Author: Eiko

Tags: p-adic, hodge theory, comparison theorems, number theory

Time: 2024-11-21 20:06:02 - 2024-11-21 23:41:13 (UTC)

These are notes taken for the talk p-adic Hodge Theory for local systems.

p-adic Hodge Theory For Local Systems

Motivation

is to find a p-adic analogue of the de Rham cohomology and Hodge decomposition. Consider any smooth complex projective variety $M$,

$$H_{sing}^n(M,\mathbb{Q})\otimes \mathbb{C}\cong H_{dR}^n(M)=\underset{p+q=n}{⨁}{H}^q(X,{\mathrm{\Omega }}^p)$$

$p$-adic setup

Let $K/{\mathbb{Q}}_p$ be a finite extension. For a smooth proper variety $X/K$ we can consider the following cohomologies:

$$\{\begin{array}{l}{H}_{et}^n(X):=H_{et}^n(X_{\bar{K}},{\mathbb{Q}}_p)\\ H_{dR}^n(X):=H_{dR}^n(X/K)\end{array}$$

We want an isomorphism somewhere in the cohomology or in the decomposition. Observe that in the first isomorphism in order to compare cycles and differentials you have to put periods in, and base change from rational to complex numbers because these periods are generally transcendental.

Fontain Defined De Rham Period Ring $B_{dR}\supset \bar{K}$

Theorem (de Rham comparison)

$$H_{et}^n(X){\otimes }_{{\mathbb{Q}}_p}{B}_{dR}\cong H_{dR}^n(X/K){\otimes }_K{B}_{dR}$$

this is the analogue of the first isomorphism above.

Corollary

$$H_{et}^n(X){\otimes }_{{\mathbb{Q}}_p}{\mathbb{C}}_p\cong \underset{i+j=n}{⨁}{H}^j(X,{\mathrm{\Omega }}^i){\otimes }_K{\mathbb{C}}_p(-i)$$

where ${\mathbb{C}}_p$ is the completion of an algebraic closure of ${\mathbb{Q}}_p$ (or $K$).

Galois representations

We want to understand these the comparison theorems from the point of view of Galois representations. The Galois group $G_K:=\mathrm{Gal}(\bar{K}/K)$ acts on the etale cohomology and on the de Rham cohomology. The comparison theorems are about the action of $G_K$ on these cohomologies.

• $H_{et}^n(X)$ has action of $G_K$, so does $B_{dR}$.

A Galois representation $V\in {\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_{{\mathbb{Q}}_p}$ of $G_K$ is called de Rham, if $V{\otimes }_{{\mathbb{Q}}_p}{B}_{dR}\cong B_{dR}^{{\dim }_{{\mathbb{Q}}_p}V}$ as $G_K$-representations.

$$\{\text{de Rham Galois representations}\}\subset \{\text{Galois representations}\}$$

Local systems

Observation. Galois groups are etale fundamental groups for specturm of fields

$$\mathrm{Gal}(\bar{K}/K)\cong {\pi }_1^{et}(\mathrm{Spec}K)$$

So the above is all about representations of ${\pi }_1^{et}(\mathrm{Spec}(K))$. How about changing $\mathrm{Spec}(K)$ to a general variety $S$? This is the $p$-adic Hodge theory for ${\mathbb{Q}}_p$-representations of ${\pi }_1^{et}(S)$, (or local systems on $S$).

Example

Consider a smooth proper map $f:X\to S$ with $S$ a smooth curve over $K$. Then $f$ induces a local system $R^n{f}_{\ast }{\mathbb{Q}}_p$ on $S$ (I think here ${\mathbb{Q}}_p$ is talking about etale sheaf), whose fibre at $s\in S$ is the etale cohomology of fibre $H_{et}^n(X_s)$ (integration on the fibre? owo)

This can be viewed as a family of Galois representations parametrized by $S$, since the fibres of local systems are exactly the Galois representations of the etale fundamental group ${\pi }_{et}^1(s)=G_{k(s)}$ of the fibre.

Theorem (Liu-Zhu) If ${\mathcal{L}}_s$ is de-Rham, then any stalk of $\mathcal{L}$ is de-Rham.

So we can see de-Rham property as an invariant or property of this smaller category of representations.

For general Galois representations there is a general way to associate a invariant called the generalized Hodge-Tate weights, we can ask how these invariants behave for general local systems.

Theorem (Continued) The set of generalized Hodge-Tate weights for ${\mathcal{L}}_s$ is constant on $S$.