jacquet-Langlands transfer operator $\displaystyle T_{b,\mu}^{G\to G_{b}}$ on $\mathrm{tr.dist}_{\mathrm{ell}}\rho$ can be described in a similar way to a Hecke operator where we pull back to the moduli of local Shtukas, multiply by a kernel function, and then push forward.

On the other side, one needs to compute $\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]$. Here we use that $R\Gamma(G,b,\mu)[\rho]=h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$. This is a version of the expression of the cohomology of the moduli stack of local shtukas that we previously discussed where $q^{*}\mathcal{S}_{\mu}$ is the pullback to the Hecke stack of the sheaf corresponding to $\mu$ provided by the geometric Satake equivalence and before pushing forward via $h_{\rightarrow}$ we are pulling back to the degree $1$ part of the Hecke stack, which is why we have $j^{*}$ (the embedding of this degree $1$ part) and $h_{\rightarrow}'$ denotes that we are pushing forward from this degree $1$ part.

Hansen, Kaletha, and Weinstein then apply a categorical version of the Lefschetz-Verdier trace formula (using a framework developed by Qing Lu and Weizhe Zheng) to be able to relate $\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]=\mathrm{tr.dist}_{\mathrm{ell}}h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$ to $\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho$.

Let us discuss briefly the setting of this categorical trace. We consider a category $\mathrm{CoCorr}$ whose objects are pairs $(X,A)$ where $X$ is an Artin v-stack over $\ast$ and $A\in D_{et}(X,\Lambda)$. The morphisms in this category are given by a pair of maps $c_{1},c_{2}:C\to X$ where $c_{2}$ is smooth-locally representable in diamonds, together with a map $u:c_{1}^{*}A\to c_{2}^{!}A$. We also write $c$ for the pair $(c_{1},c_{2})$. Given an endomorphism $f:(X,A)\to (X,A)$ the categorical trace of $f$ is given by $(\mathrm{Fix}(c),\omega)$ where $\mathrm{Fix}(c)$ is the pullback of $c:C\to X\times X$ and $\Delta_{X}: X\to X\times X$ and $\omega\in H^{0}(\mathrm{Fix}(c),K_{X})$ (here $K_{X}$ is the dualizing sheaf, which may obtained as the right-derived pullback of $\Lambda$ via the structure morphism of $X$). In the special case where the correspondence $c$ arises form an automorphism $g$ of $X$, and $g^{*}A=A$, then one may think of $\mathrm{Fix}(c)$ as the fixed points of $g$ and the categorical trace gives an element of $\Lambda$ (the local term) for each fixed point.

For Hansen, Kaletha, and Weinstein’s application, they consider $f$ to be the identity. The categorical trace is then given by $(\mathrm{In}(X),\mathrm{cc}_{X}(A))$, where $\mathrm{In}(X)=X\times_{X\times X}X$ is the inertia stack, classifying pairs $(x,g)$ with $g$ an automorphism of $x$, and $\mathrm{cc}_{X}(A)\in H^{0}(\mathrm{In}(X),K_{\mathrm{In}(X)})$ is called the characteristic class.

The idea now is that certain properties of the setting we are considering (such as universal local acyclicity) allow us to identify the trace distribution $\mathrm{tr.dist}_{\mathrm{ell}}h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$ as a characteristic class $\mathrm{cc}_{\mathrm{Bun}_{G}^{1}}(h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho))$. From there we can use properties of the abstract theory to relate it to $\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho$ (for instance, we can use a Kunneth formula for the characteristic class to decouple the parts involving $\rho$ and $\mathcal{S}_{\mu}$, and relate the former to pulling back to the moduli stack of local shtukas, and relate the part involving the latter to multiplication by the kernel function).

### VII. The Spectral Action

We have seen that the machinery of excursion operators gives us the automorphic to Galois direction of the local Langlands correspondence. We now describe one possible approach to obtain the other (Galois to automorphic) direction. We are going to use the language of the categorical geometric Langlands correspondence mentioned at the end of in The Global Langlands Correspondence for Function Fields over a Finite Field.

Recall our construction of the moduli stack of local $\ell$-adic Galois representations in Moduli Stacks of Galois Representations. Using the same strategy we can construct a moduli stack of L-parameters, which we shall denote by $Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G}$. This notation comes from the fact that in Fargues and Scholze’s work the L-parameters can be viewed as 1-cocycles.

Let $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ denote the subcategory of compact objects in $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$. The categorical local Langlands correspondence in this case is the following conjectural equivalence of categories:

$\displaystyle D(\mathrm{Bun}_{G},\Lambda)^{\omega}\cong D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$

Here the right-hand side is the derived category of bounded complexes on $Z^{1}(W_{E},\widehat{G}$ with quasicompact support, coherent cohomology, and nilpotent singular support. We will leave the definition of these terms to the references, but we will think of $D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ as being a derived category of coherent sheaves on $Z^{1}(W_{E},\widehat{G})$.

We now outline an approach to proving the categorical local Langlands correspondence. Let $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ be the category of perfect complexes on $Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G}$. Then there is an action of $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ on $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, called the spectral action, such that composing with the map $\mathrm{Rep}(\widehat{G})^{I}\to \mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})^{BW^{I}}$ gives us the action of the Hecke operator.

The idea is that the spectral action gives us a functor from $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, sending an object $M$ of $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to the object $M\ast \mathcal{W}_{\psi}$ of $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, where $\mathcal{W}_{\psi}$ is the Whittaker sheaf (the sheaf on $\mathrm{Bun}_{G}$ corresponding to the representation $\mathrm{c-Ind}_{U(F)}^{B(F)}\psi$, where $B$ is a Borel subgroup of $G$, $U$ is the unipotent radical of $B$, and $\psi$ is a character of $U$). The hope is then that this functor can be extended from $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to all of $D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$, and that it will provide the desired equivalence of categories.

Now we discuss how this spectral action is constructed. Let us first consider the following more general situation. Let $L$ be a field of characteristic $0$, let $H$ be a split reductive group, and let $W$ be a discrete group. We write $BH$ and $BW$ for their corresponding classifying spaces. Let $\mathcal{C}$ be an idempotent-complete, $L$-linear stable $\infty$-category.

For all $I$, a $W$-equivariant, exact tensor action of $\mathrm{Rep}(H)$ on $\mathcal{C}$ is a functor

$\displaystyle \mathrm{Rep}(H^{I})\times \mathcal{C}\to\mathcal{C}^{BW^{I}}$

natural in $I$, exact as an action of $\mathrm{Rep}(H)$ after forgetting the $BW^{I}$-equivariance, and such that the action of $BW^{I}$ is compatible with the tensor product.

Now what we want to show is that a $W$-equivariant, exact tensor action of $\mathrm{Rep}(H)$ on $\mathcal{C}$ is the same as an $L$-linear action of $\mathrm{Perf}(\mathrm{Maps}(BW,BH))$ on $\mathcal{C}$.

To prove the above statement, Fargues and Scholze use the language of higher category theory. Let $\mathrm{An}$ be the $\infty$-category of anima, which is obtained from simplicial sets by inverting weak equivalences. The specific anima that we are interested in is $BW$, which is obtained by taking the nerve of the category $[\ast/W]$. An important property of $\mathrm{An}$ is that it is freely generated under sifted colimits by the full subcategory of finite sets.

We now define two functors $F_{1}$ and $F_{2}$ from $\mathrm{An}^{\mathrm{op}}$ to $\mathrm{An}$. The functor $F_{1}$ sends a finite set $S$ to the exact $L$-linear actions of $\mathrm{Perf}(\mathrm{Maps}(S,BH))$ on $\mathcal{C}$, which is equivalent to the exact $L$-linear monoidal functors from $\mathrm{Perf}(\mathrm{Maps}(S,BH))$ to $\mathrm{End}(\mathcal{C})$. The functor $F_{2}$ sends a finite set $S$ to the $S$-equivariant exact actions of $\mathrm{Rep}(H)$ on $\mathcal{C}$, which is equivalent to natural transformations from the functor $I\mapsto\mathrm{Hom}(S,I)$ to the functor $I\mapsto\mathrm{Fun}(\mathrm{Rep}(H^{I}),\mathrm{End}(\mathcal{C}))$.

There is a natural transformation from $F_{1}$ to $F_{2}$ that happens to be an isomorphism on finite sets. Now since the category $\mathrm{An}$ is generated by finite sets under sifted colimits, all we need is for the functors $F_{1}$ and $F_{2}$ to preserve sifted colimits.

For $F_{2}$ this follows from the fact that $S\mapsto S^{I}$ preserves sifted colimits. For $F_{1}$, this comes from the fact that $\mathrm{Maps}(S,BH)\cong [\mathrm{Spec}(A)/H^{S'}]$ for some animated $L$-algebra $A$ and some set $S'$, and then looking at the structure of $\mathrm{Perf}([\mathrm{Spec}(A)/H^{S'}])$ and $\mathrm{IndPerf}([\mathrm{Spec}(A)/H^{S'}])$.

Now that we have our abstract theory let us go back to our intended application. Let $W_{E}$ be the Weil group of $F$. It turns out that every L-parameter $\varphi:W_{E}\to \widehat{G}$ factors through a quotient $W_{E}/P$, where $P$ is some open subgroup of the wild inertia. This means that $Z^{1}(W_{E},\widehat{G})$ is the union of all $Z^{1}(W_{E}/P,\widehat{G})$ over all such $P$ (compare also with the construction in Moduli Stacks of Galois Representations), and this also means that we can focus our attention on $Z^{1}(W_{E}/P,\widehat{G})$.

We can actually go further and replace $W_{E}/P$ with its subgroup $W$ generated by the elements $\sigma$ and $\tau$ satisfying $\sigma\tau\sigma^{-1}=\tau^{q}$, together with the wild inertia (we have also already considered this in Moduli Stacks of Galois Representations, where we called it $\mathrm{WD}/Q$), and get the same moduli space, i.e. $Z^{1}(W_{E}/P,\widehat{G})\cong Z^{1}(W,\widehat{G})$.

