# Moduli Stacks of (phi, Gamma)-modules

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

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

# p-adic Hodge Theory: An Overview

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References: