In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

Solution 1:

So, the problem as I see it is as follows. The term '$p$-adic Hodge theory' can mean one of two things:

1. The study of $p$-adic representations of $p$-adic fields.
2. Comparison theorems.

What you're really asking about is the second of these two. Comparison theorems are theorems telling us the geometric $p$-adic representations are commensurate with the type of representations one might study in the pursuit of 1.

The first comparison theorem is due to Faltings, building on the work of those such as Tate (who first observed the decomposition below for abelian varieties), and others:

Theorem(Faltings): Let $K$ be a $p$-adic field, and $X/K$ be smooth proper. Then, there is an isomorphism $$\mathbb{C}_K\otimes_{\mathbb{Q}_p}H^n_\text{et}(X_{\overline{K}},\mathbb{Q}_p)\cong \bigoplus_q\left(\mathbb{C}_K(-q)\otimes_K H^q(X,\Omega^{n-q}_{X/K})\right)$$ in the category $\text{Rep}_{\mathbb{C}_K}(G_K)$.

Maybe some explanation of terms are in order here. A $p$-adic field is just a characteristic $0$ complete field with perfect residue field of characteristic $p$ (e.g. $\mathbb{Q}_p$ or $\mathbb{Q}_p^\text{ur}$). As you might guess, $\mathbb{C}_K=\widehat{\overline{K}}$ (i.e. if $K=\mathbb{Q}_p$ then $\mathbb{C}_K=\mathbb{C}_p$). Note that since $G_K:=\text{Gal}(\overline{K}/K)$ acts on $\overline{K}$ by isometries, it extends to an action of $G_K$ on $\mathbb{C}_K$.

The category $\text{Rep}_{\mathbb{C}_K}(G_K)$ denotes the category of $\mathbb{C}_K$-semilinear representations of $G_K$. This means an action of $G_K$ on a finite dimensional $\mathbb{C}_K$-vector space $V$, such that instead of $g(\alpha x)=\alpha g(x)$, $\alpha\in\mathbb{C}_K$, and $v\in V$ (i.e. $\mathbb{C}_K$-linearity), we have that $g(\alpha x)=g(\alpha)g(x)$ (i.e. $\mathbb{C}_K$-semilinearity). This is analogous to a complex vector space with a conjugate-linear action of a group.

Finally, I should tell you how $G_K$ acts on both sides of this isomorphism. On the left it acts diagonally (both have a $G_K$ action). On the right hand side, it just acts on the factors $\mathbb{C}_K(-q)$ (this is the Tate-twist, as I'm sure you know).

Now, this theorem looks very similar to normal Hodge theory for $X/\mathbb{Q}$ a smooth projective variety:

$$H^n_\text{sing}(X^\text{an},\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}\cong\bigoplus_q \left(\mathbb{C}\otimes_{\mathbb{Q}}H^q(X,\Omega^{n-q}_{X/\mathbb{Q}})\right)$$ (where I've made judicious use of GAGA, of course).

So, how to comparison theorems like Faltings' (Faltings's?) theorem above, relate to $p$-adic Hodge theory as stated in 1.? Well, as $p$-adic Hodge theorists, we are trying to study the category $\text{Rep}_{\mathbb{Q}_p}(G_K)$ of $\mathbb{Q}_p$-linear (terrible notation, I know) $G_K$-representations. These are often way too hard to understand in the abstract since the wild inertia group of $G_K$ can act highly non-trivially (see, also in this vein, Grothendieck's $\ell$-adic monodromy theorem). So, we often times want to focus our attention on simpler types of representations.

One way of specifying these simpler types of representations is hoping that when we pass from the hard category $\text{Rep}_{\mathbb{Q}_p}(G_K)$ to the easier category $\text{Rep}_{\mathbb{C}_p}(G_K)$ (by $V\leadsto V\otimes_{\mathbb{Q}_p}\mathbb{C}_K$) things behave nicely. In particular, we may hope that upon moving to this simpler category $\text{Rep}_{\mathbb{C}_K}(G_K)$ our representation breaks into very simple pieces. The simplest such pieces (as codified by the Sen-Tate theorems: $H^0(G_K,\mathbb{C}_K(r))=0$ for $r\ne 0$, and $H^0(G_K,\mathbb{C}_K)=K$) are those which are just sums of representations of the form $\mathbb{C}_K(q)$, for $q\in\mathbb{Z}$.

