Potential semi-stability of p-adic representations coming from algebraic geometry

Last week, I participated in a summer school on arithmetic geometry and Langlands program. Although this is not my field of interests, I always give some respect those who work in those fields. I also did gave a motivational TA talk about couting points on elliptic curves. To be honest, I do not expect to learn much from this school but I got enough inspiration to read arithmetic stuff, which I am going to write.

Since I am an algebraic geometer, I'd like to present things in a geometric way. Let $p$ be a prime number and denote $\mathbb{Q}_p$ the $p$-adic local field. Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $G_K = \mathrm{Gal}(\overline{K}/K)$ the absolute Galois group. A $\ell$-adic representation of $G_K$ is a continuous morphism (of groups)

$$G_K \longrightarrow \mathrm{GL}_n(V),$$ where $V$ is a finite-dimensional $\mathbb{Q}_{\ell}$-vector space with $\ell \neq p$ a prime. We denote by $K^{unr}$ the maximal unramified extension and $I_K = \mathrm{Gal}(\overline{K}/K^{unr})$ the inertia group. We say that 

• $V$ has good reduction if $I_K$ acts trivially on $V$.
• $V$ has potentially good reduction if there exists a finite extension $L/K$ so that $V$ as a representation of $G_L$ has good reduction.
• $V$ is semi-stable if the image of $I_K$ in $\mathrm{GL}(V)$ is unipotent.
• $V$ is potentially semi-stable if there exists a finite extension $L/K$ so that $V$ as a representation of $G_L$ is semi-stable.

We are going to see that, examples coming from algebraic geometry is potentially semi-stable.

1. (Cyclotomic character) For each $n \geq 0$, we see that the roots of unity $\mu_{\ell^n}(\overline{K}) \simeq (\mathbb{Z}/\ell^n\mathbb{Z})$ (non-canonically, as it depends on a choice of root of unity) and they form an inverse system $\mu_{\ell^{n+1}} \longrightarrow \mu_{\ell^n}$ and each of them carries an action of $G_K$. Thus, there is an action $G_K \longrightarrow \varprojlim \mu_{\ell^n}(\overline{K})$. By convention, we set the RHS to be $\mathbb{Z}_{\ell}(1)$, the Tate twist. It is a free $\mathbb{Z}_l$-module of rank $1$. Hence a morphism $G_K \longrightarrow \mathbb{Z}_{\ell}^{\times}$, called the cyclotomic character. Tensoring with $\mathbb{Q}_{\ell}$ we obtain $G_K \longrightarrow \mathbb{Q}_{\ell}^{\times}$.
   
2. (Elliptic curves) Let $E/K$ be an elliptic curve, then $G_K$ acts on $E(\overline{K})$. In particular, $G_K$ acts on $E[\ell^n] = \left \{P \in E(\overline{K}) \mid \ell^n P = 0 \right \}$. We know that $E[\ell^n]$ is a free $\mathbb{Z}/\ell^n\mathbb{Z}$-module of rank $2$ and the family $E[\ell^{n+1}] \longrightarrow E[\ell^n]$ forms an inverse system; hence they induce a represention (after taking limit and tensoring with $\mathbb{Q}_{\ell}$) $G_K \longrightarrow \mathrm{GL}_2(\mathbb{Q}_l)$.
   
3. (Etale cohomology) Let $X/K$ be a $K$-variety. Let $X_{\overline{K}}$ be its base change to $\overline{K}$ and $p \colon X_{\overline{K}} \longrightarrow X$ the projection. For each $g \in G_K = \mathrm{Gal}(\overline{K}/K)$, we get an isomorphism $$H^i_{et}(X_{\overline{K}},p^*\mathcal{F}) \longrightarrow H^i_{et}(X_{\overline{K}},g^*p^*\mathcal{F})$$ for any sheaf $\mathcal{F}$ on $X$. In the special case when $\mathcal{F} = \mathbb{Z}/\ell^n\mathbb{Z}$ the constant sheaf, we get an automorphism of $H^i_{et}(X_{\overline{K}},\mathbb{Z}/\ell^n\mathbb{Z})$ and hence there is an action of $G_K$ on $H^i_{et}(X_{\overline{K}},\mathbb{Z}/\ell^n\mathbb{Z})$. By taking projective limit over the system $$H^i_{et}(X_{\overline{K}},\mathbb{Z}/\ell^{n+1}\mathbb{Z}) \longrightarrow H^i_{et}(X_{\overline{K}},\mathbb{Z}/\ell^n\mathbb{Z})$$ and tensoring with $\mathbb{Q}_{\ell}$, we obtain the so-called $\ell$-adic representation given by the action of $G_K$ on $H^i_{et}(X_{\overline{K}},\mathbb{Q}_{\ell})$. If $X$ is proper and smooth over $K$, these representations are finite-dimensional. One can also consider cohomology with compact support. For instance, putting $X = \mathbb{P}^1$, one obtains the "dual" of the cyclotomic character.
   

