2 Preliminaries on fundamental groups and $p$ -adic cohomology

The purpose of this section is to recall some general facts about unipotent fundamental groups, as well as some of the material from [Reference Lazda and PálLP16] concerning $p$ -adic cohomology over equicharacteristic local fields. So let $X/F$ be a smooth, geometrically connected, quasi-projective variety and $x\in X(F)$ . Then, for any $\ell \neq \operatorname{char}(F)$ , the $\ell$ -adic version of the unipotent fundamental group has two different but equivalent interpretations. First of all, we may consider the étale fundamental group $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X,\bar{x})$ based at a geometric point above $x$ , together with the homotopy exact sequence

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})\rightarrow \unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X,\bar{x})\rightarrow G_{F}\rightarrow 1.\end{eqnarray}$$

The point $x$ induces a splitting of this exact sequence, and this gives rise to an action of $G_{F}$ on $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})$ . Applying the Malcev completion functor over $\mathbb{Q}_{\ell }$ then gives a pro-unipotent group scheme $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}$ over $\mathbb{Q}_{\ell }$ which by functoriality is acted on by $G_{F}$ .

Alternatively, we can use a Tannakian approach. That is, we consider the Tannakian category $\operatorname{Uni}_{\mathbb{Q}_{\ell }}(X_{F^{\text{sep}}})$ of unipotent lisse $\mathbb{Q}_{\ell }$ -sheaves on $X$ . The point $x$ gives rise to a fibre functor $x^{\ast }:\operatorname{Uni}_{\mathbb{Q}_{\ell }}(X_{F^{\text{sep}}})\rightarrow \operatorname{Vec}_{\mathbb{Q}_{\ell }}$ and hence we may define $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}$ to be the (pro-unipotent) affine group scheme representing automorphisms of $x^{\ast }$ . By functoriality $G_{F}$ acts on $\operatorname{Uni}_{\mathbb{Q}_{\ell }}(X_{F^{\text{sep}}})$ , preserving $x^{\ast }$ , and hence by Tannaka duality on $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}$ . As the notation suggests, this coincides with the previous construction by [Reference IllusieSGA5, Exposé VI, § 1.4.2].

While the first definition does not behave well when $\ell =\operatorname{char}(F)$ , the second has a clear analogue, at least when $F=k$ is finite, replacing lisse $\mathbb{Q}_{\ell }$ -sheaves on $X_{F^{\text{sep}}}$ by overconvergent isocrystals on $X/K$ , the category of which we will denote by $\operatorname{Isoc}^{\dagger }(X/K)$ . Let $\operatorname{Uni}_{K}(X)$ denote its full subcategory of unipotent objects. Again, the point $x$ induces a fibre functor $x^{\ast }:\operatorname{Uni}_{K}(X)\rightarrow \operatorname{Vec}_{K}$ and by definition $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/K,x)$ is the (pro-unipotent) affine group scheme representating automorphisms of $x^{\ast }$ . The Frobenius pullback functor $F^{\ast }:\operatorname{Uni}_{K}(X)\rightarrow \operatorname{Uni}_{K}(X)$ is an autoequivalence, and induces an isomorphism $F_{\ast }:\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/K,x)\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/K,x)$ .

When $F=k(\!(t)\!)$ is equicharacteristic local, we need to use the machinery of [Reference Lazda and PálLP16] to define the $p$ -adic unipotent fundamental groups. Since we will also need the results from [Reference Lazda and PálLP16] concerning $p$ -adic cohomology for varieties over such $F$ , we will take the opportunity now to quickly recap some of the main points from [Reference Lazda and PálLP16]. There we constructed, for any variety $X/F$ , cohomology groups $H_{\text{rig}}^{i}(X/{\mathcal{E}}_{K}^{\dagger })$ as $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over the bounded Robba ring ${\mathcal{E}}_{K}^{\dagger }$ , satisfying all the expected properties of an ‘extended’ Weil cohomology theory. In other words, we have finite dimensionality, vanishing in the expected degrees, versions with compact support or support in a closed subscheme, Poincaré duality, Künneth formula, etc. We also constructed a category $\operatorname{Isoc}^{\dagger }(X/{\mathcal{E}}_{K}^{\dagger })$ of overconvergent isocrystals on $X$ relative to ${\mathcal{E}}_{K}^{\dagger }$ , and in [Reference LazdaLaz16, Corollary 2.2] it was proved that this category is Tannakian. The point $x$ provides a fibre functor

$$\begin{eqnarray}x^{\ast }:\operatorname{Uni}_{{\mathcal{E}}_{K}^{\dagger }}(X)\rightarrow \operatorname{Vec}_{{\mathcal{E}}_{K}^{\dagger }}\end{eqnarray}$$

from the subcategory of unipotent isocrystals, and again we may define $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)$ to be the corresponding pro-unipotent group scheme over ${\mathcal{E}}_{K}^{\dagger }$ . It was also shown in [Reference LazdaLaz16, § 5] how to put a canonical ‘ $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module structure’ on $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)$ , that is, a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module structure on its Hopf algebra, which simply as a $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module (i.e. forgetting the Hopf algebra structure) is a direct limit of its finite-dimensional sub- $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules.

Base changing the whole situation to ${\mathcal{R}}_{K}$ , we therefore obtain a non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{R}}_{K},x):=\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)\otimes _{{\mathcal{E}}_{K}^{\dagger }}{\mathcal{R}}_{K}\end{eqnarray}$$

over ${\mathcal{R}}_{K}$ , as well as abelian ones,

$$\begin{eqnarray}H_{\text{rig}}^{i}(X/{\mathcal{R}}_{K}):=H_{\text{rig}}^{i}(X/{\mathcal{E}}_{K}^{\dagger })\otimes _{{\mathcal{E}}_{K}^{\dagger }}{\mathcal{R}}_{K}.\end{eqnarray}$$

These should be considered $p$ -adic analogues of the $\ell$ -adic Galois representations $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}$ and $H_{\acute{\text{e}}\text{t}}^{i}(X_{F^{\text{sep}}},\mathbb{Q}_{\ell })$ , respectively.

3 The formalism of $\ell$ -independence and statement of conjectures

In this section we will review some of the formalism behind the notion of $\ell$ -independence over finite and equicharacteristic local fields, following [Reference DeligneDel80, Reference FontaineFon94]. For this section, $F$ will stand for either a finite field $k$ or a local field with residue field $k$ .

Since $W_{F}$ is pro-discrete, we may form the associated group scheme $W_{F}^{\text{alg}}$ over $\mathbb{Q}$ by writing $W_{F}=\mathop{\varprojlim }\nolimits_{i}G_{i}$ in the category of groups, viewing each discrete group $G_{i}$ as a group scheme over $\mathbb{Q}$ and then setting $W_{F}^{\text{alg}}=\mathop{\varprojlim }\nolimits_{i}G_{i}$ in the category of group schemes.

If $F$ is local then there is a natural action of $W_{F}^{\text{alg}}$ on the additive group $\mathbb{G}_{a}$ by setting

$$\begin{eqnarray}gxg^{-1}=q^{-v(g)}x,\end{eqnarray}$$

where $v(g)$ is such that $g$ maps to $\operatorname{Frob}_{k}^{v(g)}$ in $W_{k}$ .

Note that the usage of the term ‘Weil–Deligne’ group to refer to $W_{F}^{\text{alg}}$ over finite fields is not at all standard, and we employ it here only so as to be able to make uniform statements for finite and local fields.

We will now let $E$ be a field of characteristic $0$ , and let ${\mathcal{C}}\rightarrow \operatorname{Aff}_{E}$ be a category fibred over the category of affine $E$ -schemes. In other words, for every $E$ -algebra $R$ we have a category ${\mathcal{C}}_{R}$ and for every morphism $R\rightarrow R^{\prime }$ we have a pullback (or ‘base extension’) functor ${\mathcal{C}}_{R}\rightarrow {\mathcal{C}}_{R^{\prime }}$ . The reader should keep in mind the following examples:

1. – the category of (quasi-)coherent ${\mathcal{O}}$ -modules;

2. – the category of pro-nilpotent Lie ${\mathcal{O}}$ -algebras;

3. – the category of (quasi-coherent) Hopf ${\mathcal{O}}$ -algebras;

4. – the category of pro-unipotent group schemes.