Let $F_{n}$ be the free group on $n$ generators. For every map $F_{n}\to W$, we have a map

$\displaystyle Z^{1}(W,\widehat{G})\to Z^{1}(F_{n},\widehat{G})$

The category $\lbrace (n,F_{n})\rbrace$ is a sifted category, and upon taking sifted colimits, we obtain an isomorphism

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$

There is also a version of this statement that involves higher category theory. It says that the map

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$

is an isomorphism in the stable $\infty$-category $\mathrm{IndPerf}(B\widehat{G})$. Furthermore the category $\mathrm{Perf}(B\widehat{G})$ generates $\mathrm{Perf}(Z^{1}(W_{E}/P,\widehat{G})/\widehat{G})$ under cones and retracts, and $\mathrm{IndPerf}(Z^{1}(W_{E}/P,\widehat{G})/\widehat{G})$ identifies with the $\infty$-category of $\mathcal{O}(Z^{1}(W_{E}/P, \widehat{G})$-modules inside $\mathrm{IndPerf}(B\widehat{G})$.

If we take invariants under the action of $\widehat{G}$, we then have

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))^{\widehat{G}}\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))^{\widehat{G}}$

Note that $\mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))$ is precisely the same data as the algebra of excursion operators. We can see this using the fact that $(Z^{1}(F_{n},\widehat{G}))$ is isomorphic to $\widehat{G}^{n}$, and $\mathcal{O}(Z^{1}(F_{n},\widehat{G}))^{\widehat{G}}$ is functions on $\widehat{G}^{n}$ which are invariant under the action of $\widehat{G}$. But this is the same as the data of an excursion operator $(I,V,\alpha,\beta,(\gamma_{i})_{i\in I})$ ($I$ here has $n$ elements), because such a function is of the form $\langle \beta,\alpha((\gamma_{i})_{i\in I})\rangle$.

Now that we have our description of $\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$ as $\mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))$, we can now apply the abstract theory developed earlier to obtain our spectral action.

Let us now focus on the case of $G=\mathrm{GL}_{n}$ and relate the spectral action to the more classical language of Hecke eigensheaves (see also The Global Langlands Correspondence for Function Fields over a Finite Field). Let $L$ be an algebraically closed field over $\mathbb{Q}_{\ell}$. Given an L-parameter $\varphi:W_{E}\to\mathrm{GL}_{n}(L)$, we have an inclusion $i_{\varphi}:\mathrm{Spec}(L)\to Z^{1}(W_{E},\widehat{G})_{L}$ and a sheaf $i_{\varphi *}L$ on $Z^{1}(W_{E},\widehat{G})_{L}$. For any $A\in D(\mathrm{Bun}_{G},\Lambda)$ we can take the spectral action $i_{\varphi *}L \ast A$. This turns out to be a Hecke eigensheaf! However, it is often going to be zero. Still, in work by Johannes Anschütz and Arthur-César Le Bras, they show that the above construction can give an example of a nonzero Hecke eigensheaf, by relating the spectral action to an averaging functor, which is an idea that comes from the work of Edward Frenkel, Dennis Gaitsgory, and Kari Vilonen on the geometric Langlands program.

### VIII. The p-adic local Langlands correspondence

The work of Fargues and Scholze deals with the “classical” (i.e. $\ell\neq p$) local Langlands correspondence. As we have seen for example in Completed Cohomology and Local-Global Compatibility, the p-adic local Langlands correspondence (i.e. $\ell=p$) is much more complicated and mysterious compared to the classical case. Still, one might wonder whether the machinery we have discussed here can be suitably modified to obtain an analogous “geometrization” of the p-adic local Langlands correspondence.

Since we are dealing with what we might call p-adic, instead of $\ell$-adic, Galois representations, we would have to replace $Z^{1}(W_{E},\widehat{G})$ with the moduli stack of $(\varphi, \Gamma)$-modules (also known as the Emerton-Gee stack, see also Moduli Stacks of (phi, Gamma)-modules).

We still would like to work with the derived category of some sort of sheaves on $\mathrm{Bun}_{G}$. This is because, in work of Pierre Colmez, Gabriel Dospinescu, and Wieslawa Niziol (and also in related work of Peter Scholze which uses a different approach), the p-adic etale cohomology of the Lubin-Tate tower has been used to realize the p-adic local Langlands correspondence, and we have already seen that the Lubin-Tate tower is related to $\mathrm{Bun}_{G}$ and the Hecke stack. Since p-adic etale cohomology is the subject of p-adic Hodge theory (see also p-adic Hodge Theory: An Overview), we might also expect ideas from p-adic Hodge theory to become relevant.

So now have to find some sort of p-adic replacement for $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$. It is believed that the correct replacement might be the derived category of almost solid modules, whose theory is currently being developed by Lucas Mann. Some of the ideas are similar to that used by Peter Scholze to formulate p-adic Hodge theory for rigid-analytic varieties (see also Rigid Analytic Spaces), but also involves many new ideas. Let us go through each of the meanings of the words in turn.

The “almost” refers to theory of almost rings and almost modules developed by Gerd Faltings (see also the discussion at the end of Adic Spaces and Perfectoid Spaces). For an $R$-module $M$ over a local ring $R$, we say that $M$ is almost zero if it is annihilated by some element of the maximal ideal of $R$. We define the category of almost $R$-modules (or $R^{a}$-modules) to be the category of $R$-modules modulo the category of almost zero modules.

The “solid” refers to the theory of solid rings and solid modules discussed earlier, although we will use the later language developed by Dustin Clausen and Peter Scholze. Let $A$ be a ring. We define the category of condensed $A$-modules, denoted $\mathrm{Cond}(A)$, to be the category of sheaves of $A$-modules on the category of profinite sets. Given a profinite set $S=\varprojlim S_{i}$, we define $A_{\blacksquare}[S]$ to be the limit $\varinjlim_{A'}\varprojlim_{i}A'[S_{i}]$, where $A'$ runs over all finite-type $\mathbb{Z}$-algebras contained in $A$, and we define the category of solid $A$-modules, denoted $A_{\blacksquare}-\mathrm{Mod}$, to be the subcategory of $\mathrm{Cond}(A)$ generated by $A_{\blacksquare}[S]$. The idea of condensed mathematics is to incorporate topology – for instance the category of compactly generated weak Hausdorff spaces, which forms most of the topological spaces we care about, embeds fully faithfully into the category of condensed sets. On the other hand, condensed abelian groups, rings, modules, etc. have nice algebraic properties, for instance when it comes to forming abelian categories, which topological abelian groups, rings, modules, etc. do not have. The solid rings and solid modules corresponds to “completions”, and in particular they have a reasonable “completed tensor product” that will become useful to us later on when forming derived categories.

Finally, the “derived category” refers to the same idea of a category of complexes with morphisms up to homotopy and quasi-isomorphisms inverted, as we have previously discussed, except, however, that we need to actually not completely forget the homotopies; in fact we need to remember not only the homotopies but the “homotopies between homotopies”, and so on, and for this we need to formulate derived categories in the language of infinity category theory. The reason why we need to this is because our definition will involve “gluing” derived categories, and for this we need to remember the homotopies, including the higher ones.

Let us now look at how Mann constructs this derived category of almost solid modules. Let $\mathrm{Perfd}_{pi}^{\mathrm{aff}}$ be the category of affinoid perfectoid spaces $X=\mathrm{Spa}(A,A^{+})$ together with a pseudouniformizer $\pi$ of $A$. We define a functor $X\mapsto D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ as the sheafification of the functor $\mathrm{Spa}(A,A^{+})\to D_{\blacksquare}^{a}(A^{+}/\pi)$ (the derived category of almost solid $A^{+}/\pi$-modules) on $\mathrm{Perfd}_{\pi}^{\mathrm{aff}}$ equipped with the pro-etale topology.

If $X=\mathrm{Spa}(A,A^{+})$ is weakly perfectoid of finite type over some totally disconnected space, then $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ is just $D_{\blacksquare}^{a}(A^{+}/\pi)$. More generally, $X$ will gave a pro-etale cover by some $Y$ which is weakly perfectoid of finite type over some totally disconnected space, and $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ can be expressed as the limit $\varprojlim_{n} D_{\blacksquare}^{a}(B_{n}^{+}/\pi)$, where $Y_{n}=\mathrm{Spa}(B_{n},B_{n}^{+})$, and $Y_{n}$ runs over is the degree $n$ part of the Cech nerve of $Y$.

Now let $X$ be a small v-stack. There is a unique hypercomplete (this means it satisfies descent along all hypercovers, which are generalizations of the Cech nerve) sheaf on $X_{v}$ that agrees with the functor $Y\mapsto D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)$ for every affinoid perfectoid space $Y$ in $X_{v}$. We define $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+})$ to be the global sections of this sheaf. This is the construction that we want to apply to $X=\mathrm{Bun}_{G}$.

The derived category of almost solid modules comes with a six-functor formalism (see also Perverse Sheaves and the Geometric Satake Equivalence). Let $Y\to X$ be a map. The derived pullback $^{*}:D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)$ is the restriction map of the sheaf $D_{\blacksquare}^{a}$. The derived pushforward $^{*}:D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ is defined to be the right adjoint to the derived pullback. The derived tensor product $-\otimes-$ and derived Hom $\underline{\mathrm{Hom}}(-,-)$ are inherited from $D_{\blacksquare}^{a}(A^{+}/\pi)$.

