Moduli Stacks of (phi, Gamma)-modules

November 26, 2021February 22, 2022 / Anton Hilado / 1 Comment

In Moduli Stacks of Galois Representations we discussed the moduli stack of representations of the absolute Galois group $G_{K}:=\mathrm{Gal}(\overline{K}/K)$ when $K$ is a finite extension of $\mathbb{Q}_{p}$ , for representations valued in some $\mathbb{Z}_{\ell}$ -algebra, where $\ell$ is a prime number different from $p$ . When $\ell=p$ however, it turns out using the same strategy as for $\ell\neq p$ can lead to some moduli stacks that are difficult to study.

Instead, we are going to use the theory of $(\varphi,\Gamma)$ -modules as an intermediary. Let $k$ be the residue field of $K$ , and let $\mathbf{A}_{K}$ be the p-adic completion of the Laurent series field $W(k)((T))$ (where $W(k)$ here denotes the ring of Witt vectors over $k$ ). For future reference, let us now also state that we will use $\mathbf{A}_{K}^{+}$ to denote $W(k)\otimes_{\mathbb{Z}_{p}} A[[T]]$ . An etale $(\varphi,\Gamma)$ -module over $\mathbf{A}_{K}$ is a finite $\mathbf{A}_{K}$ -module equipped with commuting semilinear actions of $\varphi$ (an endomorphism coming from the Frobenius of $W(k)$ ) and $\Gamma$ (the subgroup of $\mathrm{Gal}(K(\zeta_{p^{\infty}})/K)$ isomorphic to $\mathbb{Z}_{p}$ , see also Iwasawa theory, p-adic L-functions, and p-adic modular forms). One of the important facts about etale $(\varphi,\Gamma)$ -modules is the following:

The category of etale $(\varphi,\Gamma)$ -modules is equivalent to the category of continuous $G_{K}$ -modules over finite $\mathbb{Z}_{p}$ -modules.

This equivalence is given more explicit as follows. Let $\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}$ be the p-adic completion of the ring of integers of the maximal unramified extension of $\mathbf{A}_{K}[1/p]$ in $W(\mathcal{O}_{\mathbb{C}_{p}}^{\flat})$ . Then to obtain a $G_{K}$ -module $V$ from a $\varphi,\Gamma$ -module $M$ , we take

$\displaystyle V=(\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}\otimes_{\mathbf{A}_{K}} M)^{\varphi=1}$

and to obtain the $(\varphi,\Gamma)$ -module $M$ from the $G_{K}$ – module $V$ , we take

$\displaystyle M=(\widehat{\mathbf{A}}_{K}^{\mathrm{ur}}\otimes_{\mathbb{Z}_{p}} V)^{G_{K_{\mathrm{cyc}}}}$

where $K_{\mathrm{cyc}}$ is $K$ adjoin all the $p$ -power roots of unity. Note that if we wanted bonafide Galois representations instead of $G_{K}$ -modules we can invert $p$ , and our Galois representations will be over $\mathbb{Q}_{p}$ . They will be equivalent to $(\varphi,\Gamma)$ -modules over $\mathbf{A}_{K}[1/p]$ .

More generally we can consider etale $(\varphi,\Gamma)$ -modules with coefficients in $A$ , where $A$ is some $\mathbb{Z}_{p}$ -algebra. This means they are finite $\mathbf{A}_{K,A}$ -modules, where $\mathbf{A}_{K,A}$ is the p-adic completion of $W(k)\otimes_{\mathbb{Z}_{p}}A((T))$ , instead of $\mathbf{A}_{K}$ -modules (again for future reference, we note that $\mathbf{A}_{K,A}^{+}$ will be used for $W(k)\otimes_{\mathbb{Z}_{p}}A[[T]]$ ). The category of etale $(\varphi,\Gamma)$ -modules with coefficients in $A$ is equivalent to the category of continuous $G_{K}$ -representations over finite $A$ -modules.

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

The ind-algebraicity of the moduli stack $\mathcal{X}_{d}$

As stated earlier, the moduli stack $\mathcal{X}_{d}$ is an ind-algebraic stack. This means it can be written as the 2-colimit $\varinjlim_{i\in I} \mathcal{X}_{d,i}$ in the 2-category of stacks of 2-directed system $\lbrace\mathcal{X}_{d,i}\rbrace_{i\in I}$ of algebraic stacks $\mathcal{X}_{d,i}$ . (Recall from Algebraic Spaces and Stacks that an algebraic stack is an fppf stack $\mathcal{Y}$ such that there exists a map from an affine scheme $U$ to $\mathcal{Y}$ and this map is representable by algebraic spaces, surjective, and smooth).

To understand why $\mathcal{X}_{d}$ is an ind-algebraic stack, we need to understand it as the scheme-theoretic image of a certain map of certain moduli stacks. The idea is that the target stacks (which is going to be the moduli stack $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ of $\varphi$ -modules with a semilinear action of the “discretization” of $\Gamma$ , more on this later) is ind-algebraic and we can deduce the ind-algebraicity of $\mathcal{X}_{d}$ from this.

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

We will need to discuss moduli stacks $\mathcal{R}_{d}$ of rank $d$ $\varphi$ -modules, moduli stacks $\mathcal{C}_{d,h}$ of rank $d$ $\varphi$ -modules of $F$ -height at most $h$ , and moduli stacks $W_{d,h}$ of rank $d$ weak Wach modules of $T$ -height at most $h$ .

We define $\mathcal{R}_{d}^{a}$ to be the stack over $\mathbb{Z}/p^{a}\mathbb{Z}$ such that for any $\mathbb{Z}/p^{a}\mathbb{Z}$ -algebra $A$ , $\mathcal{R}_{d}^{a}(A)$ is the groupoid of all $\varphi$ -modules which are projective and of rank $d$ . We have that $\mathcal{R}_{d}^{a}$ is also a stack over $\mathbb{Z}_{p}$ , and we can define $\mathcal{R}_{d}=\varinjlim_{a}\mathcal{R}_{d}^{a}$ , which is a stack over $\mathbb{Z}_{p}$ which we may think of as the moduli space of $\varphi$ -modules which are projective and of rank $d$ .

Let $F$ be a polynomial in $W(k)[T]$ which is congruent to a power of $T$ modulo $p$ and let $h$ be a nonnegative integer. A $\varphi$ -module of $F$ -height at most $h$ over $\mathbf{A}_{K,A}^{+}$ is a finitely generated $T$ -torsion free $\mathbf{A}_{K,A}^{+}$ -module $\mathfrak{M}$ together with a $\varphi$ -semilinear map $\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$ such that the map $1\otimes \varphi_{\mathfrak{M}}:\varphi^{*}\mathfrak{M}\to\mathfrak{M}$ is injective, and whose cokernel is annihilated by $F^{h}$ . We let $\mathcal{C}_{d,h}$ be the stack such that $\mathcal{C}_{d,h}(A)$ is the groupoid of $\varphi$ -module of $F$ -height at most $h$ over $\mathbf{A}_{K,A}^{+}$ which are projective of rank $d$ .

In the special case that the polynomial $F$ is the minimal polynomial of the uniformizer of $K$ , a $\varphi$ -module of $F$ -height at most $h$ over $\mathbf{A}_{K,A}^{+}$ is also called a Breuil-Kisin module of height at most $h$ . We will encounter Breuil-Kisin modules again later.