Theorem 1 (Grothendieck). Let $X$ be a smooth, projective variety over $K$, then the $\ell$-adic representations $H^i_{et}(X_{\overline{K}},\mathbb{Q}_{\ell})$ are potentially semi-stable.

Before digging into the proof, let me stress that some of the above terminologies are from geometry. Let $\mathcal{O}_K$ denotes the ring of integers and $k$ its residue field. The triplet $(K,\mathcal{O}_K,k)$ is something for which I would call "a disk" in the sense that:

• $\mathcal{O}_K$ is the full disk.
• $K$ the generic fiber, is understood as a punctured disk.
• $k$ the special fiber, is nothing but the origin. 

For any variety $\mathcal{X} \longrightarrow \mathrm{Spec}(\mathcal{O}_K)$, we draw the commutative diagram by base change $\require{AMScd}$

$$\begin{CD} \mathcal{X}_{\eta} @>>> \mathcal{X} @<<< \mathcal{X}_s\\ @VVV @VVV @VVV \\ \eta = \mathrm{Spec}(K) @>>> \mathrm{Spec}(\mathcal{O}_K) @<<< s = \mathrm{Spec}(k) \end{CD}$$ In some sense, given a variety $X/K$, we expect it comes from a variety $\mathcal{X}/\mathcal{O}_K$, i.e., $\mathcal{X}_{\eta} = X$ and the special fiber $\mathcal{X}_s$ has good geometric properties. In such a case, we refer $\mathcal{X}$ as a model of $X$. More precisely, we say that

• $X$ has good reduction if there exists a smooth proper model $\mathcal{X}/\mathcal{O}_K$. Equivalently, we demand that $\mathcal{X}_s/k$ is smooth. 
• $X$ has semi-stable reduction if there exists a flat, proper model $\mathcal{X}/\mathcal{O}_K$ whose special fiber $\mathcal{X}_s$ has simple normal crossings.

If $X$ has good reduction or has semi-stable reduction, so do the corresponding $\ell$-adic representions $H^i_{et}(X_{\overline{K}},\mathbb{Q}_{\ell})$. The converse, however, is far from being true, even for elliptic curves. In some sense, a model (together with its special fiber) controls the generic fiber. This is the whole philosophy of nearby cycles functors because the special fiber is usually simpler than the generic fiber; specially in the case when resolutions of singularities are available, so one can transform problems on generic fibers to ones on special fibers. Grothendieck's theorem on potential semi-stability can be stated as follows (with some improvements by Gabber)

Theorem 2 (Grothendieck, Gabber). Let $\mathcal{X}/\mathcal{O}_K$ be a $\mathcal{O}_K$-variety. Under the assumption of theorem 2 for the generic fiber $X = \mathcal{X}_{\eta}$, there exists an open subgroup $I_1 \subset I$, independent of $l$, such that for all $g \in I_1$, $(g-1)^{i+1}=0$ on $H^i_c(X_{\overline{K}},\mathbb{Z}/\ell^n\mathbb{Z})$ or $H^i(X_{\overline{K}},\mathbb{Z}/\ell^n\mathbb{Z})$. 

Let us now go into the details.

Proposition 3. Let $\rho \colon G_K \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell})$ be a $l$-adic representation, then there exists an open subgroup $I_1 \subset I$ such that for all $g \in I_1$, $\rho(g)$ is unipotent.