The remaining two functors in the six-functor formalism are the “shriek” functors $f_{!}$ and $f^{!}$. If $f:Y\to X$ is a “nice” enough map, we have a factorization of $f$ into a composition $g\circ j$ where $j:Y\to Z$ is etale and $g:Z\to X$ is proper, and we define

$\displaystyle f_{!}:=g_{*}\circ j_{!}$

where $j_{!}$ is the right-adjoint to $j_{*}$. We then define $f^{!}:D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ to be the right-adjoint to $f_{!}$. The six-functor formalism satisfies certain important properties, such as functoriality of $f_{*}, f^{*}, f_{!}, f^{!}$, proper base change for $f_{!}$, and a projection formula for $f_{!}$. In Lucas Mann’s thesis, he uses the six-functor formalism he has developed to prove Poincare duality for a rigid-analytic variety $X$ of pure dimension $d$ over an algebraically closed nonarchimedean field $K$ of mixed characteristic:

$\displaystyle H_{et}^{i}(X,\mathbb{F}_{\ell})\otimes_{\mathbb{F}_{\ell}} H_{et}^{2d-i}(X,\mathbb{F}_{\ell})\to \mathbb{F}_{\ell}(-d)$

As of the moment, there are still many questions regarding a possible geometrization of the p-adic local Langlands program. As more developments are worked out, we hope to be able to discuss them in future posts on this blog, together with the different aspects of the theory that has already been developed, and the many other different future directions that it may lead to.

References:

Geometrization of the Local Langlands Correspondence by Laurent Fargues and Peter Scholze

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

The Geometric Langlands Conjecture (notes from Oberwolfach Arbeitsgemeinschaft)

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

Etale Cohomology of Diamonds by Peter Scholze

On the Kottwitz Conjecture for Local Shtuka Spaces by David Hansen, Tasho Kaletha, and Jared Weinstein

Averaging Functors in Fargues’ Program for GL_n by Johannes Anschütz and Arthur-César Le Bras

Cohomologue p-adique de la Tour de Drinfeld: le Cas de la Dimension 1 by Pierre Colmez, Gabriel Dospinescu, and Wiesława Nizioł

Lectures on Condensed Mathematics by Peter Scholze

# Perverse Sheaves and the Geometric Satake Equivalence

The idea behind “perverse sheaves” originally had its roots in the work of Mark Goresky and Robert MacPherson on “intersection homology”, but has since taken a life of its own after the foundational work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne and has found many applications in mathematics. In this post, we will describe what perverse sheaves are, and state an important result in representation theory called the geometric Satake equivalence, which makes use of this language.

A perverse sheaf is a certain object of the “derived category of sheaves with constructible cohomology”, satisfying certain conditions. This is quite a lot of new words, but we shall be defining them in this post, starting with “constructible”.

Let $X$ be an algebraic variety with a stratification, i.e. a decomposition

$\displaystyle X=\coprod_{\lambda\in\Lambda}X_{\lambda}$

of $X$ into a finite disjoint union of connected, locally closed, smooth subsets $X_{\lambda}$ called strata, such that the closure of any stratum is a union of strata.

A sheaf $\mathcal{F}$ on $X$ is constructible if its restriction $\mathcal{F}\vert_{X_{\lambda}}$ to any stratum $X_{\lambda}$ is locally constant (for every point $x$ of $X_{\lambda}$ there is some open set $V$ containing $x$ on which the restriction $\mathcal{F}\vert_{V}$ to $U$ is a constant sheaf). A locally constant sheaf which is finitely generated (its stalks are finitely generated modules over some ring of coefficients) is also called a local system. Local systems are quite important in arithmetic geometry – for instance, local sheaves on $X$ correspond to representations of the etale fundamental group $\pi_{1}(X)$. The character sheaves discussed at the end of The Global Langlands Correspondence for Function Fields over a Finite Field are also examples of local systems (in fact, perverse sheaves, which we shall define later in this post, can be viewed as a generalization of local systems and are also important in the geometric Langlands program).

Now let us describe roughly what a derived category is. Given an abelian category (for example the category of abelian groups, or sheaves of abelian groups on some space $X$) $A$, we can think of the derived category $D(A)$ as the category whose objects are the cochain complexes in $A$, but whose morphisms are not quite the morphisms of cochain complexes in $A$, but instead something “looser” that only reflects information about their cohomology.

Let us explain what we mean by this. Two morphisms between cochain complexes in $A$ may be “chain homotopic”, in which case they induce the same morphisms of the corresponding cohomology groups. Therefore, as an intermediate step in constructing the derived category $D(A)$, we first create a category $K(A)$ where the objects are the cochain complexes in $A$, but where the morphisms are the equivalence classes of morphisms of cochain complexes in $A$ where the equivalence relation is that of chain homotopy. The category $K(A)$ is called the homotopy category of cochain complexes (in $A$).

Finally, a morphism of chain complexes in $A$ is called a quasi-isomorphism if it induces an isomorphism of the corresponding cohomology groups. Therefore, since we want the morphisms of $D(A)$ to reflect the information about the cohomology, we want the quasi-isomorphisms of chain complexes in $A$ to actually become isomorphisms in the category $D(A)$. So as our final step, to obtain $D(A)$ from $K(A)$, we “formally invert” the quasi-isomorphisms.

We do not yet have everything we need to define what a perverse sheaf is, but we have mentioned previously that they are an object of the derived category of sheaves on an algebraic variety $X$ with constructible cohomology. We denote this latter category $D_{c}^{b}(X)$ (this is used if there is some stratification of $X$ for which we have this category; if the stratification $\Lambda$ is specified, we say $\Lambda$-constructible instead of constructible, and we denote the corresponding category by $D_{\Lambda}^{b}(X)$).

Let us say a few things about the category $D_{c}^{b}(X)$. Having “constructible cohomology” means that the cohomology sheaves of $D_{c}^{b}(X)$ are complexes of sheaves, we can take their cohomology, and this cohomology is valued in sheaves (these sheaves are what we call cohomology sheaves) which are constructible, i.e. on each stratum $X_{\lambda}$ they are local systems. The category $D_{c}^{b}(X)$ is also equipped with a very useful extra structure (which we will also later need to define perverse sheaves) called the six-functor formalism.

These six functors are $R\mathrm{Hom}$, $\otimes^{\mathbb{L}}$, $Rf_{*}$, $Rf^{*}$, $Rf_{!}$, and $Rf^{!}$, the first four being the derived functors corresponding to the usual operations of Hom, tensor product, pushforward, and pullback, respectively, and the last two are the derived “shriek” functors (see also The Hom and Tensor Functors and Direct Images and Inverse Images of Sheaves). The functor $\otimes^{\mathbb{L}}$ makes $D_{c}^{b}(X)$ into a symmetric monoidal category, and $R\mathrm{Hom}$ is its right adjoint. The functor $Rf_{*}$ is right adjoint to $Rf^{*}$, and similarly $Rf_{!}$ is right adjoint to $Rf^{!}$. In the case that $f$ is proper, $Rf_{!}$ is the same as $Rf_{*}$, and in the case that $f$ is etale, $Rf^{!}$ is the same as $Rf^{*}$. We note that it is quite common in the literature to omit the $R$ from the notation, and to let the reader infer that the functor is “derived” from the context (i.e. it is a functor between derived categories).

A derived category is but a specific instance of the even more abstract concept of a triangulated category, which we have defined already, together with the related concepts of a t-structure and the heart of a t-structure, in The Theory of Motives.

In fact we will need the concept of a t-structure to define perverse sheaves. Let us now define this t-structure on the derived category of constructible sheaves. Let $X=\coprod_{\lambda\in\Lambda} X_{\lambda}$ be an algebraic variety with its stratification, and for every stratum $X_{\lambda}$ let $d_{\lambda}$ denote its dimension. We write $D_{\mathrm{const}}^{b}$ for the subcategory of $D_{\Lambda}^{b}$ whose cohomology sheaves are locally constant, and for any object $\mathfrak{F}$ of some derived category we write $\mathcal{H}^{i}(\mathfrak{F})$ for its $i$-th cohomology sheaf. We define

$\displaystyle ^{p}D_{\lambda}^{\leq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i> d_{\lambda}\rbrace$

$\displaystyle ^{p}D_{\lambda}^{\geq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i< -d_{\lambda}\rbrace$

Now let $i_{\lambda}:X_{\lambda}\to X$ be the inclusion of a stratum $X_{\lambda}$ into $X$. We further define

$\displaystyle ^{p}D^{\leq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}^{b}:Ri_{\lambda}^{*}\frak{F}\in ^{p}D_{\lambda}^{\leq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

$\displaystyle ^{p}D^{\geq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}:Ri_{\lambda}^{!}\frak{F}\in ^{p}D_{\lambda}^{\geq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

This defines a t-structure, and we define the category of perverse sheaves on $X$, denoted $\mathrm{Perv}(X)$, as the heart of this t-structure.

With the definition of perverse sheaves in hand we can now state the geometric version of the Satake correspondence (see also The Unramified Local Langlands Correspondence and the Satake Isomorphism). Let $k$ be either $\mathbb{C}$ or $\mathbb{F}_{q}$, and let $K=k((t))$, and let $\mathcal{O}=k[[t]]$. Let $G$ be a reductive group. The loop group $LG$ is defined to be the scheme whose $k$-points are $G(K)$ and the positive loop group $L^{+}G$ is defined to be the scheme whose $k$-points are $G(\mathcal{O})$. The affine Grassmannian is then defined to be the quotient $LG/L^{+}(G)$.

The geometric Satake equivalence states that there is equivalence between the category of perverse sheaves $\mathrm{Perv}(\mathrm{Gr}_{G})$ on the affine Grassmannian $\mathrm{Gr}_{G}$ and the category $\mathrm{Rep}(^{L}G)$ of representations of the Langlands dual group $^{L}G$ of $G$. It was proven by Ivan Mirkovic and Kari Vilonen using the Tannakian formalism (see also The Theory of Motives) but we will not discuss the details of the proof further here, and leave it to the references or future posts.