We have the following important properties of the stacks $\mathcal{C}_{d,h}^{a}$ and $\mathcal{R}_{d}^{a}$ :

The moduli stack $\mathcal{C}_{d,h}^{a}$ is an algebraic stack of finite presentation over $\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$ , with affine diagonal.
The moduli stack $\mathcal{R}_{d}^{a}$ is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
The morphism $\mathcal{C}_{d,h}^{a}\to \mathcal{R}_{d}^{a}$ is representable by algebraic spaces, proper, and of finite presentation.
The diagonal morphism $\Delta:\mathcal{R}_{d}^{a}\to\mathcal{R}_{d}^{a}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathcal{R}_{d}^{a}$ is representable by algebraic spaces, affine, and of finite presentation.

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

The moduli stack $\mathcal{C}_{d,h}$ is an p-adic formal algebraic stack of finite presentation over $\mathrm{Spf}(\mathbb{Z}_{p})$ , with affine diagonal.
The moduli stack $\mathcal{R}_{d}$ is a limit-preserving ind-algebraic stack whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
The morphism $\mathcal{C}_{d,h}\to \mathcal{R}_{d}$ is representable by algebraic spaces, proper, and of finite presentation.
The diagonal morphism $\Delta:\mathcal{R}_{d}\to\mathcal{R}_{d}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathcal{R}_{d}$ is representable by algebraic spaces, affine, and of finite presentation.

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

Now let $\gamma$ be a topological generator of $\Gamma$ , and let $\Gamma_{\mathrm{disc}}=\langle\gamma\rangle$ , so that $\Gamma_{\mathrm{disc}}\cong\mathbb{Z}$ . Let

$\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}:=\mathcal{R}_{d}\times_{\Delta,\mathcal{R}_{d}\times\mathcal{R}_{d},\Gamma_{\gamma}}\mathcal{R}_{d}$

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

A rank $d$ projective weak Wach module of $T$ -height at most $h$ and level at most $s$ over $\mathbf{A}_{K,A}^{+}$ is a rank $d$ projective $\varphi$ -module $\mathfrak{M}$ of $T$ -height at most $h$ over $\mathbf{A}_{K,A}^{+}$ , such that $\mathfrak{M}[1/T]$ has a semilinear action of $\Gamma_{\mathrm{disc}}$ satsifying $(\gamma^{p^{s}}-1)\mathfrak{M}\subseteq T\mathfrak{M}$ .

Let $\mathcal{W}_{d,h,s}$ be the moduli stack of rank $d$ projective weak Wach modules, of $T$ -height at most $h$ , and level at most $s$ . This is a p-adic formal algebraic stack of finite presentation over $\mathbb{Z}_{p}$ . To show this we make the following steps.

We consider the fiber product $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}\times_{\mathcal{R}_{d}}\mathcal{C}_{d,h}$ . This is the moduli stack of rank $d$ projective $\varphi$ -modules $\mathfrak{M}$ over $\mathbf{A}_{K,A}^{+}$ of $T$ -height at most $h$ , equipped with a semilinear action of $\Gamma_{\mathrm{disc}}$ on $\mathfrak{M}[1/T]$ . It is a p-adic formal algebraic stack of finite presentation over $\mathrm{Spf}(\mathbb{Z}_{p})$ .

Now consider $\mathcal{W}_{d,h}$ , the moduli stack of rank $d$ projective weak Wach modules of height at most $h$ . We have an isomorphism $\varinjlim_{s}\mathcal{W}_{d,h,s}\xrightarrow{\sim}\mathcal{W}_{d,h}$ , and $\mathcal{W}_{d,h}$ has a closed immersion of finite presentation into the fiber product $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}\times_{\mathcal{R}_{d}}\mathcal{C}_{d,h}$ .

We let $\mathcal{W}_{d,h,s}^{a}:=\mathcal{W}_{d,h,s}\times_{\mathrm{Spf}(\mathbb{Z}_{p})}\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$ . This is a closed substack of $\mathcal{W}_{d,h,s}$ . We define $\mathcal{X}_{d,h,s}^{a}$ to be the scheme-theoretic image of the composition $\mathcal{W}_{d,h,s}^{a}\hookrightarrow\mathcal{W}_{d,h,s}\to\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ . The stack $\mathcal{X}_{d,h,s}^{a}$ is a closed substack of $\mathcal{X}_{d}$ , and in fact we will see that $\varinjlim\mathcal{X}_{d,h,s}^{a}$ is isomorphic to $\mathcal{X}_{d}$ .

Let us explain very briefly how the last statement works. The existence of a morphism from $\mathcal{X}_{d,h,s}^{a}$ to $\mathcal{X}_{d}$ (which factors through $\mathcal{X}_{d}^{a})$ basically comes down to being able to extend the action of $\Gamma_{\mathrm{disc}}$ to a continuous action of $\Gamma$ .

Now to show that the morphism $\varinjlim\mathcal{X}_{d,h,s}^{a}\to\mathcal{X}$ , we have to show that for any $\mathbb{Z}/p^{a}\mathbb{Z}$ -algebra $A$ any morphism $\mathrm{Spec}(A)\to\mathcal{X}_{d}$ must factor through $\mathcal{X}_{d,h,s}^{a}$ , for some $h$ and some $s$ . It is in fact enough to show this for $B$ such that there is a scheme-theoretically dominant map $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ and such that if $M$ is the $(\varphi,\Gamma)$ -module corresponding to $\mathrm{Spec}(A)\to\mathcal{X}_{d}$ , then $M_{B}$ is free. The freeness of $M_{B}$ allows us to find a $\varphi$ -invariant lattice $\mathfrak{M}$ inside it which corresponds to a weak Wach module over $B$ . Associating $M_{B}$ to $\mathfrak{M}$ gives us a map $\mathcal{W}_{d,h,s}^{a}\to\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ . Recalling that the scheme-theoretic image of $\mathcal{W}_{d,h,s}^{a}$ in $\mathcal{R}_{d}^{\Gamma_{\mathrm{disc}}}$ is $\mathcal{X}_{d,h,s}^{a}$ , we see that our map $\mathrm{Spec}(B)\to\mathcal{X}_{d}$ factors through $\mathcal{X}_{d,h,s}^{a}$ and thus $\varinjlim\mathcal{X}_{d,h,s}^{a}\to\mathcal{X}$ is an isomorphism. The existence of $B$ satisfying such properties is guaranteed by the work of Emerton and Gee.

Crystalline moduli stacks

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

Let $A_{\mathrm{inf},A}$ denote $\varprojlim_{a}(\varprojlim_{i}(W_{a}(\mathcal{O}_{\mathbb{C}_{p}}^{\flat})\otimes_{\mathbb{Z}_{p}}A)/v^{i})$ , where $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ denotes the tilt of the ring of integers of the p-adic complex numbers (see also Perfectoid Fields) and $v$ is an element of the maximal ideal of $W_{a}\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ whose image in $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ is nonzero. A Breuil-Kisin-Fargues module of height at most $h$ with $A$ -coefficients is a finitely generated $A_{\mathrm{inf},A}$ -module $\mathfrak{M}^{\mathrm{inf}}$ together with a $\varphi$ -semilinear map $\varphi_{\mathfrak{M}^{\mathrm{inf}}}:\mathfrak{M}^{\mathrm{inf}}\to\mathfrak{M}^{\mathrm{inf}}$ such that the map $1\otimes \varphi_{\mathfrak{M}^{\mathrm{inf}}}:\varphi^{*}\mathfrak{M}^{\mathrm{inf}}\to\mathfrak{M}^{\mathrm{inf}}$ is injective, and whose cokernel is annihilated by $E(u)^{h}$ , where $E(u)$ is the minimal polynomial of the uniformizer of $K$ . A Breuil-Kisin-Fargues $G_{K}$ -module of height at most $h$ is a Breuil-Kisin-Fargues module of height at most $h$ together with a semilinear $G_{K}$ action that commutes with $\varphi$ .