We let $K^{tr}$ be the maximal tamely ramified extension of $K$, obtained by adding all $n$-th roots of a uniformizer of $K$ to $K^{unr}$, with $n$ prime to $p$. We consider the wild inertia group $P_K =  \mathrm{Gal}(\overline{K}/K^{tr})$, which is a pro $p$-group. There is an isomorphism 

$$I_K/P_K \longrightarrow \varprojlim_{(n,p)=1}\mu_n(\overline{K}) \simeq \prod_{q \neq p}\mathbb{Z}_q(1), \sigma \longmapsto \left(\frac{\sigma(\sqrt[q]{\pi})}{\sqrt[q]{\pi}} \right),$$ where $\pi$ denotes a uniformizer of $K$. Let $P_{K,\ell}$ be the inverse image of $\prod_{q \neq p,l} \mathbb{Z}_q$ in $I_K$ via the canonical surjection $I_K \longrightarrow I_K/P_K$. The group $P_{K,\ell}$ can be identified with the group $\mathrm{Gal}(\overline{K}/K^{tr,\ell})$, where $K^{tr,\ell}$ is the $\ell$-part of $K^{tr}$, which is obtained by adding all $\ell^n$-roots of a uniformizer to $K^{unr}$. We note that $\mathrm{Gal}(K^{tr,\ell}/K^{unr}) \simeq \mathbb{Z}_{\ell}(1)$ via

$$\sigma \longmapsto \left(\frac{\sigma(\sqrt[l^n]{\pi})}{\sqrt[l^n]{\pi}} \right),$$ and no power of $\ell$ can kill any element in $P_{K,\ell}$; heuristically speaking, the order of $P_{K,\ell}$ is prime to $\ell$. The following filtration maybe helpful

$$1 \subset P_K \subset P_{K,\ell} \subset I_K \subset G_K$$ and conversely

$$K \subset K^{unr} \subset K^{tr,\ell}  \subset K^{tr} \subset \overline{K}.$$ We have the following key step in proving arithmetic local monodromy theorem.

Lemma 4. Let $\rho \colon G_K \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell})$ be a $\ell$-adic represention, then $\rho(P_{K,\ell})$ is finite. 

Proof.  We will show that $\mathrm{Ker}(\rho_{\mid P_{K,\ell}})$ is open (hence it has finite index because $P_{K,\ell}$ is compact). By looking at valuations, we see that 

$$K_m = \left \{1 + \ell^m \mathrm{M}_n(\mathbb{Z}_{\ell}) \right \} \subset \mathrm{GL}_n(\mathbb{Q}_{\ell})$$ provided that $m \geq 1$. We have a descending sequence $K_{m+1} \subset K_m$ of open subgroups and each quotient $K_m/K_{m+1}$ is an abelian group killed by $l$. Let $V = (\rho_{\mid P_{K,\ell}})^{-1}(K_1)$ then $V$ is an open subgroup of $P_{K,\ell}$. We claim that $V = \mathrm{Ker}(\rho_{\mid P_{K,\ell}})$. At first, it is clear that $(\rho_{\mid P_{K,\ell}})(V) \subset K_1$ and $\mathrm{Ker}(\rho_{\mid P_{K,\ell}}) \subset V$. Since $V$ is an open subgroup of $P_{K,\ell}$, no power of $\ell$ can kill elements in $V$. As a consequence, the image of $(\rho_{\mid P_{K,\ell}})(V)$ in $K_1/K_2$ must be trivial as the latter group is killed by $\ell$. This proves that $(\rho_{\mid P_{K,\ell}})(V) \subset K_2$. Proceed the same way, we see that 

$$(\rho_{\mid P_{K,\ell}})(V) \subset \bigcap_{m \geq 1}K_m = 1.$$ Thus, $V \subset \mathrm{Ker}(\rho_{\mid P_{K,\ell}})$ and hence equality holds. 

We need some technical definition.

Definition 5. Let $t$ be a topological generator of $\mathbb{Z}_{\ell}(1)$. for any $b \in \mathbb{Z}_{\ell}$, we see $t^b$ to be the limit $\varprojlim_{m \in \mathbb{Z}, m \to b} t^m$. (The same definition makes sense for $b \in \hat{\mathbb{Z}}$)

Proposition 3 can be rephrased (and generalized a bit) as follows.

Proposition 3'. Assume that no finite extension of $k$ contains all $l^n$-roots of unity, then any $\ell$-adic representation $\rho: G_K \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell})$ is potentially semi-stable. 