As we have seen in The Global Langlands Correspondence for Function Fields over a Finite Field, the geometric Satake equivalence is important in being able to define the excursion operators in Vincent Lafforgue’s approach to the global Langlands correspondence for function fields over a finite field. It has (in possibly different variants) also found applications in other parts of arithmetic geometry, for example in certain approaches to the local Langlands correspondence, as well as the study of Shimura varieties. We shall discuss more in future posts on this blog.

References:

Perverse sheaf on Wikipedia

Constructible sheaf on Wikipedia

Derived category on Wikipedia

Satake isomorphism on Wikipedia

An illustrated guide to perverse sheaves by Geordie Williamson

Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson and Anna Romanov

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

Perverse sheaves and fundamental lemmas (notes by Chao Li from a course by Wei Zhang)

Perverse sheaves in representation theory (notes by Chao Li from a course by Carl Mautner)

Geometric Langlands duality and representations of algebraic groups over commutative rings by Ivan Mirkovic and Kari Vilonen

# Adic Spaces and Perfectoid Spaces

At the end of Formal Schemes we hinted at the concept of adic spaces, which subsumes both formal schemes and rigid analytic spaces (see also Rigid Analytic Spaces). In this post we will define what these are, give some examples, and introduce and discuss briefly a very special type of adic spaces, the perfectoid spaces, which generalizes what we discussed in Perfectoid Fields.

We begin by discussing the rings that we will need to construct adic spaces. A topological ring (see also Formal Schemes) $A$ is called a Huber ring if it contains an open subring $A_{0}$ which is adic with respect to a finitely generated ideal of definition $I$ contained in $A_{0}$. This means that the nonnegative powers of $I$ form a basis of open neighborhoods of $0$. The subring $A_{0}$ is called a ring of definition for $A$.

Here are some examples of Huber rings:

• Any ring $A$, equipped with the discrete topology, with the ring of definition $A_{0}=A$, and the ideal of definition $I=(0)$.
• The p-adic numbers $A=\mathbb{Q}_{p}$, with the p-adic topology, with the ring of definition $A_{0}=\mathbb{Z}_{p}$, and the ideal of definition $I=(p)$.
• The field of formal Laurent series $A=k((x))$ over some field $k$, with the metric topology given by the nonarchimedean valuation defined by the order of vanishing at $0$, with the ring of definition $A_{0}=k[[x]]$, and the ideal of definition $I=(x)$.
• Generalizing the previous two examples, any nonarchimedean field $K$ is an example of a Huber ring, with ring of definition $A_{0}=\lbrace x:\vert x\vert\leq 1\rbrace$ and ideal of definition $I=(\varpi)$ for some $\varpi$ satisfying $0<\vert\varpi\vert <1$.

A subset $S$ of a Huber ring, or more generally a topological ring, is called bounded if, for any open neighborhood $U$ of $0$, we can always find another open neighborhood $V$ of $0$ such that all the products of elements of $V$ with elements of $S$ are contained inside $U$. An element of a Huber ring is called power bounded if the set of all its nonnegative powers is bounded. For a Huber ring $A$ we denote the set of power bounded elements by $A^{\circ}$. Any element of the ring of definition will always be power bounded.

With the definition of power bounded elements in hand we give two more examples of Huber rings:

• Let $K$ be a nonarchimedean field as in the previous example, and let $\varpi$ again be an element such that $0<\vert\varpi\vert<1$. Its set of power bounded elements is given by $K^{\circ}=\lbrace x:\vert x\vert\leq 1\rbrace$. Now let $A=K^{\circ}[[T_{1},\ldots, T_{n}]]$ with the $I$-adic topology (see also Formal Schemes), where $I$ is the ideal $(\varpi, T_{1},\ldots,T_{n})$. Then $A$ is a Huber ring with ring of definition $A_{0}=A$ and ideal of definition $I$.
• Let $K$, $K^{\circ}$, and $\varpi$ be as above. Consider the Tate algebra $A= K\langle T_{1},\ldots,T_{n}\rangle$ (see also Rigid Analytic Spaces), a topological ring whose topology is generated by a basis of open neighborhoods of $0$ given by $\varpi^{n} A$. Then $A$ is a Huber ring with ring of definition given by $A_{0}=K^{\circ}\langle T_{1},\ldots,T_{n}\rangle$ and ideal of definition given by $(\varpi)$.

A subring $A^{+}$ of a Huber ring $A$ which is open, integrally closed, and power bounded is called a ring of integral elements. A Huber pair is a pair $(A,A^{+})$ consisting of a Huber ring $A$ and a ring of integral elements $A^{+}$ contained in $A$. Note that the set of power bounded elements, $A^{\circ}$, is itself an example of a ring of integral elements! In fact, in many examples that we will consider the relevant Huber pair will be of the form $(A,A^{\circ})$.

Now we introduce the adic spectrum of an Huber pair $(A,A^{+})$, denoted $\mathrm{Spa}(A,A^{+})$. They will form the basic building blocks of adic spaces, like affine schemes are to schemes or affinoid rigid analytic spaces are to rigid analytic spaces. We will proceed in the usual manner; first we define the underlying set, then we put a topology on it, and then construct a structure sheaf – except that in the case of adic spaces, what we will construct is merely a structure presheaf and may not always be a sheaf! Then we will define more general adic spaces to be something that locally looks like the adic spectrum of some Huber pair.

The underlying set of the adic spectrum $\mathrm{Spa}(A,A^{+})$ is the set of equivalence classes of continuous valuations $\vert\cdot\vert$ on $A$ such that $\vert a\vert\leq 1$ whenever $a$ is in $A^{+}$. From now on we will change our notation and let $x$ denote a continuous valuation, and we write $f$ for an element of $A$, so that we can write $\vert f(x)\vert$ instead of $\vert a\vert$, to drive home the idea that these (equivalence classes of) continuous valuations are the points of our space, on which elements of our ring $A$ are functions.

The underlying topological space of $\mathrm{Spa}(A,A^{+})$ is then obtained from the above set by equipping it with the topology generated by the subsets of the form

$\displaystyle \lbrace x: \vert f(x)\vert \leq \vert g(x)\vert \neq 0\rbrace$

for all $f,g\in A$.

Let us now define the structure presheaf. First let us define rational subsets. Let $T$ be a subset of $A$ such that the set consisting of all products of elements of $T$ with elements of $A$ is an open subset of $A$. We define the rational subset

$\displaystyle U\left(\frac{T}{s}\right):=\lbrace x:\vert t(x)\vert \leq\vert s(x)\vert\neq 0\rbrace$

for all $t\in T$. If $U$ is a rational subset of the Huber pair $(A,A^{+})$, then there is a Huber pair $(\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }(U),\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }^{+}(U))$ such that the map $\mathrm{Spa}(A,A^{+})\to \mathrm{Spa}(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$ factors through $U$ and this map is universal among such maps.

Now we define our structure presheaf by assigning to any open set $W$ the Huber pair $(\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W) , \mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W))$ where $\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W):=\varprojlim \mathcal{O}(U)$ where the limit is taken over all inclusions of rational subsets $U\subseteq W$, and $\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W)$ is similarly defined.

Again, the structure presheaf of $\mathrm{Spa}(A,A^{+})$ may not necessarily be a sheaf; in the case that it is, we say that the Huber pair $(A,A^{+})$ is sheafy. In this case we will also refer to $\mathrm{Spa}(A,A^{+})$ (the underlying topological space together with the structure sheaf) as an affinoid adic space. We can now define more generally an adic space as the data of a topological space $X$, a structure sheaf $\mathcal{O}_{X}$, and for each point $x$ of $X$, an equivalence class of continuous valuations on the stalk $\mathcal{O}_{X,x}$, such that it admits a covering of $U_{i}$‘s giving rise to the data of a structure sheaf and a collection of valuations, all of which is isomorphic to that given by an affinoid adic space.

Recall that we said above that the set of power-bounded elements, $A^{\circ}$, is an example of a ring of integral elements. Therefore $(A,A^{\circ})$ is an example of a Huber pair. It is convention that, if our Huber pair is given by $(A,A^{\circ})$ we write $\mathrm{Spa}(A)$ instead of $\mathrm{Spa}(A,A^{\circ})$. Let us now look at some examples of adic spaces.

Consider $\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ (which by the previous paragraph we may also simply write as $\mathrm{Spa}(\mathbb{Q}_{p})$, since $\mathbb{Z}_{p}$ is the set of power-bounded elements of $\mathbb{Q}_{p}$). Then the underlying topological space of $\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ consists of one point, corresponding to the usual p-adic valuation on $\mathbb{Q}_{p}$.

Next we consider $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ (which by the same idea as above we may write as $\mathrm{Spa}(\mathbb{Z}_{p}$). The underlying topological space of $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ now consists of two points, one of which is open, and one of which is closed. The open (or “generic”) point corresponds once again to the usual p-adic valuation $\mathbb{Q}_{p}$ restricted to $\mathbb{Z}_{p}$. The closed point is the valuation which sends any $\mathbb{Z}_{p}$ which contains a power of $p$ to $0$, and sends everything else to $1$.

More complicated is $\mathrm{Spa}(\mathbb{Q}_{p}\langle T\rangle,\mathbb{Z}_{p}\langle T\rangle)$, also known as the adic closed unit disc. We can compare this with the closed unit disc discussed in Rigid Analytic Spaces. In that post we the underlying set of the closed unit disc was given by the set of maximal ideals of $\mathbb{Q}_{p}\langle T\rangle$. But every such maximal ideal gives rise to a continuous valuation on $\mathbb{Q}_{p}\langle T\rangle$. So every point of the rigid analytic closed unit disc gives rise to a point of the adic closed unit disc. But the adic closed unit disc has more points!