Let us note that given a Breuil-Kisin module $\mathfrak{M}$ , we can obtain a Breuil-Kisin-Fargues module $\mathfrak{M}^{\mathrm{inf}}$ by taking $\mathfrak{M}^{\mathrm{inf}}=A_{\mathrm{inf},A}\otimes_{\mathbf{A}_{K,A}^{+}}\mathfrak{M}$ . To be able to take the tensor product we need a map from $\mathbf{A}_{K,A}^{+}$ to $A_{\mathrm{inf},A}$ , which in this case is provided by sending the element $T$ in $\mathbf{A}_{K,A}^{+}\cong W(k)\otimes_{\mathbb{Z}_{p}}A[[T]]$ to a compatible system of p-power roots of the uniformizer $\pi$ in $A_{\mathrm{inf},A}$ (we also say that we are in the “Kummer case“, as opposed to the “cyclotomic case” where p-power roots of unity are used; in the literature the symbol $\mathfrak{S}_{A}$ is also used in place of $\mathbf{A}_{K,A}^{+}$ , which is reserved for the cyclotomic case; note also that $G_{K_{\mathrm{cyc}}}$ will be replaced by $G_{K_{\infty}}$ in this case, $K_{\infty}$ being $K$ adjoin all p-power roots of $\pi$ ).

There is a notion of a Breuil-Kisin-Fargues $G_{K}$ -module of height at most $h$ admitting all descents. This means that, for every $\pi$ a uniformizer of $K$ and every $\pi^{\flat}$ the p-power roots of $\pi$ in $\mathcal{O}_{\mathbb{C}_{p}}^{\flat}$ , we can find a Breuil-Kisin module $\mathfrak{M}_{\pi^{\flat}}$ inside the part of the Breuil-Kisin-Fargues module $\mathfrak{M}^{\mathrm{inf}}$ fixed by the absolute Galois group of the field obtained by adjoining all p-power roots of $\pi$ to $K$ (satisfying some conditions related to certain submodules being independent of the choice of $\pi$ and $\pi^{\flat}$ ). If $\mathfrak{M}^{\mathrm{inf}}$ is a Breuil-Kisin-Fargues $G_{K}$ -module and $L$ is a finite extension of $K$ , we say that $\mathfrak{M}^{\mathrm{inf}}$ admits all descents over $L$ if the Breuil-Kisin Fargues $G_{L}$ -module obtained by restricting the $G_{K}$ action to $G_{L}$ admits all descents.

Let $\mathfrak{M}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$ -module of height at most $h$ admitting all descents. We say that $\mathfrak{M}^{\mathrm{inf}}$ is crystalline if, for all $g\in G_{K}$ and for any choice of $\pi$ and $\pi^{\flat}$ we have

$\displaystyle (g-1)(\mathfrak{M}_{\pi^{\flat}})\subset \varphi^{-1}([\varepsilon]-1)[\pi^{\flat}]\mathfrak{M}^{\mathrm{inf}}$ .

As the name implies, the importance of the crystalline condition is that it gives rise to crystalline Galois representations (see p-adic Hodge Theory: An Overview). To obtain a Galois representation from a Breuil-Kisin-Fargues $G_{K}$ -module $\mathfrak{M}^{\mathrm{inf}}$ of height at most $h$ admitting all descents, first we take $M=W(\mathbb{C}_{p}^{\flat})\otimes_{A_{\mathrm{inf}}}\mathfrak{M}^{\mathrm{inf}}$ . Then $M$ is a $(G_{K},\varphi)$ -module. Then we can take $M^{\varphi=1}$ to get a $G_{K}$ -module, and finally we can tensor with $\mathbb{Q}_{p}$ to get a Galois representation, which we shall denote by $V(M)$ . As hinted at earlier, the Galois representation $V(M)$ will be crystalline if and only if $\mathfrak{M}^{\mathrm{inf}}$ is crystalline. Furthermore $V(M)$ will have Hodge-Tate weights in the range $[0,h]$ if and only if $\mathfrak{M}^{\mathrm{inf}}$ has height at most $h$ .

Let $\mathcal{C}_{d,\mathrm{crys},h}^{a}$ be the limit-preserving category of groupoids over $\mathrm{Spec}(\mathbb{Z}/p^{a}\mathbb{Z})$ such that $\mathcal{C}_{d,\mathrm{crys},h}^{a}(A)$ , for $A$ a finite type $\mathbb{Z}/p^{a}\mathbb{Z}$ -algebra, is the groupoid of Breuil-Kisin-Fargues $G_{K}$ -modules with $A$ -coefficients of height at most $h$ , admitting all descents, and crystalline. We let $\mathcal{C}_{\mathrm{d,crys},h}:=\varinjlim_{a}\mathcal{C}_{d,\mathrm{crys},h}^{a}$ .

There is a map from $\mathcal{C}_{d,\mathrm{crys},h}$ to $\mathcal{X}_{d}$ given by sending a Breuil-Kisin-Fargues $G_{K}$ -module $\mathfrak{M}$ to the $(\varphi,\Gamma)$ -module $\mathfrak{M}^{\mathrm{inf}}\otimes_{\mathbf{A}_{\mathrm{inf},A}}W(C^{\flat})_{A}$ . We now let $\mathcal{C}_{d,\mathrm{crys},h}^{\mathrm{fl}}$ be the maximal substack of $\mathcal{C}_{d,\mathrm{crys},h}$ which is flat over $\mathrm{Spf}(\mathbb{Z}_{p})$ , and define $\mathcal{X}^{\mathrm{crys},h}$ to be the scheme-theoretic image of $\mathcal{C}_{d,\mathrm{crys},h}^{\mathrm{fl}}$ under the map from $\mathcal{C}_{d,\mathrm{crys},h}$ to $\mathcal{X}_{d}$ as described above.

Let us now introduce the notion of Hodge types. A Hodge type is a set of tuples $\underline{\lambda}=\lbrace\lambda_{\sigma,i}\rbrace_{\sigma:K\hookrightarrow\overline{\mathbb{Q}}_{p},1\leq i\leq d}$ of nonnegative integers such that $\lambda_{\sigma,i}\leq\lambda_{\sigma,i+1}$ for all $\sigma$ and all $1\leq i\leq d-1$ . A Hodge type is regular if $\lambda_{\sigma,i}<\lambda_{\sigma,i+1}$ for all $\sigma$ and all $1\leq i\leq d-1$ .

We also have the notion of an inertial type, which is defined to be a $\overline{\mathbb{Q}}_{p}$ -representation of the inertia group $I_{K}$ which extends to a representation of the Weil group $W_{K}$ with open kernel (which implies that it has finite image).