It turns out that there is a nice way of canonically phrasing when an element $V$ of $\text{Rep}_{\mathbb{C}_K}(G_K)$ breaks into these very simple pieces. Namely, let's define a functor

$$D:\text{Rep}_{\mathbb{C}_K}(G_K)\to \text{Gr}_{K,f}$$

where the right-hand category is the category of finite-dimensional graded $K$-spaces. The functor acts as follows:

$$V\leadsto \bigoplus_{q\in\mathbb{Z}}\left(\mathbb{C}_K(q)\otimes_{\mathbb{C}_K} V\right)^{G_K}\cong \left(B_\text{HT}\otimes_{\mathbb{C}_K} V\right)^{G_K}$$

where $B_{\text{HT}}$ is the graded ring $\mathbb{C}_K[T,T^{-1}]$ with the $G_K$-action $g(T^i)=\chi_p(g)^i T^i$ (where $\chi_p$ is the $p$-adic cyclotomic character of $G_K$). Note that this is very not obviously a finite-dimensional $K$-space--this follows, once again, from the theorem of Sen-Tate and a clever lemma of Serre-Tate.

In fact, there is actually a canonical injective embedding

$$\bigoplus_q\left(\mathbb{C}_K(-q)\otimes\left(\mathbb{C}_K(q)\otimes_{\mathbb{C}_K}V\right)^{G_K}\right)\hookrightarrow V$$

which is an isomorphism if and only if $V$ decomposes as a direct sum of these simple pieces $\mathbb{C}_K(q)$ (this is something to be checked). If this holds, we call $V$ Hodge-Tate. We call $W\in\text{Rep}_{\mathbb{Q}_p}(G_K)$ Hodge-Tate if moving to $\text{Rep}_{\mathbb{C}_K}(G_K)$ (i.e. $W\leadsto W\otimes_{\mathbb{Q}_p}\mathbb{C}_K$) is Hodge-Tate.

If, for $W\in\text{Rep}_{\mathbb{Q}_p}(G_K)$ we're willing to denote $D(W\otimes_{\mathbb{Q}_p}\mathbb{C}_K)$ by just $D(W)$, then Faltings theorem can then be stated in the following pleasant way:

Theorem(Faltings): Let $K$ be a $p$-adic field, and $X/K$ a smooth proper variety. Then, $H^n_\text{et}(X_{\overline{K}},\mathbb{Q}_p)$ is Hodge-Tate, and $$D\left(H^n_\text{et}(X_{\overline{K}},\mathbb{Q}_p)\right)\cong H^n_\text{Hodge}(X/K)$$

where

$$H^n_\text{Hodge}(X/K)=\bigoplus_q H^{n-q}(X,\Omega^q_{X/K})$$

(note these are all graded vector spaces!).

Thus, from this perspective, the comparision theorems (those which most resemble normal Hodge theory) factor into the theory of $p$-adic Galois representations by saying that geometric representations are 'nice' (i.e. are Hodge-Tate).

Faltings comparison theorem is just the first in a long line of Comparision theorems ($C_\text{HT}$,$C_{\text{dR}}$,$C_{\text{cris}}$, and $C_{\text{st}}$ to name a few) which say that geometric representations (in various forms) are nicer, and nicer types of $p$-adic representations. These correspond replacing the ring $B_{\text{HT}}$ above with 'finer rings' (called, imaginatively, $B_{\text{dR}},B_\text{cris},B_\text{st}$) and replacing $\text{Gr}_{K,f}$ with more structured categories. These are called period rings, the explanation of which could be the subject of another post.

Hope this helps!