An example of a point of the adic closed unit disc is as follows. Let $\Gamma$ be the ordered abelian group $\mathbb{R}_{>0}\times \gamma^{\mathbb{Z}}$, where $\gamma$ is such that $a<\gamma<1$ for all real numbers $a<1$ in this order. Define a continuous valuation $\vert\cdot\vert_{x^{-}}$ on $\mathbb{Q}_{p}\langle T\rangle$ as follows:

$\displaystyle \vert \sum_{n=0}a_{n}T^{n}\vert_{x^{-}}=\sup_{n\geq 0}\vert a_{n}\vert\gamma^{n}$

This valuation defines a point $x^{-}$ of the adic closed unit disc. This valuation sees $T$ as being infinitesimally less than $1$, i.e. $\vert T(x^{-})\vert=\vert T\vert_{x^{-}}<1$, but $\vert T(x^{-})\vert>a$ for all $a<1$ in $\mathbb{Q}_{p}$. This point $x^{-}$ serves a useful purpose. Recall in Rigid Analytic Spaces that we were unable to disconnect the closed unit disc into two open sets (the “interior” and the “boundary”) because of the Grothendieck topology. In this case we do not have a Grothendieck topology but an honest-to-goodness actual topology, but still we will not be able to disconnect the adic closed unit disc into the analogue of these open sets. This is because the disjoint union of the open sets $\cup_{n\geq 1}\vert T^{n}(x)\vert<\vert p\vert$ and $\vert T(x)\vert=1$ will not miss the point $x^{-}$, so just these two will not cover the adic closed unit disc.

Finally let us consider $\mathrm{Spa}(\mathbb{Z}_{p}[[ T]],\mathbb{Z}_{p}[[T]])$. This is the adic open unit disc. This has a map to $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$, and the preimage of the generic point of $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ is called the generic fiber (this generic fiber may also be thought of as the adic open unit disc over $\mathbb{Q}_{p}$, which makes it more comparable to the example of the adic closed unit disc earlier). The adic open unit disc has many interesting properties (for example it is useful to study in closer detail if one wants to study the fundamental curve of p-adic Hodge theory, also known as the Fargues-Fontaine curve) but we will leave this to future posts.

Let us now introduce a very special type of adic space. First we define a very special type of Huber ring. We say that a Huber ring $A$ is Tate if it contains a topologically nilpotent unit (also called a pseudo-uniformizer). An element $\varpi$ is topologically nilpotent if its sequence of powers $\varpi, \varpi^{2},\ldots$ converges to $0$. For example, the Huber ring $\mathbb{Q}_{p}$ (as discussed above) is Tate, with pseudo-uniformizer given by $p$.

If, in addition to being Tate, the Huber ring $A$ is complete, uniform (which means that $A^{\circ}$ is bounded in $A$), and contains a pseudo-uniformizer $\varpi$ such that $\varpi^{p}\vert p$ in $A^{\circ}$ and the p-th power map map $A/\varpi\to A/\varpi^{p}$ is an isomorphism, then we say that $A$ is perfectoid. As can be inferred from the name, this generalizes the perfectoid fields we introduced in Perfectoid Fields. We recall the important property of perfectoid fields (which we can now generalize to perfectoid rings) – if $R$ is perfectoid, then the category of finite etale $R$-algebras is equivalent to the category of finite etale $R^{\flat}$-algebras, where $R^{\flat}$ is the tilt of $R$. For fields, this manifests as an isomorphism of their absolute Galois groups, which generalizes the famous Fontaine-Wintenberger theorem.

A perfectoid space is an adic space which can be covered by affinoid adic spaces $\mathrm{Spa}(A,A^{+})$, where $A$ is perfectoid. If $X$ is a perfectoid space, we can associate to it its tilt $X^{\flat}$, by taking the tilts of the affinoid adic spaces that cover $X$ and gluing them together. In fact, for a fixed perfectoid space $X$, there is an equivalence of categories between perfectoid spaces over $X$, and perfectoid spaces over $X^{\flat}$. This is the geometric version of the equivalence of categories of finite etale algebras over a perfectoid ring and its tilt. In addition, although we will not do it in this post, one can define the etale sites of $X$ and $X^{\flat}$, and these will also be equivalent.

To end this post, we mention some properties of perfectoid spaces that make it useful form some applications. It turns out that if $X$ is a smooth rigid analytic space, it always has a pro-etale cover by affinoid perfectoid spaces. A pro-etale map $U\to X$ may be thought of as a completed inverse limit $\varprojlim_{i} U_{i}\to X$, where each $U_{i}\to X$ is an etale map. An example of a pro-etale cover is as follows. If we let $\mathbb{Q}_{p}^{\mathrm{cycl}}$ be the perfectoid field given by the completion of $\cup_{n}\mathbb{Q}_{p}(\mu_{p^{n}})$ (this is somewhat similar to the example involved in the Fontaine-Wintenberger theorem in Perfectoid Fields), then $\mathrm{Spa}(\mathbb{Q}_{p}^{\mathrm{cycl}})$ is a pro-etale cover of $\mathrm{Spa}(\mathbb{Q}_{p})$. To see why this is pro-etale, note that a finite separable extension of fields is etale, and $\mathbb{Q}_{p}^{\mathrm{cycl}}$ is the completion of the infinite union (direct limit) of such finite separable extensions $\mathbb{Q}_{p}(\mu_{p^{n}})$ of $\mathbb{Q}_{p}$, but looking at the adic spectrum means the arrows go the other way, which is why we think of it as an inverse limit.

Another property of perfectoid spaces is the following. If $U$ is a perfectoid affinoid space over $\mathbb{C}_{p}$, then for all $i>0$ $H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+})$ (this is the cohomology of the sheaf of functions bounded by $1$ on the etale site of $X$) is annihilated by the maximal ideal of $\mathcal{O}_{\mathbb{C}_{p}}$. We also say that $H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+})$ is almost zero.

Together, what these two properties tell us is that we can compute the cohomology of a smooth rigid analytic space via the Cech complex associated to its covering by perfectoid affinoid spaces. This has been applied in the work of Peter Scholze to the mod p cohomology of the rigid analytic space associated to a Siegel modular variety, in order to relate it to Siegel cusp forms (see also Siegel modular forms). In this case the covering by perfectoid affinoid spaces is provided by a Siegel modular variety at “infinite level”, which happens to have a map (called the period map) to a Grassmannian (the moduli space of subspaces of a fixed dimension of some fixed vector space), and there are certain properties that we can then make use of (for instance, the line bundle on the Siegel modular variety whose sections are cusp forms can be obtained via pullback from a certain line bundle on the Grassmannian) together with p-adic Hodge theoretic arguments to relate the mod p cohomology to Siegel cusp forms.

All this has the following stunning application. Recall that in we may obtain Galois representations from cusp forms (see for example Galois Representations Coming From Weight 2 Eigenforms). This can also be done for Siegel cusp forms more generally. These cusp forms live on a modular curve or Siegel modular variety, which are obtained as arithmetic manifolds, double quotients $\Gamma\backslash G(\mathbb{R})/K$ of a real Lie group $G$ (in this case the symplectic group) by a maximal compact open subgroup $K$ and an arithmetic subgroup $\Gamma$. But they are also algebraic varieties, so can be studied using the methods of algebraic geometry (see also Shimura Varieties). For example, we can use etale cohomology to obtain Galois representations.

But not all arithmetic manifolds are also algebraic varieties! For instance we have Bianchi manifolds, which are double quotients $\Gamma\backslash\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2)$, where $\Gamma$ can be, say, a congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z}[i])$ (or we can also replace $\mathbb{Z}[i]$ with the ring of integers of some other imaginary quadratic field). The groups involve look complex, but the theory of algebraic groups and in particular the method of Weil restriction allows us to look at them as real Lie groups. This is not an algebraic variety (one way to see this is that $\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2)$ is hyperbolic 3-space, so a Bianchi manifold has 3 real dimensions and as such cannot be related to an algebraic variety the way a complex manifold can).

Still, it has been conjectured that the singular cohomology (in particular its torsion subgroups) of such arithmetic manifolds which are not algebraic varieties can still be related to Galois representations! And for certain cases this has been proved using the following strategy. First, these arithmetic manifolds can be found as an open subset of the boundary of an appropriate compactification of a Siegel modular variety. Then, methods from algebraic topology (namely, the excision long exact sequence) allow us to relate the cohomology of the arithmetic manifold to the cohomology of the Siegel modular variety.

On the other hand, by our earlier discussion, the covering of the (rigid analytic space associated to the) Siegel modular variety by affinoid perfectoid spaces given by the Siegel modular variety at infinite level, together with the period map of the latter to the Grassmannian, allows one to show that the mod p cohomology of Siegel modular varieties is related to Siegel cusp forms, and it is known how to obtain Galois representations from these. Putting all of these together, this allows us to obtain Galois representations from the cohomology of manifolds which are not algebraic varieties.

A deeper look at aspects of perfectoid spaces, as well as their generalizations and applications (including a more in-depth look at the application to the mod p cohomology of Siegel modular varieties discussed in the previous couple of paragraphs), will hopefully be discussed in future posts.

References:

Perfectoid space on Wikipedia

Berkeley lectures on p-adic geometry by Peter Scholze and Jared Weinstein

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

Lecture notes on perfectoid Shimura varieties by Ana Caraiani

On torsion in the cohomology of locally symmetric varieties by Peter Scholze

p-adic Hodge theory for rigid analytic varieties by Peter Scholze

# The Unramified Local Langlands Correspondence and the Satake Isomorphism

In The Local Langlands Correspondence for General Linear Groups we gave the statement of the local Langlands correspondence for the groups $\mathrm{GL}_{n}(F)$ for $F$ a p-adic field. In this post we will consider a special case of this correspondence in more detail, called the unramified case (we shall define what this is shortly), and we take the opportunity to introduce an important concept in the theory, that of the Satake isomorphism (which we will state, but not prove).

Let us continue to stick with $\mathrm{GL}_{n}(F)$, although what we discuss here also generalizes to other reductive groups. Let $\mathcal{O}_{F}$ be the ring of integers of $F$.