We can associate to a Breuil-Kisin-Fargues $G_{K}$ -module a Hodge type and an inertial type as we now discuss. We let $E$ be a finite extension of $\mathbb{Q}_{p}$ large enough so that it contains all the embeddings of $K$ into $\mathbb{C}_{p}$ . Let $A^{\circ}$ be a p-adically complete flat $\mathcal{O}_{E}$ -algebra topologically of finite type over $\mathcal{O}_{E}$ , and let $A=A^{\circ}[1/p]$ . Let $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$ module with $A^{\circ}$ coefficients admitting all descents over $L$ . We write $\mathfrak{M}_{A^{\circ}}$ for the associated Breuil-Kisin module, and define $\mathrm{Fil}^{i}\varphi^{*}\mathfrak{M}_{A^{\circ}}:=(1\otimes\varphi_{\mathfrak{M}_{A^{\circ}}})^{-1} (E(u)^{i}\varphi^{*}\mathfrak{M}_{A^{\circ}})$ . We write

$\displaystyle D_{\mathrm{dR}}(\mathfrak{M}^{\mathrm{inf}})=((\varphi^{*}\mathfrak{M}_{A^{\circ}}/E(u)\varphi^{*}\mathfrak{M}_{A^{\circ}})\otimes_{A^{\circ}}A)^{\mathrm{Gal}(L/K)}$

and

$\displaystyle \mathrm{Fil}^{i}D_{\mathrm{dR}}(\mathfrak{M}^{\mathrm{inf}})=(\mathrm{Fil}^{i}(\varphi^{*}\mathfrak{M}_{A^{\circ}}/E(u)\mathrm{Fil}^{i-1}\varphi^{*}\mathfrak{M}_{A^{\circ}})\otimes_{A^{\circ}}A)^{\mathrm{Gal}(L/K)}$

We have the decomposition $K\otimes_{\mathbb{Q}_{p}}A=\prod_{\sigma:K\hookrightarrow E}A$ . We have idempotents $e_{\sigma}$ corresponding to each factor of this decomposition, and we have the decomposition

$\displaystyle D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})=\prod_{\sigma:K\hookrightarrow E} e_{\sigma}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$

of the $K\otimes_{\mathbb{Q}_{p}}A$ -module $D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$ into $A$ -modules $e_{\sigma}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$ .

Now let $\underline{\lambda}$ be a Hodge type. We say that a Breuil-Kisin-Fargues $G_{K}$ -module $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ has Hodge type $\underline{\lambda}$ if $e_{\sigma}\mathrm{Fil}^{i}D_{\mathrm{dR}}(\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}})$ has constant rank equal to $\#\lbrace j\vert\lambda_{\sigma\vert K,j}\geq i\rbrace$ .

Now on to inertial types. Let $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ be a Breuil-Kisin-Fargues $G_{K}$ -module admitting all descents over $L$ and let $\mathfrak{M}_{A^{\circ},\pi^{\flat}}$ be the associated Breuil-Kisin module. Consider $\overline{\mathfrak{M}}_{A^{\circ}}=\mathfrak{M}_{A^{\circ},\pi^{\flat}}/[\pi^{\flat}]\mathfrak{M}_{A^{\circ},\pi^{\flat}}$ , a submodule of $W(\overline{k})\otimes_{A_{\mathrm{inf}},A}\mathfrak{M}^{\mathrm{inf}}$ . Let $\ell$ be the residue field of $L$ and let $L_{0}=W(\ell)[1/p]$ . We have a $W(\ell)\otimes_{\mathbb{Z}_{p}}A$ -semilinear action of $\mathrm{Gal}(L/K)$ on $\mathfrak{M}_{A^{\circ},\pi^{\flat}}$ induced from the action of $G_{K}$ on $\mathfrak{M}_{A^{\circ}}^{\mathrm{inf}}$ , which in turn induces an action of $I_{L/K}$ on the $L_{0}\otimes A$ -module $\overline{\mathfrak{M}}_{A^{\circ}}\otimes_{A^{\circ}} A$ .

Fix an embedding $\sigma:L_{0}\hookrightarrow E$ . As before we have a corresponding idempotent $e_{\sigma}$ . Now let $\tau$ be an inertial type. Given a Breuil-Kisin-Fargues $G_{K}$ -module we say that it has inertial type $\tau$ if as an $I_{L/K}$ -module, $e_{\sigma} \overline{\mathfrak{M}}_{A^{\circ}}\otimes_{A^{\circ}} A$ is isomorphic to the base change of $\tau$ to $A$ .

We now define $\mathcal{C}_{d,\mathrm{crys},h}^{L/K,\mathrm{fl},\underline{\lambda},\tau}$ to be the moduli stacks of Breuil-Kisin-Fargues $G_{K}$ -modules of rank $d$ , height at most $h$ , and admitting all descents to $L$ , that give rise to Galois representations which become crystalline over $L$ and with associated Hodge type $\underline{\lambda}$ and inertial type $\tau$ . We define $\mathcal{X}_{d,\mathrm{crys}}^{\underline{\lambda},\tau}$ to be the scheme-theoretic image of $\mathcal{C}_{d,\mathrm{crys},h}^{L/K,\mathrm{fl},\underline{\lambda},\tau}$ in $\mathcal{X}$ .

It is known, via what we know about the corresponding versal rings $R_{d,\mathrm{crys}}^{\underline{\lambda},\tau}$ , that the moduli stacks $\mathcal{X}_{d,\mathrm{crys}}^{\underline{\lambda},\tau}\otimes_{\mathrm{Spf}\mathcal{O}}\mathbb{F}$ are equidimensional of dimension equal to the quantity

$\displaystyle \sum_{\sigma}\#\lbrace1\leq i<j\leq d\vert\lambda_{\sigma,i}\geq\lambda_{\sigma,j}\rbrace$

In particular, if $\underline{\lambda}$ is a regular Hodge type, then this quantity is equal to $[K:\mathbb{Q}_{p}]d(d-1)/2$ . This plays a role in the formulation of the geometric Breuil-Mezard conjecture as we shall see later.

The reduced substack $\mathcal{X}_{d,\mathrm{red}}$

Let us now consider the reduced substack $\mathcal{X}_{d,\mathrm{red}}$ . This is an algebraic stack of finite presentation over $\mathbb{F}_{p}$ , equidimensional of dimension $[K:\mathbb{Q}_{p}]d(d-1)/2$ , and its irreducible components are labeled by Serre weights.

To see more explicitly the geometry of $\mathcal{X}_{d,\mathrm{red}}$ let us focus on the case $K=\mathbb{Q}_{p}$ and $d=1,2$ .

For $d=1$ we are looking at characters $G_{K}\to\overline{\mathbb{F}}_{p}^{\times}$ . These are of the form $\mathrm{ur}_{a}\overline{\varepsilon}^{i}$ . In the picture of $(\varphi,\Gamma)$ -modules, these are obtained from the trivial $(\varphi,\Gamma)$ -module over $\mathbf{A}_{\mathbb{Q}_{p},\mathbb{F}_{p}}$ by twisting $\varphi$ by $a$ and twisting $\Gamma$ by $\overline{\varepsilon}^{i}$ . For each $i$ the representations are therefore parametrized by $\mathbb{G}_{m}$ , but we also have automorphisms parametrized by $\mathbb{G}_{m}$ .

For $d=2$ , the irreducible representations are of the form $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega_{2}^{i}$ . These form a $0$ -dimensional substack inside $\mathcal{X}_{2}$ . The reducible ones which are of the form

$\displaystyle \begin{pmatrix}\mathrm{ur}_{ab}\overline{\varepsilon}^{i}&*\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{j}\end{pmatrix}$

will belong to the irreducible component of $\mathcal{X}_{2,\mathrm{red}}$ labeled by the Serre weight $\mathrm{Sym}^{i-j-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{j}$ (this is unambiguous except in the case where $i-j=1$ or $i-j=p$ , in which case the component labeled by $i-j=1$ is one where the representations with $a=1$ are dense, and the component labeled by $i-j=p$ is one where the representations with $a\neq 1$ are dense). Such a representation will correspond to a closed point if it is semisimple.

