p-adic Hodge Theory: An Overview

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

p-adic Hodge theory on Wikipedia

de Rham cohomology on Wikipedia

Hodge theory on Wikipedia

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

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

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

Reciprocity laws and Galois representations: recent breakthroughs Jared Weinstein

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