Let us now explain what “unramified” means for both sides of the correspondence. We say that an irreducible admissible representation of $\mathrm{GL}_{n}(F)$ is unramified if there exists a nonzero vector that is fixed by $\mathrm{GL}_{n}(\mathcal{O}_{F})$. Meanwhile, we say that a Weil-Deligne representation (see also Weil-Deligne Representations) is unramified if it factors as $W_{F}\twoheadrightarrow\mathbb{Z}\hookrightarrow \mathrm{GL}_{2}(\mathbb{C})$, and the monodromy operator $N$ is zero. Let us note that an unramified Weil-Deligne representation is determined by where the Frobenius element (which maps to $1$ under the map to $\mathbb{Z})$ gets sent to, up to conjugacy. Hence the unramified Weil-Deligne representations are in bijection with conjugacy classes of diagonalizable elements in $\mathrm{GL}_{n}(\mathbb{C})$.

Then the unramified local Langlands correspondence is the following statement:

There is a bijection between the set of unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and set of unramified Weil-Deligne representations.

In this post we will prove this statement, assuming the Satake isomorphism. To explain what the Satake isomorphism is, let us first discuss a generalization of the Hecke algebra (see also Hecke Operators).

The spherical Hecke algebra $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$ is the algebra of compactly supported locally constant functions on $\mathrm{GL}_{n}(F)$ which are bi-invariant under the action (invariant under the left and right action) of $\mathrm{GL}_{n}(\mathcal{O}_{F})$. The “multiplication” on this algebra is given by convolution, i.e., given two elements $f_{1}$ and $f_{2}$ of the spherical Hecke algebra, their “product” is given by

$\displaystyle (f_{1}\cdot f_{2})(g)=\int_{\mathrm{GL}_{n}(F)}f_{1}(x)f_{2}(g^{-1}x)dx$

There is an action of the spherical Hecke algebra (more generally there is also an action of compactly supported locally constant functions of $G$, without the bi-invariance condition) on a representation $\pi$ of $\mathrm{GL}_{n}(F)$ as follows. Let $f$ be an element of the spherical Hecke algebra and let $v$ be a vector in the vector space on which the representation $\pi$ acts. Then

$\displaystyle \pi(f)v=\int_{\mathrm{GL}_{n}(F)} f(g)\pi(g)(v) dg$.

This action makes the representation $\pi$ into an $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-module. The importance of the spherical Hecke algebra to the unramified local Langlands correspondence is that there is a bijection between the set of unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and set of irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules.

The spherical Hecke algebra $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$ is commutative, and from this it follows that the irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules are in bijection with maps $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\to\mathbb{C}$.

Let $T$ be a maximal torus in $\mathrm{GL}_{n}(F)$ (see also Reductive Groups Part I: Over Algebraically Closed Fields) and let $X_{\bullet}(T)$ be the set of all cocharacters of $X_{\bullet}(T)$. Let $\mathbb{C}[X_{\bullet}(T)]$ be the ring formed by adjoining the elements of $X_{\bullet}(T)$ as formal variables to $\mathbb{C}$. Recall that the Weyl group $W$ is defined as the quotient of the normalizer of $T$ in $\mathrm{GL}_{n}(F)$ by the centralizer of $T$ in $\mathrm{GL}_{n}(F)$ (see also Reductive Groups Part I: Over Algebraically Closed Fields). The ring $\mathbb{C}[X_{\bullet}(T)]$ has an action of $W$ which comes from the action of $W$ on $T$. We denote the invariants of this action by $\mathbb{C}[X_{\bullet}(T)]^{W}$.

Now we can state the Satake isomorphism as follows:

$\displaystyle \mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\cong \mathbb{C}[X_{\bullet}(T)]^{W}$.

Let us now see how the Satake isomorphism helps us prove the unramified local Langlands correspondence. We define the dual torus $\widehat{T}$ to be $\mathrm{Spec}(\mathbb{C}[X_{\bullet}(T)])$. Then homomorphisms $\mathbb{C}[X_{\bullet}(T)]^{W}\to\mathbb{C}$ correspond to $W$-conjugacy classes of elements in $\widehat{T}(\mathbb{C})$. These conjugacy classes, in turn, are in bijection with the conjugacy classes of diagonalizable elements in $\mathrm{GL}_{n}(\mathbb{C})$. But these conjugacy classes are in bijection with the unramified Weil-Deligne representations, as mentioned earlier. At the same time, by the Satake isomorphism $\mathbb{C}[X_{\bullet}(T)]^{W}\to\mathbb{C}$ corresponds to maps $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\to\mathbb{C}$, and therefore to irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules, and therefore finally to unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$. This gives us the unramified local Laglands correspondence for $\mathrm{GL}_{n}(F)$.

As mentioned earlier, all of this can also be applied to more general reductive groups, with appropriate generalizations of what it means for an irreducible admissible representation to be unramified. In this case, the conjugacy classes involved will be that of the Langlands dual group.

There is also a “geometric” version of the Satake isomorphism which relates the representations of the Langlands dual group to the category of perverse sheaves on a very special geometric object called the affine Grassmannian. We will discuss more of this in future posts.

References:

Satake isomorphism on Wikipedia

Unramified representations and the Satake isomorphism by James Newton

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

Langlands correspondence and Bezrukavnikov’s equivalence by Anna Romanov and Geordie Williamson

# A Guide to Arithmetic Geometry

I have added a new page on the blog called A Guide to Arithmetic Geometry. It is a compilation of some of my older posts about arithmetic geometry on this blog that is ordered in a way that I think would be suitable for a beginner who is interested in arithmetic geometry and would like some kind of broad overview. I have made this page because my newer posts have been more advanced and specialized which might make the more beginner-friendly posts harder to find. There are of course no guarantees that this “guide” is the only way (or even a good way) to go through arithmetic geometry and I have posted some links to posts of a similar nature written by other arithmetic geometers as well. Hopefully people will find this useful!

# Taylor-Wiles Patching

In Galois Deformation Rings we mentioned the idea of “modularity lifting“, which forms one part of the approach to proving that a Galois representation arises from a modular form, the other part being residual modularity. In that post we also mentioned “R=T” theorems, which are in turn the approach to proving modularity lifting, the “R” standing for the Galois deformation rings that were the main topic of that post, and “T” standing for (a certain localization of) the Hecke algebra. In this post, we shall discuss R=T theorems in a little more detail, and discuss the ideas involved in its proof. We shall focus on the weight $2$ cusp forms (see also Galois Representations Coming From Weight 2 Eigenforms), although many of these ideas can also be generalized to higher weights.

### A review of Galois deformation rings and Hecke algebras

Let us recall again the idea behind R=T theorems. We recall from Galois Deformation Rings that if we have a fixed residual representation $\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathbb{F}$ (here $\mathbb{F}$ is some finite field of characteristic $p$), we have a Galois deformation ring $R_{\overline{\rho}}$, with the defining property that maps from $R_{\overline{\rho}}$ into some complete Noetherian local $W(\mathbb{F})$-algebra $A$ correspond to certain Galois representations over $A$, namely those which “lift” the residual representation $\overline{\rho}$. If we compose these maps with maps from $A$ into $\overline{\mathbb{Q}}_{p}$, we get maps that correspond to certain Galois representations over $\overline{\mathbb{Q}}_{p}$.

In addition, since we want to match up Galois representations with modular forms (cusp forms of weight $2$ in particular this post), we will want to impose certain conditions on the Galois representations that are parametrized by our deformation ring $R_{\overline{\rho}}$. For instance, it is known that p-adic Galois representations that arise from a cusp form of weight $2$ and level $\Gamma=\Gamma(N)$ are unramified at all the primes except $p$ and the ones that divide $N$. There is a way to construct a modification of our deformation ring $R_{\overline{\rho}}$ so that the Galois representations it parametrizes satisfies these conditions (also known as deformation conditions or deformation problems). We shall denote this modified deformation ring simply by $R$.

On the other hand, maps from the Hecke algebra to some coefficient field (we will choose this to be $\overline{\mathbb{Q}}_{p}$; conventionally this is $\mathbb{C}$, but $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ are isomorphic as fields) correspond to systems of eigenvalues coming from modular forms.

Now the idea is to match up these maps, since then it would be the same as matching Galois representations and modular forms; however, we note that currently our maps from $R_{\overline{\rho}}$ only correspond to Galois representations that come from lifting our fixed Galois representation $\overline{\rho}$ and we have not made any such restriction on the maps from our Hecke algebra, so they don’t quite match up yet.

### Galois representations valued in localizations of the Hecke algebra

What we will do to fix this is to come up with a maximal ideal of the Hecke algebra that corresponds to $\overline{\rho}$, and, instead of considering the entire Hecke algebra, which is too large, we will instead consider the localization of it with respect to this maximal ideal. We have, following the Hodge decomposition (for weights $k>2$, a generalization of this is given by a theorem of Eichler and Shimura)

$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong S_{2}(\Gamma,\mathbb{C})\oplus \overline{S_{2}(\Gamma,\mathbb{C})}$

where $M_{2}(\Gamma,\mathbb{C})$ (resp. $S_{2}(\Gamma,\mathbb{C})$) is the space of modular forms (resp. cusp forms) of weight $2$ and level $\Gamma$. The advantage of expressing modular forms in this form is that we shall be able to consider them “integrally”. We have that

$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathbb{C}$

Now let $E$ be a finite extension of $\mathbb{Q}_{p}$, with ring of integers $\mathcal{O}$, uniformizer $\varpi$ and residue field $\mathbb{F}$ (the same field our residual representation $\overline{\rho}$ takes values in). We can now consider

$\displaystyle H^{1}(Y(\Gamma), \mathcal{O})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathcal{O}$

Let $\Sigma$ be the set consisting of the prime $p$ and the primes dividing the level, which we shall assume to be squarefree (these conditions put us in the minimal case of Tayor-Wiles patching – though the strategy holds more generally, we assume these conditions to simplify our discussion). We have a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ acting on $H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p})$, and similarly a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acting on $H^{1}(Y(\Gamma), \mathcal{O})$. Recall that these are the subrings of their respective endomorphism rings generated by the Hecke operators $T_{\ell}$ and $S_{\ell}$ for all $\ell\not\in \Sigma$ (see also Hecke Operators and Galois Representations Coming From Weight 2 Eigenforms). The eigenvalue map

$\displaystyle \lambda_{g}:\mathbb{T}(S(\Gamma,\overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$

which associates to a Hecke operator its eigenvalue on some cusp form $g\in S(\Gamma,\overline{\mathbb{Q}}_{p})$ extends to a map

$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$.