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

Suppose we have a family of $d$ -dimensional Galois representations $\overline{\rho}_{T}$ parametrized by a reduced finite scheme $T$ (this family corresponds to a map $T\to\mathcal{X}_{\mathrm{red}}$ ). Let $\overline{\alpha}$ be a fixed Galois representation of dimension $a$ .

The theory of the Herr complex then allows us to find a bounded complex of finite rank locally free $\mathcal{O}_{T}$ -modules

$C_{T}^{0}\to C_{T}^{1}\to C_{T}^{2}$

whose cohomology computes $H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ (the finite type points of $H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ correspond to the usual Galois cohomology). If $\mathrm{Ext}^{2}(\overline{\alpha},\overline{\rho}_{T})$ is locally free of some rank $r$ , then we have a bounded complex of finite rank locally free $\mathcal{O}_{T}$ -modules

$C_{T}^{0}\to Z_{T}^{1}$

where $Z_{T}^{1}$ is defined to be the kernel of the map $C_{T}^{1}\to C_{T}^{2}$ . There is a surjection $Z_{T}^{1}\twoheadrightarrow H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ .

Let $V$ be the vector bundle over $T$ corresponding to $Z_{T}^{1}$ , and let $\overline{\rho}_{V}$ be the pullback of $\overline{\rho}_{T}$ to $V$ . Then we can use the surjection $Z_{T}^{1}\twoheadrightarrow H^{1}(G_{K},\overline{\rho}_{T}\otimes\overline{\alpha}^{\vee})$ to construct a “universal extension” $\mathcal{E}_{V}$ that fits into the following exact sequence:

$\displaystyle 0\to\overline{\rho}_{V}\to\mathcal{E}_{V}\to\overline{\alpha}\to 0$

This universal extension $\mathcal{E}_{V}$ is a family of Galois representations parametrized by $V$ , i.e. a map $\mathcal{E}_{V}\to \mathcal{X}_{d+a, \mathrm{red}}$ . Being able to construct families of higher-dimensional Galois representations as extensions of lower-dimensional ones helps us study the moduli stacks of Galois representations for any dimension, and is used for instance, to prove the earlier stated facts about the dimension and irreducible components of these moduli stacks.

The “coarse moduli space” and the Bernstein center

Let us now look at a “coarse moduli space” $X$ associated to $\mathcal{X}_{d}^{\mathrm{det}=\psi}$ . This coarse moduli space $X$ is a moduli space of pseudorepresentations. The associated reduced space $X_{\mathrm{red}}$ should be a chain of projective lines, as we shall shortly explain.

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

$\displaystyle \begin{pmatrix}\mathrm{ur}_{a}\overline{\varepsilon}^{i}&0\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{j}\end{pmatrix}$

and from our earlier discussion we can associate to it the Serre weight $\mathrm{Sym}^{i-j-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{j}$ . But we can also see this as

$\displaystyle \begin{pmatrix}\mathrm{ur}_{a}\overline{\varepsilon}^{j}&0\\0&\mathrm{ur}_{b}\overline{\varepsilon}^{i}\end{pmatrix}$

and now the associated Serre weight is $\mathrm{Sym}^{j-i-1}\overline{\mathbb{F}}^{2}\otimes \mathrm{det}^{i}$ . Therefore there are two Serre weights that we can associate to this reducible mod p Galois representation! Now if we fix the determinant of our Galois representation to be, say $\overline{\varepsilon}$ , then besides the two Serre weights our Galois representation only depends on the parameter $a$ (because in this case we must have $\mathrm{ur}_{b}=\mathrm{ur}_{a}^{-1}$ ).

We can consider our two Serre weights now to be the $0$ and $\infty$ of a projective line (these points will also correspond to irreducible representations) and the points of the projective line in between these gives us the values of the parameter $a$ which parametrize the reducible representations. But a “ $0$ ” Serre weight could also be considered as the “ $\infty$ ” Serre weight associated to another family of Galois representations. Therefore we have a chain of projective lines parametrizing our semisimple Galois representations. This is the reduced space $X_{\mathrm{red}}$ of our “coarse moduli space” $X$ .

One interesting application of these ideas, which is currently part of ongoing work by Andrea Dotto, Matthew Emerton, and Toby Gee, is that the category of mod p representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ (which we shall denote by $\mathcal{A}$ ) forms a stack over the Zariski site of $X_{\mathrm{red}}$ !

That is, to every Zariski open set of $X_{\mathrm{red}}$ , we can associate a category $\mathcal{A}_{U}$ and these categories glue together well and form a stack over the Zariski site of $X_{\mathrm{red}}$ . To define these categories $\mathcal{A}_{U}$ we need to use the theory of “blocks” developed by Vytautas Paskunas. Namely, Paskunas showed that the category of locally admissible representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ decomposes into “blocks” labeled by semisimple Galois representations $G_{\mathbb{Q}_{p}}\to\mathrm{GL}_{2}(\overline{\mathbb{F}})$ .

We can now construct the category $\mathcal{A}_{U}$ as follows. Let $Y$ be a closed subset of $X_{\mathrm{red}}$ . Then we define $\mathcal{A}_{Y}$ to be the full sub category of $\mathcal{A}$ consisting of all representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ whose irreducible subquotients live in blocks labeled by the $\overline{\mathbb{F}}_{p}$ -points of $Y$ (since these correspond to semisimple Galois representations, which in turn label the blocks). Then for $U$ an open subset of $X_{\mathrm{red}}$ , we define $\mathcal{A}_{U}$ to be the Serre quotient $\mathcal{A}/\mathcal{A}_{Y}$ , where $Y=X_{\mathrm{red}}\subset U$ .

If $\mathcal{C}$ is an additive category, the Bernstein center of $\mathcal{C}$ , denoted $Z(\mathcal{C})$ , is defined to be the ring of endomorphisms of the identity functor $\mathcal{C}\to\mathcal{C}$ .

It is expected that $\mathcal{A}$ , the category of mod p representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , forms a sheaf over $X$ , and the Bernstein center $Z(\mathcal{A})$ coincides with the structure sheaf $\mathcal{O}_{X}$ of $X$ .

Relation to p-adic local Langlands and modularity

Let us now discuss how the moduli stack $\mathcal{X}_{d}$ is related to the p-adic local Langlands correspondence and questions of modularity.

We want there to be a sheaf $\mathcal{M}$ on $\mathcal{X}_{d}$ which realizes the p-adic local Langlands correspondence. In the case $K=\mathbb{Q}_{p}$ and $d=2$ , we can apply a construction of Colmez to the universal $(\varphi,\Gamma)$ -module on $\mathcal{X}_{2}$ and obtain a quasi-coherent sheaf $\mathcal{M}=(D\boxtimes\mathbb{P}^{1})/(D^{\natural}\boxtimes\mathbb{P}^{1})$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ -representations on it. Being a quasi-coherent sheaf it has an action of the structure sheaf $\mathcal{O}_{\mathcal{X}_{2}}$ , which is expected to be the same as the action of the Bernstein center of the category of smooth $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ -representations on $\mathbb{Z }_{p}$ -modules which are locally $p$ -power torsion.

In the case when either $K=\mathbb{Q}_{p}$ or $d\neq 2$ , there is so far no known satisfactory analogue of Colmez’ construction. However, it is believed that if there is such a sheaf $\mathcal{M}$ it must coincide with a certain patched module construction $M_{\infty}$ , which is a module over the deformation ring $R_{\overline{\rho}}$ , after pulling back over the map $\mathrm{Spf}(R_{\overline{\rho}})\to\mathcal{X}_{d}$ .