In particular, given an object $X\in {\mathcal{C}}_{E}$ , it makes sense to speak of the functor on $E$ -algebras,

$$\begin{eqnarray}\text{}\underline{\operatorname{Aut}}_{E}(X)(R)=\operatorname{Aut}_{{\mathcal{C}}_{R}}(X\otimes R).\end{eqnarray}$$

Note that when ${\mathcal{C}}$ is the category of coherent ${\mathcal{O}}$ -modules, one recovers the usual notion of an $E$ -valued Weil(–Deligne) representation; in general the point of this slightly fiddly approach is in order to have a well-defined notion for non-abelian objects such as Hopf algebras or group schemes. We can now finally make precise the concept of $\ell$ -independence for (not necessarily abelian) Weil–Deligne representations.

When ${\mathcal{C}}$ is the category of coherent ${\mathcal{O}}$ -modules, we also have a weaker notion of compatibility described as follows. If $F$ is finite then we define the ‘monodromy’ filtration $M_{\bullet }$ on a representation $V$ of $W_{F}^{\prime }$ to be the trivial one, namely, $0$ in negative degrees and $V$ otherwise. If $F$ is local then we will let $M_{\bullet }$ be the usual monodromy filtration coming from viewing the $\mathbb{G}_{a}$ action as a nilpotent endomorphism $N:V\rightarrow V$ .

Definition 3.6. Let $\{V_{i}\}_{i\in I}$ be a family of finite-dimensional $E_{i}$ -valued Weil–Deligne representations as above. We say that $\{V_{i}\}_{i\in I}$ is weakly  $E$ -compatible if for all $k$ the character

$$\begin{eqnarray}\operatorname{Tr}(-|\operatorname{Gr}_{k}^{M}V_{i}):W_{F}\rightarrow E_{i}\end{eqnarray}$$

of the $k$ th graded piece of the monodromy filtration has values in $E$ and is independent of $i$ .

Note that the difference between weak compatibility and compatibility is essentially that of Frobenius semisimplicity. For a representation $V$ of $W_{F}^{\prime }$ , we will let $V^{F\text{-}\text{ss}}$ denote its Frobenius semisimplification. Thus when $F=k$ is finite this is just the semisimplification, and when $F$ is local we semisimplify the underlying $W_{F}$ -representation.

Now, if $X/F$ is a variety and $\ell \neq p$ , then one can view the $i$ th étale cohomology group $H_{\acute{\text{e}}\text{t}}^{i}(X_{F^{\text{sep}}},\mathbb{Q}_{\ell })$ as a continuous representation of $G_{F}$ , and hence as an $\mathbb{Q}_{\ell }$ -valued algebraic representation of $W_{F}^{\prime }$ . When we consider it as the latter we will generally write $H_{\ell }^{i}(X)$ . For $\ell =p$ and $F=k$ finite, the natural Frobenius action on the rigid cohomology $H_{\text{rig}}^{i}(X/K)$ allows us to consider it as a $K$ -valued algebraic representation $H_{p}^{i}(X)$ of $W_{F}^{\prime }$ . For $\ell =p$ and $F\cong k(\!(t)\!)$ equicharacteristic local, we may consider the rigid cohomology groups $H_{\text{rig}}^{i}(X/{\mathcal{R}}_{K})$ of $X$ as $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over the Robba ring ${\mathcal{R}}_{K}$ . Using Marmora’s functor [Reference MarmoraMar08] from $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{R}}_{K}$ to $K^{\text{un}}$ -valued Weil–Deligne representations, we may therefore consider the Weil–Deligne representation $H_{p}^{i}(X)$ associated to $H_{\text{rig}}^{i}(X/{\mathcal{R}}_{K})$ . When $\ell =p$ and $F$ is mixed characteristic local, the étale cohomology $H_{\acute{\text{e}}\text{t}}^{i}(X_{F^{\text{sep}}},\mathbb{Q}_{p})$ is de Rham as a representation of $G_{F}$ , hence we may follow Fontaine’s construction [Reference FontaineFon94] to produce a $K^{\text{un}}$ -valued Weil–Deligne representation, which we will again denote by $H_{p}^{i}(X)$ . Of course, we may also consider versions $H_{c,\ell }^{i}(X)$ with compact support. For cohomology, then, the $\ell$ -independence conjectures (first formulated for $F$ local in mixed characteristic by Fontaine in [Reference FontaineFon94]) are as follows.