Now since $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acts on $H^{1}(\Gamma, \mathcal{O})$ we will also have an eigenvalue map

$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathcal{O}$

compatible with the above, in that applying $\lambda_{g}$ followed by embedding the resulting eigenvalue to $\overline{\mathbb{Q}}_{p}$ is the same as composing the map from $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ into $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ first then applying the eigenvalue map. Now we can compose the eigenvalue map to $\mathcal{O})$ with the reduction mod $\varpi$ so that we get $\displaystyle \overline{\lambda}_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathbb{F}$.

Now let $\mathfrak{m}$ be the kernel of $\overline{\lambda}_{g}$. This is a maximal ideal of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$. In fact, we can associate to $\lambda_{g}$ a residual representation $\overline{\rho}_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{F})$, such that the characteristic polynomial of the $\mathrm{Frob}_{\ell}$ is given by $X^{2}-\lambda_{g}(T_{\ell})X+\ell \lambda_{g}(S_{\ell})$.

Now let $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ be the completion of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ with respect to $\mathfrak{m}$. It turns out that there is a Galois representation $\rho_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{T}(\Gamma)_{\mathfrak{m}})$ which lifts $\overline{\rho}_{\mathfrak{m}}$. Furthermore, $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ is a complete Noetherian local $\mathcal{O}$-algebra!

Putting all of these together, what this all means is that if $\overline{\rho}=\overline{\rho}_{\mathfrak{m}}$, there is a map $R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$. Furthermore, this map is surjective. Again, the fact that we have this surjective map reflects that fact that we can obtain Galois representations (of a certain form) from modular forms. Showing that this is an isomorphism amounts to showing that Galois representations of this form always come from modular forms.

### Taylor-Wiles patching: Rough idea behind the approach

So now, to prove our “R=T” theorem, we need to show that this map is actually an isomorphism.

Let $M=H^{1}(Y(\Gamma),\mathcal{O})$. The idea is that $R$ will have an action on $M$, which will factor through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$. If we can show that $M$ is free as an $R$-module, then since this action factors through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ via a surjection, then the map from $R$ to $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ must be an isomorphism.

This, by itself, is still too difficult. So what we will do is build an auxiliary module, sometimes called the patched module and denoted $M_{\infty}$, which is going to be a module over an auxiliary ring we shall denote by $R_{\infty}$, from which $M$ and $R$ can be obtained as quotients by a certain ideal. The advantage is that we can bring another ring in play, the power series ring $\mathcal{O}[[x_{1},\ldots,x_{q}]]$, which maps to $R_{\infty}$ (in fact, two copies of it will map to $R_{\infty}$, which is important), and we will use what we know about power series rings to show that $M_{\infty}$ is free over $R_{\infty}$, which will in turn show that $M$ is free over $R$.

In turn, $M_{\infty}$ and $R_{\infty}$ will be built as inverse limits of modules and rings $R_{Q_{n}}$ and $M_{Q_{n}}$. The subscript $Q_{n}$ refers to a set of primes , called “Taylor-Wiles primes” at which we shall also allow ramification (recall that initially we have imposed the condition that our Galois representations be unramified at all places outside of $p$ and the primes that divide the level $N$). As we shall see, these Taylor-Wiles primes will be specially selected so that we will be able to construct $M_{\infty}$ and $R_{\infty}$ with the properties that we will need. This passage to the limit in order to make use of what we know about power series is inspired by Iwasawa theory (see also Iwasawa theory, p-adic L-functions, and p-adic modular forms).

### Taylor-Wiles primes

A Taylor-Wiles prime of level $n$ is defined to be a prime $v$ such that the norm $q_{v}$ is congruent to $1$ mod $p^{n}$, and such that $\overline{\rho}(\mathrm{Frob}_{v})$ has distinct $\mathbb{F}$-rational eigenvalues. For our purposes we will need, for every positive integer $n$, a set $Q_{n}$ of Taylor-Wiles primes of cardinality equal to the dimension of the dual Selmer group of $R$ (which we shall denote by $q$), and such that the dual Selmer group of $R_{Q_{n}}$ is trivial. It is known that we can always find such a set $Q_{n}$ for every positive integer $n$.

Let us first look at how this affects the “Galois side”, i.e. $R_{Q_{n}}$. There is a surjection $R_{Q_{n}}\twoheadrightarrow R$, but the important property of this, that is due to how the Taylor-Wiles primes were selected, is that the dimensions of their tangent spaces (which is going to be equal to the dimension of the Selmer group as discussed in More on Galois Deformation Rings) are the same.

Now it so happens that, when we are considering $2$-dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the dimensions of the Selmer group and the dual Selmer group will be the same. This is what is known as the numerical coincidence, and is quite special to our case. In general, for instance when instead of $\mathbb{Q}$ we have a more general number field $F$, this numerical coincidence may not hold (we will briefly discuss this situation at the end of this post). The numerical coincidence, as well as the fact that the dimension of the tangent spaces of $R$ and $R_{Q_{n}}$ remain the same, are both consequences of the Wiles-Greenberg formula, which relates the Selmer group and the dual Selmer group.

Now let us look at the “automorphic side”, i.e. $M_{Q_{n}}$. We call this the automorphic side because they are localizations of spaces of modular forms (which are automorphic forms). We first need to come up with a new kind of level structure.

Letting $Q_{n}$ be some set of Taylor-Wiles primes, we define $\Gamma_{0}(Q_{n})=\Gamma\cap\Gamma_{0}(\prod_{v\in Q_{n}}v)$ and we further define $\Gamma_{Q_{n}}$ to be such that the quotient $\Gamma_{0}(Q_{n})/\Gamma_{Q_{n}}$ is isomorphic to the group $\Delta_{Q_{n}}$, defined to be the product over $v\in Q_{n}$ of the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$.

We define a new Hecke algebra $\mathbb{T}_{Q_{n}}$ obtained from $\mathbb{T}$ by adjoining new Hecke operators $U_{v}$ for every prime $v$ in $Q_{n}$. We define a maximal ideal $\mathfrak{m}_{Q_{n}}$ of $\mathbb{T}_{Q_{n}}$ generated by the elements of $\mathfrak{m}$ and $U_{v}-\alpha_{v}$ again for every prime $v$ in $Q_{n}$.

We now define $M_{Q_{n}}$ to be $H^{1}(Y(\Gamma_{Q}),\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$. This has an action of $\Delta_{Q_{n}}$ and is therefore a $\mathcal{O}[\Delta_{Q_{n}}]$-module. In fact, $M_{Q_{n}}$ is a free $\mathcal{O}[\Delta_{Q_{n}}]$-module. This will become important later. Another important property of $M_{Q_{n}}$ is that its $\Delta_{Q_{n}}$-coinvariants are isomorphic to $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$.

Now $R_{Q_{n}}$ also has the structure of a $\mathcal{O}[\Delta_{Q_{n}}]$-algebra. If we take $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(R_{Q_{n}})$ and restrict it to $\mathrm{Gal}(\overline{\mathbb{Q}}_{v}/\mathbb{Q}_{v})$ (for$v$ in $Q_{n}$), we get that the resulting local representation is of the form $\eta_{1}\oplus\eta_{2}$, where $\eta_{1}$ and $\eta_{2}$ are characters. Using local class field theory (see also The Local Langlands Correspondence for General Linear Groups), we obtain a map $\mathbb{Z}_{v}^{\times}\to R_{Q_{n}}^{\times}$. This map factors through the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$. Thus given $Q_{n}$ we have a map $\Delta_{Q_{n}}\to R_{Q_{n}}$.

Now it so happens that the action of $\Delta_{Q_{n}}$ on $M_{Q_{n}}$ factors through the map to $R_{Q_{n}}$. So therefore we have

$\displaystyle \mathcal{O}[\Delta_{Q_{n}}]\to R_{Q_{n}}\to\mathbb{T}_{Q_{n}}\curvearrowright M_{Q_{n}}$

### Taylor-Wiles patching: The patching construction

Now we will perform the patching construction, which means taking the inverse limit over $n$. First we must show that this is even possible, i.e. that we have an inverse system. We can formalize this via the notion of a patching datum.

We let $S_{\infty}$ denote $\mathcal{O}[[(\mathbb{Z}_{p})^{q}]]\cong \mathcal{O}[[x_{1},\ldots,x_{q}]]$ and let $\mathfrak{a}$ denote the ideal $(x_{1},\ldots,x_{q})$. Let us also define $R_{\infty}$ to be another power series ring $\mathcal{O}[[y_{1},\ldots,y_{q}]]$ but in a different set of variables of the same number. In the non-minimal case they might look quite different, but in either case there will be a map from $S_{\infty}$ to $R_{\infty}$; this may be thought of as the limiting case of the map from $\mathcal{O}[\Delta_{Q_{n}}]$ to $R_{Q_{n}}$ discussed earlier.