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

Let $\underline{\lambda}$ be a Hodge type and let $\tau$ be an inertial type. Let $\xi_{\sigma}=(\xi_{\sigma,1},\ldots,\xi_{\sigma,d})$ , where $\xi_{\sigma,i}=\lambda_{\sigma,i}-(d-i)$ . Let $M_{\xi_{\sigma}}$ be the algebraic $\mathcal{O}_{K}$ -representation of $\mathrm{GL}_{d}(\mathcal{O}_{K})$ with highest weight $\xi_{\sigma}$ , and let $L_{\underline{\lambda}}=M_{\xi\sigma}\otimes_{\mathcal{O}_{K},\sigma}\mathcal{O}_{E}$ .

Now to the inertial type $\tau$ , there is an “inertial local Langlands correspondence” that associates to $\tau$ a smooth admissible representation $\sigma^{\mathrm{crys}}(\tau)$ of $\mathrm{GL}_{d}(\overline{\mathbb{Q}}_{p})$ over $\overline{\mathbb{Q}_{p}}$ . Let $\sigma^{\mathrm{crys},\circ}(\tau)$ be a $\mathrm{GL}_{d}(\mathcal{O}_{K})$ -stable $\mathcal{O}_{E}$ -lattice in $\sigma^{\mathrm{crys}}(\tau)$ , and let $\sigma^{\mathrm{crys}}(\lambda,\tau)=L_{\underline{\lambda}}\otimes_{\mathcal{O}_{E}}\sigma^{\mathrm{crys}}(\lambda,\tau)$ . Finally, we let $\overline{\sigma}^{\mathrm{crys}}(\lambda,\tau)$ be the semisimplification of $\sigma^{\mathrm{crys}}(\lambda,\tau)\otimes\mathbb{F}$ . We may now view this as an $\mathbb{F}$ representation of $\mathrm{GL}_{d}(k)$ , where $k$ is the residue field of $\mathcal{O}_{K}$ . Now let $F_{\underline{k}}$ be the irreducible $\mathbb{F}$ representation of $\mathrm{GL}_{d}(k)$ associated to the tuple $\underline{k}$ (these are higher-dimensional versions of the Serre weights discussed in The mod p local Langlands correspondence for GL_2(Q_p)). We have the decomposition

$\displaystyle \sigma^{\mathrm{crys}}(\lambda,\tau)=\bigoplus F_{\underline{k}}^{n_{\underline{k}}^{\mathrm{crys}}(\lambda,\tau)}$

Let $\mathcal{M}(\sigma^{\circ}(\lambda,\tau)):=\mathrm{Hom}_{\mathrm{GL}_{d}(\mathcal{O}_{K})}(\sigma^{\circ}(\lambda,\tau)^{\vee},\mathcal{M})$ . Let $\mathcal{Z}(\sigma^{\circ}(\lambda,\tau))$ be the support of $\mathcal{M}(\sigma^{\circ}(\lambda,\tau))$ on $\mathcal{X}_{d}$ . It is expected that $\mathcal{Z}(\sigma^{\circ}(\lambda,\tau))=\mathcal{Z}(\mathcal{X}_{d}^{\mathrm{crys},\underline{\lambda},\tau})_{\mathbb{F}}$ .

Let $\mathcal{M}(F_{\underline{k}}):=\mathrm{Hom}_{\mathrm{GL}_{d}(\mathcal{O}_{K})}(F_{\underline{k}}^{\vee},\mathcal{M})$ and let $\mathcal{Z}(F_{\underline{k}})$ be the support of $\mathcal{M}(F_{\underline{k}})$ on $\mathcal{X}_{d}$ . The geometric Breuil-Mezard conjecture states that

$\displaystyle \mathcal{Z}(\sigma^{\mathrm{crys}}(\lambda,\tau))=\sum_{\underline{k}} n_{\underline{k}}^{\mathrm{crys}}(\lambda,\tau)\mathcal{Z}(F_{\underline{k}})$ .

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

References:

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

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

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

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

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

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

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

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

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

The mod p local Langlands correspondence for GL_2(Q_p)

November 4, 2021 / Anton Hilado / 2 Comments

In The Local Langlands Correspondence for General Linear Groups, we stated the local Langlands correspondence for $\mathrm{GL}_n(F)$ where $F$ is a finite extension of $\mathbb{Q}_{p}$ . We recall that it states that there is a one-to-one correspondence between irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and F-semisimple Weil-Deligne representations of $\mathrm{Gal}(\overline{F}/F)$ . Here all the relevant representations are over $\mathbb{C}$ .

In this post, we will consider representations over a finite field $\mathbb{F}$ of characteristic $p$ . While we may hope that some sort of “mod p local Langlands correspondence” will also hold in this case, at the moment all we know is the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , which we will discuss in this post. It is a sort of stepping stone to the “p-adic local Langlands correspondence” (where the representations are over some extension of $\mathbb{Q}_{p}$ ), which, as in the mod p case, is only known for the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ . The p-adic local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ plays an important role in the proofs of the known cases of the Fontaine-Mazur conjecture, which concerns when a Galois representation comes from the etale cohomology of some variety.

Let us start by discussing some representation theory of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$ . This will be somewhat similar to our discussion in The Local Langlands Correspondence for General Linear Groups, as we will see later when we state the classification of the irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , but we will also need some new ingredients.

A Serre weight is an absolutely irreducible representation (absolutely irreducible means it is irreducible over the algebraic closure of $\mathbb{F}$ ) of $\mathrm{GL}_{2}(\mathbb{F}_{p})$ over $\mathbb{F}$ . This is the same as an absolutely irreducible smooth representation of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ over $\mathbb{F}$ .

Serre weights are completely classified and can be explicitly described. Let $r\in\lbrace 0,\ldots,p-1\rbrace$ and let $s\in \lbrace 0,\ldots, p-2\rbrace$ . Then a Serre weight is always of the form $\mathrm{Sym}^{r}\mathbb{F}^{2}\otimes\mathrm{det}^{s}$ .

The name “Serre weight” originates from its relationship to Serre’s modularity conjecture, which is a conjecture about when a residual representation comes from a modular form, and what the level and the weight of the modular form should be. Avner Ash and Glenn Stevens made the observation that a residual representation $\overline{\rho}$ is modular of weight $k$ ( $k\geq 2$ ) and level $\Gamma_{1}(N)$ if and only if $H^{1}(\Gamma_{1}(N),\mathrm{Sym}^{k-2}\mathbb{F}^{2})_{\mathfrak{m}_{\overline{\rho}}}$ (here $\mathfrak{m}_{\overline{\rho}}$ is a certain maximal ideal of the Hecke algebra associated to $\overline{\rho}$ ) is nonzero.

For convenience, from here on in this post we shall consider Serre weights not just as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ but as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}$ , which sends the uniformizer of $\mathbb{Q}_{p}$ to $1$ .

Serre weights are important because from them we can obtain induced representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ . Let $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ be the representation of $\mathrm{GL} _{2}(\mathbb{Q}_{p})$ coming from compactly supported functions from $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ to $\sigma$ which satisfy $f(kg)=\sigma(k)f(g)$ .

The endomorphisms of $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ form the Hecke algebra, which is isomorphic to $\mathbb{F}[T]$ . In other words, we can consider $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ as a module over $\mathbb{F}[T]$ , and we can take the quotient $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ . This quotient is irreducible, and it is an important class of absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ called the supersingular representations.

The rest of the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ are very similar to what we discussed in The Local Langlands Correspondence for General Linear Groups. Namely, they can be obtained from principal series representations, which are induced representations of characters from the Borel subgroup $B(\mathbb{Q}_{p})$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ (i.e. the upper-triangular matrices in $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ ).