Proof. After making a finite extension is necessary, we may even assume that $\rho(P_{K,\ell})$ is trivial. In this way, any $\ell$-adic representation can be viewed as a morphism $\overline{\rho} \colon \colon \mathrm{Gal}(K^{tr,\ell}/K) \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell})$. We then have an exact sequence

$$1 \longrightarrow \mathrm{Gal}(K^{tr,\ell}/K^{unr}) \simeq \mathbb{Z}_{\ell}(1) \longrightarrow \mathrm{Gal}(K^{tr,\ell}/K) \longrightarrow \mathrm{Gal}(k^{sep}/k) \longrightarrow 1$$ with $k$ being the residue field of $K$. Let $t$ be a topological generator of $\mathbb{Z}_{\ell}(1)$. We choose a finite extension $E$ of $\mathbb{Q}_{\ell}$ so that the characteristic polynomial of $\overline{\rho}(t)$ splits into linear factors in $E$. Then $W = E \otimes_{\mathbb{Q}_{\ell}} \mathbb{Q}_{\ell}^n$ admits an induced action from $\mathbb{Q}_{\ell}^n$ via 

$$g(\lambda \otimes v) = \lambda \otimes gv.$$ We use the same nation for the induced action $\overline{\rho} \colon G_K \longrightarrow \mathrm{GL}_n(E)$. Let $a \in E$ be an eigenvalue of $\overline{\rho}(t)$ and $v \neq 0 \in W$ a corresponding eigenvector, i.e., 

$$\overline{\rho}(t)(v) = av.$$ For any $g \in \mathrm{Gal}(K^{tr,\ell}/K)$, we claim that $gtg^{-1}=t^{\chi_{\ell}(g)}$, where $\chi_{\ell} \colon \mathrm{Gal}(K^{tr,\ell}/K) \longrightarrow \mathbb{Z}_{\ell}^{\times}$ is some character function. Indeed, let

$$L = K[X]/(X^{\ell^n} - \pi) = K(\sqrt[\ell^n]{\pi}) \subset K^{tr,\ell}$$ and $\alpha= \overline{X} \in L$ then $\overline{\rho}(g^{-1})(\alpha) = \zeta_{\ell^n}\alpha$ for some $\ell^n$-root of unity $\zeta_{\ell^n}$. Note that $t \in I_K/P_{K,\ell} \simeq \mathbb{Z}_{\ell}(1)$, it must act trivially on $\zeta_{\ell^n}$ because the extension $K[X]/(X^{\ell^n}-1) \subset K^{unr}$ (again, $\ell \neq p$ is important). Moreover $\overline{\rho}(t)(\alpha) = \zeta_{\ell^n}^{t_n}\alpha$ for some $t_n \in \mathbb{Z}_{\ell}^{\times}$. This implies that

$$\overline{\rho}(gtg^{-1})(\alpha) = \overline{\rho}(gt)(\zeta_{\ell^n}\alpha) = \overline{\rho}(g)(\zeta_{\ell^n}^{t_n+1}\alpha) = (\zeta_{\ell^n}^{t_n})\alpha$$ and hence we can define $\chi_{\ell}(g) = t_n$. An elementary calculation shows that $$\overline{\rho}(t)(g^{-1}v) = a^{\chi_{\ell}(g)}g^{-1}v,$$ and hence $a^{\chi_{\ell}(g)}$ is also an eigenvalue.The hypothesis is equivalent to saying that $\chi_{\ell}$ takes infinitely many values and any eigenvalue of $t$ is a root of unity. In particular, some power of $t$ must act unipotently. Note that $t$ is a topological generator, the previous fact shows that some (closed) subgroup of $\mathbb{Z}_{\ell}(1)$ (given by taking the closure of some power of $t$) acts unipotently. By taking inverse image via $I_K \longrightarrow I_K/P_{K,\ell}$, we deduce that some subgroup of finite index of $I_K$ acts unipotently.

Proof of theorem 1. The idea is to show that the $\ell$-adic represention on etale cohomology coming from some other action $G_{K'} \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell})$ with $K'$ satisfying the hypothesis of proposition 3'. 

Since $X$ is projective, it can be written as $V_+(f_1,...,f_m) \subset \mathbb{P}^n_K$  for some homogeneous polynomials $f_1,...,f_m$. Let $K_0$ be the field obtained by adding all coefficients of $f_1,...,f_n$ to the prime field of $K$. Let $K_1$ be the topological closure of $K_0$ in $K$. Then $K_1$ is a complete discrete valued field with residue field $k_1$ of finite type over $\mathbb{F}_p$.

Let $k_2 = k_1^{p^{-\infty}}$ be the perfect closure of $k_1$. There exists a complete discrete valued field $K_1 \subset K_2 \subset K$ whose residue field is $k_2$. Now we have

$$X = X_0 \times_{K_0} K = (X_0 \times_{K_0} K_2) \times_{K_2} K$$ and the action of $G_K$ comes from the action of $G_{K_2}$. But for $K_2$, the contidion of proposition 3' is satisfied, so the theorem follows.