Now let $n$ be a positive integer. Let $\mathfrak{a}_{n}$ be the kernel of the surjection $S_{\infty}\twoheadrightarrow \mathcal{O}[(\mathbb{Z}/p^{n}\mathbb{Z})^{q}]$, let $S_{n}$ be $S_{\infty}/(\varpi^{n},\mathfrak{a}_{n})$, and $\mathfrak{d}_{n}$ be the ideal $(\varpi^{n},\mathrm{Ann}_{R}(M)^{n})$. Abstractly, a patching datum of level $n$ is a triple $(f_{n},X_{n},g_{n})$ where

• $f_{n}:R_{\infty}\twoheadrightarrow R/\mathfrak{d}_{n}$ is a surjection of complete Noetherian local $\mathcal{O}$ algebras
• $X_{n}$ is a $R_{\infty}\otimes_{\mathcal{O}} S_{n}$-module, finite free over $S_{n}$, such that
• $\mathrm{im}(S_{N}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(R_{\infty}\to\mathrm{End}_{\mathcal{O}}X)$
• $\mathrm{im}(\mathfrak{a}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(\mathrm{ker}(f)\to\mathrm{End}_{\mathcal{O}}X)$
• $g_{n}:X/\mathfrak{a}\xrightarrow M/(\varpi^{n})$ is an isomorphism of $R_{\infty}$-modules

We say that two patching data $(f_{n},X_{n},g_{n})$ and $(f_{n}',X_{n}',g_{n}')$ of level $n$ are isomorphic if $f_{n}=f_{n}'$ and there exists an isomorphism $X_{n}\cong X_{n}'$ compatible with $g_{n}$ and $g_{n}'$. We note the important fact that there are only finitely many isomorphism classes of patching data for any level $n$.

Now we will specialize this abstract construction to help us prove our R=T theorem. We choose

• $f_{n}:R_{\infty}\twoheadrightarrow R_{Q_{n}}\twoheadrightarrow R\twoheadrightarrow R/\mathfrak{d}_{n}$
• $X_{n}=M_{Q_{n}}\otimes_{S_{\infty}} S_{n}$
• $g_{n}$ is induced by the isomorphism between the $\Delta_{Q_{n}}$-coinvariants of $H^{1}(Y_{Q_{n}},\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$ and $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$

If we have a patching datum $D_{m}=(f_{m},X_{m},g_{m})$ of level $m$, we may form $D_{m}\mod n=D_{m,n}=(f\mod \mathfrak{d}_{n},X_{m}\otimes_{S_{m}} S_{n},g_{m}\otimes_{S_{m}}S_{n})$ which is a patching datum of level $n$.

Now recall that for any fixed $n$, we can only have a finite number of isomorphism classes of patching datum of level $n$. This means we can find a subsequence $(m_{n})_{n\geq 1}$ of $(m)_{m\geq 1}$ such that $D_{m_{n+1},n+1}\mod n\cong D_{m_{n},n}$.

We can now take inverse limits. Let $M_{\infty}=\varprojlim_{n}X_{m_{n}}$, let the surjection $R_{\infty}\twoheadrightarrow R$ be given by $\varprojlim_{n}f_{m_{n},n}$, and let the surjection $M_{\infty}\twoheadrightarrow M$ be given by $\varprojlim_{n}g_{m_{n},n}$. We have

$\displaystyle \mathcal{O}[[x_{1},\ldots,x_{g}]]\to R_{\infty}\to\mathbb{T}_{\infty}\curvearrowright M_{\infty}$

Just as $M_{Q_{n}}$ is free as a module over $\mathcal{O}[\Delta_{Q_{n}}]$, we have that $M_{\infty}$ is free as a module over $S_{\infty}$. We will now use some commutative algebra to show that $M_{\infty}$ is a free $R_{\infty}$-module. The depth of a module $M'$ over a local ring $R'$ with maximal ideal $\mathfrak{m'}$ is defined to be the minimum $i$ such that $\mathrm{Ext}^{i}(R'/\mathfrak{m}',M')$ is nonzero. The depth of a module is always bounded above by its dimension.

Now the dimension of $R_{\infty}$ is $1+q$ (we know this since we defined it as a power series $\mathcal{O}[[y_{1},\ldots,y_{q}]]$). This bounds $\mathrm{dim}_{R_{\infty}}(M_{\infty})$, and by the above fact regarding the depth of a module, $\mathrm{dim}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{R_{\infty}}(M_{\infty})$. Since the action of $S_{\infty}$ on $M_{\infty}$ factors through the action of $R_{\infty}$, $\mathrm{depth}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{S_{\infty}}(M_{\infty})$. Finally, since $M_{\infty}$ is a free $S_{\infty}$-module, we have that $\mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$. In summary,

$\displaystyle 1+q=\mathrm{dim}(R_{\infty})\geq \mathrm{dim}_{R_{\infty}}(M_{\infty})\geq\mathrm{depth}_{R_{\infty}}(M_{\infty})\geq \mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$

and we can see that all of the inequalities are equalities, and all the quantities are equal to $1+q$. The Auslander-Buchsbaum formula from commutative algebra tells us that

$\displaystyle \mathrm{proj.dim}_{R_{\infty}}(M_{\infty})=\mathrm{depth}(R_{\infty})-\mathrm{depth}_{R_{\infty}}(M_{\infty})$

and since both terms on the right-hand side are equal to $1+q$, the right-hand side is zero. Therefore the projective dimension of $M_{\infty}$ relative to $R_{\infty}$ is zero, which means that $M_{\infty}$ is a projective module over $R_{\infty}$. Since $R_{\infty}$ is local, this is the same as saying that $M_{\infty}$ is a free $R_{\infty}$-module.

We have that $M\cong M_{\infty}/\mathfrak{a}M_{\infty}$ is a free module over $R_{\infty}/\mathfrak{a}R_{\infty}$. Since this action factors through maps $R_{\infty}/\mathfrak{a}R_{\infty}\to R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$ which are all surjections, they have to be isomorphisms, and we have that $M$ is a free $R$-module, and therefore $R\cong\mathbb{T}(\Gamma)_{\mathfrak{m}}$. This proves our R=T theorem.

### Generalizations and other applications of Taylor-Wiles patching

We have discussed only the “minimal case” of Taylor-Wiles patching, but one can make use of the same ideas for the non-minimal case, and one may also apply Taylor-Wiles patching to show the modularity of $2$-dimensional representations of $\mathrm{Gal}(\overline{F}/F)$ for $F$ a totally real field (in this case on the automorphic side we would have Hilbert modular forms).

However, when $F$ is a more general number field the situation is much more complicated, because one of the facts that we have used, which is vital to Taylor-Wiles patching, now fails. This is the fact that the dimension of the dual Selmer group (which is the cardinality of our sets of Taylor-Wiles primes) and the dimension of the Selmer group (which is also the dimension of the tangent space of the Galois deformation ring $R$) are equal (again this is what is known as the “numerical coincidence”). This is the important property that can fail for more general number fields. Here the dimensions of the dual Selmer group and the Selmer group may differ by some nonzero quantity $\delta$.

Moreover, in our discussion we made use of the fact that the cohomology was concentrated in a single degree. For more general number fields this is no longer true. Instead we will have some interval for which the cohomology is nonzero. However, it so happens (for certain “nice” cases) that the length of this interval is equal to $\delta+1$. This is a hint that the two complications are related, and in fact can be played off each other so that they “cancel each other out” in a sense. Instead of patching modules, in this case one patches complexes instead. These ideas were developed in the work of Frank Calegari and David Geraghty.

The method of Taylor-Wiles patching is also being put forward as an approach to the p-adic local Langlands correspondence (which is also closely related to modularity as we have seen in Completed Cohomology and Local-Global Compatibility), via the work of Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin. This is also closely related to the ideas discussed at in Moduli Stacks of (phi, Gamma)-modules (where we used the same notation $M_{\infty}$ for the patched module). Namely, we expect a coherent sheaf $\mathcal{M}$ on the moduli stack of $\varphi,\Gamma$-modules which, “locally” coincides or is at least closely related to the patched module $M_{\infty}$. This has applications not only to the p-adic local Langlands correspondence as mentioned above, but also to the closely-related Breuil-Mezard conjecture. We will discuss these ideas and more in future posts.

References:

Modularity Lifting (Course Notes) by Patrick Allen

Modularity Lifting Theorems by Toby Gee

Beyond the Taylor-Wiles Method by Jack Thorne

Motives and L-functions by Frank Calegari

Overview of the Taylor-Wiles Method by Andrew Snowden (lecture notes from the Stanford Modularity Lifting Seminar)

Reciprocity in the Langlands Program Since Fermat’s Last Theorem by Frank Calegari

Modularity Lifting Beyond the Taylor-Wiles Method by Frank Calegari and David Geraghty

Patching and the p-adic local Langlands Correspondence by Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin

# Completed Cohomology and Local-Global Compatibility

In Completed Cohomology, we mentioned that the p-adic local Langlands correspondence may be found inside the completed cohomology, and that this is used in the proof of the Fontaine-Mazur conjecture. In this post, we elaborate on these ideas. We shall be closely following the Séminaire Bourbaki article Correspondance de Langlands p-adique, compatibilité local-global et applications by Christophe Breuil.

Let us make the previous statement more precise. Let $E$ be a finite extension of $\mathbb{Q}_{p}$, with ring of integers $\mathcal{O}_{E}$, uniformizer $\varpi$, and residue field $k_{E}$. Let us assume that $\mathcal{O}_{E}$ contains the Hecke eigenvalues of a cuspidal eigenform $f$ of weight $2$. Consider the etale cohomology $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ of the open modular curve $Y(K)$ (we will define this more precisely later). Then we have that $\varinjlim_{K}H_{\mathrm{et}}^1(Y(K)\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathcal{O}_{E})\otimes_{\mathcal{O}_{E}}E$ contains $\rho_{f}\otimes_{E}\otimes_{\ell}\pi_{\ell}(\rho_{f}\vert_{\mathrm{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})})$, where $\rho_{f}$ is the p-adic Galois representation associated to $f$ (see also Galois Representations Coming From Weight 2 Eigenforms), and