To recall, the principal representations are $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$ , which means they come from functions $f:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}$ such that $f(hg)=\chi_{1}\otimes\chi_{2}(h)f(g)$ for $h\in B(\mathbb{Q}_{p})$ and $g\in\mathrm{GL}_{2}(\mathbb{Q}_{p})$ , where $\chi_{1}$ and $\chi_{2}$ are characters of $\mathbb{Q}_{p}^{\times}$ , and $\chi_{1}\otimes\chi_{2}$ as a function on $B(\mathbb{Q}_{p})$ means it sends an element $\begin{pmatrix}a& b\\0& d\end{pmatrix}$ of $B(\mathbb{Q}_{p})$ to $\chi_{1}(a)\otimes \chi_{2}(d)$ .

In the case that $\chi_{1}\neq\chi_{2}$ , the principal series representations will be absolutely irreducible, in which case we obtain another class of absolutely irreducible admissible smooth representations. Otherwise, we may obtain absolutely irreducible representations as a quotient. These will be twists (this means a tensor product by a character) of the Steinberg representation, which is defined as the quotient $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ (here $\mathbf{1}$ is the trivial representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ ). This gives a third class of absolutely irreducible admissible representations. Finally we have the characters, which give a fourth class.

In summary, the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$ can be classified into the following four kinds as follows:

One-dimensional representations (characters) $\delta: \mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
Principal series representations $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$ for $\chi_{1},\chi_{2}: \mathbb{Q}_{p}^{\times}\to\mathbb{F^{\times}}, \chi_{1}\neq\chi_{2}$
Twists of Steinberg representations $\delta\otimes\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ for $\delta:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
Supersingular representations $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ for $\sigma$ a Serre weight

Let us now discuss the other side of the correspondence, the “Galois side”. For ease of notation let us also denote $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ by $G_{\mathbb{Q}_{p}}$ .

Let $g$ be an element of the inertia subgroup of $G_{\mathbb{Q}_{p}}$ . Serre’s level 2 fundamental character $\omega_{2}$ is given by composing the map

$\displaystyle g\mapsto \frac{g(\sqrt[p^{2-1}]{-p})}{\sqrt[p^{2-1}]{-p}}$

which takes values in $\mathbb{\mu}_{p^{2}-1}$ with the isomorphism $\mu_{p^{2}-1}\xrightarrow{\sim}\mathbb{F}_{p^{2}}^{\times}$ .

Let $h$ be a natural number which is mutually prime to $p+1$ . We have that $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega_{2}^{h}$ is an irreducible 2-dimensional representation of $G_{\mathbb{Q}_{p}}$ . In fact, any absolutely irreducible representation of $G_{\mathbb{Q}_{p}}$ over $\mathbb{F}$ is of this form, possibly tensored with the unramified character $\lambda_{a}$ which takes the inverse of the Frobenius to $a\in\mathbb{F}^{\times}$ .

The mod p local Langlands correspondence is now the bijection described explicitly as follows:

To the supersingular representation $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ where $\sigma=\mathrm{Sym}^{r}\mathbb{F}^{2}$ , we associate the Galois representation $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega^{r+1}$ .

To the representation $\pi$ which is obtained as the extension $0\to\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}\varepsilon^{-1}\to\pi\to \mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{2}\otimes\chi_{1}\varepsilon^{-1}\to 0$ we associate the Galois representation $\overline{\rho}$ which is obtained as an extension $0\to\chi_{1}\to\overline{\rho}\to\chi_{2}\to 0$ . Here $\varepsilon$ is the reduction mod p of the p-adic cyclotomic character, and $\chi_{1}$ and $\chi_{2}$ are characters $\mathbb{Q}_{p}^{\times}\to\mathbb{F}^{\times}$ which are not equal to each other nor to the product of the other by the p-adic cyclotomic character or its inverse.

The p-adic local Langlands correspondence, which, as stated earlier concerns representations over some finite extension of $\mathbb{Q}_{p}$ and is important in the Fontaine-Mazur conjecture, needs to be compatible with the mod p local Langlands correspondence as well. Its statement is more involved than the mod p local Langlands correspondence, and its proof involves $(\varphi,\Gamma)$ -modules. We reserve further discussion of the p-adic local Langlands correspondence to future posts.

References:

The emerging p-adic Langlands programme by Christophe Breuil

Representations of Galois and of GL_2 in characteristic p by Christophe Breuil

Towards a modulo p Langlands correspondence for GL_2 by Christophe Breuil and Vytautas Paskunas

Completed Cohomology

November 1, 2021February 22, 2022 / Anton Hilado / 1 Comment

Let $F$ be a number field, and let $G_{F,S}$ be the Galois group over $F$ of the maximal extension of $F$ unramified outside a finite set of primes $S$ . It should follow from the Langlands correspondence that $n$ -dimensional continuous (we shall only be talking about continuous Galois representations in this post, so we omit the word “continuous” from here on) representations $\rho:G_{F,S}\to \mathrm{GL_{n}}(\overline{\mathbb{Q}}_{p})$ should correspond to certain automorphic representations $\pi$ of $\mathrm{GL}_{n}$ unramified outside $S$ (see also Automorphic Forms).

The Fontaine-Mazur-Langlands conjecture further states that such Galois representations $\rho$ that are irreducible and “geometric” (i.e. its restrictions to the primes above $p$ satisfy some conditions related to p-adic Hodge theory, see also p-adic Hodge Theory: An Overview) should match up with “algebraic” (we shall explain this shortly) cuspidal $\pi$ . Furthermore this conjecture expects that certain “Hodge numbers” associated to the Galois representation $\rho$ via p-adic Hodge theory should match up to “Hodge numbers” defined for the automorphic representation $\pi$ via its “infinitesimal character” at the archimedean primes (note that they are defined differently, since they are associated to different kinds of representations; they only share the same name because they are expected to coincide).

Generally, whether $\rho$ is “geometric” or not, its Hodge numbers going to be $p$ -adic numbers, and whether $\pi$ is “algebraic” or not, its Hodge numbers are complex numbers. However, if $\rho$ is geometric, then its Hodge numbers are integers, and if $\pi$ is algebraic, its Hodge numbers are also integers (in fact the definition of “algebraic” here just means that its Hodge numbers are integers), and this allows us to match them up.

To see things in a little more detail, let us consider the case of a $1$ -dimensional representation $\rho:G_{F,S}\to \overline{\mathbb{Q}}_{p}$ . We have seen in Galois Representations that an example of this is given by the p-adic cyclotomic character which we can also view as follows. Let $S=\lbrace p,\infty\rbrace$ . Let $G_{F,S}^{\mathrm{ab}}$ be the abelianization of $G_{F,S}$ . It follows from the Kronecker-Weber theorem that $G_{F,S}^{\mathrm{ab}}$ is isomorphic to $\mathbb{Z}_{p}^{\times}$ , and it is precisely the p-adic cyclotomic character that gives this isomorphism. Since $\mathbb{Z}_{p}^{\times}$ embeds into $\overline{\mathbb{Q}}_{p}^{\times}$ , which is also $\mathrm{GL}_{1}(\overline{\mathbb{Q}_{p}})$ , we have our $1$ -dimensional Galois representation. We can also take a power of the p-adic cyclotomic character to get another $1$ -dimensional Galois representation.

But the p-adic cyclotomic character and its powers are not the only $1$ -dimensional Galois representations. For instance, we have a map from $\mathbb{Z}_{p}^{\times}\to \mathbb{Q}_{p}^{\times}$ given by reducing $\mathbb{Z}_{p}$ mod $p^{r}$ and then composing it with the map $\chi$ that sends this element of $(\mathbb{Z}/p^{r})^{\times}$ to the corresponding $p^{r}$ -th root of unity in $\overline{\mathbb{Q}}_{p}^{\times}$ . This is a finite-order character. We also have another map from $\mathbb{Z}_{p}^{\times}\to \overline{\mathbb{Q}}_{p}^{\times}$ which sends $x$ to $x^{s}$ , for some $s$ in $\overline{\mathbb{Q}}_{p}$ such that $\vert s\vert<\frac{p}{p-1}$ . If we compose the p-adic cyclotomic character with either of these maps, we get another $1$ -dimensional Galois representation. It turns out the Hodge number of the latter representation is given by $s$ .

The $1$ -dimensional Galois representations form a rigid analytic space (see also Rigid Analytic Spaces), and their Hodge numbers form p-adic analytic functions on this space. The geometric representations are the ones that are from a power of the p-adic cyclotomic character composed with a finite-order character, and these form a countable dense subset of this rigid analytic space.

Some form of this phenomena happens more generally for higher dimensional Galois representations – they form a rigid analytic space and the geometric ones are a subset of these.

It is convenient that our Galois representations form a rigid analytic space, and suppose we want to do something similar for our automorphic representations. The problem is that the automorphic representations aren’t really “p-adic”, as we may see from the fact that their Hodge numbers are complex instead of p-adic. This is the problem that p-adically completed cohomology, also simply known as completed cohomology, aims to solve.

Let us look at how we want to find automorphic representations in cohomology. Let $G_{\infty}=\mathrm{GL_{n}}(F\otimes_{\mathbb{Q}}\mathbb{R})$ . If $F$ has $r_{1}$ real embeddings and $r_{2}$ complex embeddings, then $G_{\infty}$ will be isomorphic to $\mathrm{GL}_{n}(\mathbb{R})^{r_{1}}\times\mathrm{GL}_{n}(\mathbb{C})^{r_{2}}$ . Let $K_{\infty}^{\circ}$ be a maximal connected compact subgroup of $G_{\infty}$ . With $r_{1}$ and $r_{2}$ as earlier, $K_{\infty}^{\circ}$ will be isomorphic to $\mathrm{SO}(n)^{r_{1}}\times \mathrm{U}(n)^{r_{2}}$ .

Let $X$ be the quotient $G_{\infty}/\mathbb{R}_{>0}^{\times}K_{\infty}^{\circ}$ . This is an example of a symmetric space – for example, if $F=\mathbb{Q}$ and $n=2$ , $X$ is going to be $\mathbb{C}\setminus \mathbb{R}$ .

The space $X$ has an action of $G_{\infty}$ , and its subgroup $\mathrm{GL}_{n}(\mathcal{O}_{F})$ . Letting $N\geq 1$ , we may therefore take the quotient

$\displaystyle Y(N)=\mathrm{GL}_{n}(\mathcal{O}_{F})\backslash (X\times \mathrm{GL}_{n}(\mathcal{O}_{F}/N\mathcal{O}_{F}))$

For example, if $F=\mathbb{Q}$ and $n=2$ , then $Y(N)$ consists of copies of the (uncompactified) modular curve of level $N$ (the number of copies is equal to the number of primes less than $N$ ).

It is this space $Y(N)$ whose cohomology we are interested in. For instance $H^{i}(Y(N),\mathbb{C})$ is related to automorphic forms by a theorem of Jens Franke. However, it is complex, and not the p-adically varying one that we want. There is an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ , but the important part of this cohomology comes from the cohomology with $\mathbb{Q}$ coefficients, which is unchanged when we do this isomorphism, and therefore does not really add anything.

This is now where we introduce completed cohomology. Let us require that $N$ and $p$ be mutually prime. We define the completed cohomology $\widetilde{H}^{i}$ as follows:

$\displaystyle \widetilde{H}^{i}:=\varprojlim_{s\geq 1}\varinjlim_{r\geq 0}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

The order of the limits here is important (we will see shortly what happens when they are interchanged). By first taking the direct limit we are essentially considering the union of $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ for all $r$ with $\mathbb{Z}/p^{s}\mathbb{Z}$ coefficients. This is a very big abelian group that might not even be finitely generated. Then the inverse limit means we are taking the $p$ -adic completion – having this as the last step guarantees that the result is something that is p-adically complete (hence the name p-adically completed cohomology). So the completed cohomology $\widetilde{H}^{i}$ is a p-adically complete module over $\mathbb{Z}_{p}$ , which again may not be finitely generated. Taking the tensor product of $\widetilde{H}^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ gives us a vector space $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ which moreover is a Banach space.

Let us consider now what happens if the order of the limits were interchanged. Let us denote the result by $H^{i}$ :

$\displaystyle H^{i}:=\varinjlim_{r\geq 0}\varprojlim_{s\geq 1}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

By taking the inverse limit first we are simply considering $H^{i}(Y(Np^{r}),\mathbb{Z}_p$ , and taking the direct limit means we are taking the union of $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ for all $r$ . If we take the tensor product of $H^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ , then what we get is $H_{\mathbb{Q}_{p}}^{i}$ , the union of $H^{i}(Y(Np^{r}),\mathbb{Q}_p$ for all $r$ . Being the cohomology with characteristic zero coefficients, this may once again be related to the automorphic forms, as earlier.

Therefore, $H_{\mathbb{Q}_{p}}^{i}$ , via the Fontaine-Mazur-Langlands conjecture, should be related to the geometric Galois representations. Now it happens that we can actually embed $H_{\mathbb{Q}_{p}}^{i}$ into the completed cohomology $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ , because there is a map from $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ to $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ , and then we can take the direct limit over $r$ followed by the inverse limit over $r$ and then tensor over $\mathbb{Q}_{p}$ as previously.

This embedding of $H_{\mathbb{Q}_{p}}^{i}$ into $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should now bring to mind the picture with the geometric Galois representations which sit inside the rigid analytic space of Galois representations which may not necessarily be geometric, as discussed earlier. It is in fact a conjecture that $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should know about the rigid analytic space of Galois representations.

In the case $F=\mathbb{Q}$ and $n=2$ , the completed cohomology is some space of p-adic modular forms, and there is much that is known via the work of Matthew Emerton, who also showed that the p-adic local Langlands correspondence appears inside the completed cohomology. This has led to a proof of many cases of the Fontaine-Mazur conjecture for $2$ -dimensional odd Galois representations.

We have only provided a rough survey of the motivations behind the theory of completed cohomology in this post. We will discuss further deeper aspects of it, and its relations to the p-adic local Langlands correspondence and the Fontaine-Mazur conjecture in future posts.

References:

Completed cohomology and the p-adic Langlands correspondence by Matthew Emerton on YouTube

Completed cohomology and the p-adic Langlands program by Matthew Emerton

Completed cohomology – a survey by Frank Calegari and Matthew Emerton

Theories and Theorems

Math and Physics for Everyone

Month: November 2021

Moduli Stacks of (phi, Gamma)-modules

The ind-algebraicity of the moduli stack $\mathcal{X}_{d}$

Crystalline moduli stacks

The reduced substack $\mathcal{X}_{d,\mathrm{red}}$

The “coarse moduli space” and the Bernstein center

Relation to p-adic local Langlands and modularity

The mod p local Langlands correspondence for GL_2(Q_p)

Completed Cohomology