When $F$ is finite, $C_{\ell ,w}(X,H^{i})$ is known whenever $X$ is smooth and proper. (For $\ell \neq p$ this follows from Deligne’s proof of the Riemann hypothesis [Reference DeligneDel74]. The fact that the same is true at $\ell =p$ then follows from [Reference Katz and MessingKM74, Corollary 1].) Even in this case, however, $C_{\ell }(X,H^{i})$ is still wide open in general; by Lemma 3.7 it would follow from the Frobenius semisimplicity conjecture. Hence for $X$ proper and smooth of dimension $d$ we know $C_{\ell }(X,H^{i})$ for $i=0,1,2d-1,2d$ , as well as for abelian varieties in all degrees. Let us also remark that in [Reference Mannisto and OlssonMO] the authors prove $C_{\ell ,w}(X,H^{i})$ and $C_{\ell ,w}(X,H_{c}^{i})$ whenever $\dim X\leqslant 2$ .

When $F$ is local, a much weaker version of $C_{\ell ,w}(X,H^{i})$ and $C_{\ell ,w}(X,H_{c}^{i})$ forms part of the main result of [Reference ZhengZhe09]. However, even for smooth and projective varieties the fact that ‘purity’ for $H^{i}$ is more complicated for local fields means that there is no straightforward deduction of $C_{\ell ,w}(X,H^{i})$ from Zheng’s results. In [Reference Lazda and PálLP16, Theorem 5.85] it was proved that when $F\cong k(\!(t)\!)$ is equicharacteristic local, $C_{\ell ,w}(X,H^{i})$ holds for smooth (possibly open) curves and abelian varieties, or for smooth and proper varieties with good reduction. In fact, it was claimed there that $C_{\ell }(X,H^{i})$ holds for smooth curves and abelian varieties; however, there is a gap in the proof which we shall correct in Lemma 3.12 below.

Next let us turn to the unipotent fundamental group $\unicode[STIX]{x1D70B}_{1}$ , whose definition we recalled in the previous section. When $X/F$ is a geometrically connected variety with base point $x\in X(F)$ , and $\ell \neq p$ , we may again consider the $\ell$ -adic unipotent fundamental group $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}$ as a $G_{F}$ -representation, and we will write $\unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)$ for the associated $W_{F}^{\prime }$ -representation (with values in the fibred category of pro-unipotent group schemes). When $\operatorname{char}(F)=\ell =p$ we use the rigid fundamental group ( $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/K,x)$ when $F=k$ is finite, $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{R}}_{K},x)$ when $F\cong k(\!(t)\!)$ is local) to produce a $p$ -adic representation of $W_{F}^{\prime }$ , with values in the fibred category of pro-unipotent group schemes, which we will denote by $\unicode[STIX]{x1D70B}_{1}^{p}(X,x)$ (again using Marmora’s functor from $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules to Weil–Deligne representations when $F\cong k(\!(t)\!)$ ). When $\operatorname{char}(F)=0$ , $\ell =p$ and $X$ is a semistable curve we note that by [Reference Andreatta, Iovita and KimAIK15, Theorem 1.8] the $p$ -adic étale pro-unipotent fundamental group  $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{p}}$ is de Rham as a non-abelian representation of $G_{F}$ , hence we may again associate to it a $K^{\text{un}}$ -valued (non-abelian) Weil–Deligne representation $\unicode[STIX]{x1D70B}_{1}^{p}(X,x)$ . For more general $X$ , there are results of Déglise and Nizioł in [Reference Déglise and NiziołDN15] which again state that $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{p}}$ is de Rham.

We will write $A_{\ell ,X,x}$ for the Hopf algebra of $\unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)$ , $L_{\ell ,X,x}$ for its Lie algebra, $\hat{{\mathcal{U}}}_{\ell ,X,x}$ for the completed universal enveloping algebra of $L_{\ell ,X,x}$ and $\mathfrak{a}_{\ell ,X,x}$ for the augmentation ideal of $\hat{{\mathcal{U}}}_{\ell ,X,x}$ (these are representations in Hopf algebras, pro-nilpotent Lie algebras and associative algebras respectively, and for each $k\geqslant 1$ we may consider $\hat{{\mathcal{U}}}_{\ell ,X,x}/\mathfrak{a}_{\ell ,X,x}^{k}$ as a Weil–Deligne representation in vector spaces). The strongest form of $\ell$ -independence one can state for fundamental groups is the following.

We will also be interested in the following weakening of the above conjecture.

We would like to finish this section by fixing a hole in the proof of [Reference Lazda and PálLP16, Theorem 5.85], stating that if $F\cong k(\!(t)\!)$ is equicharacteristic local and $X/F$ is a smooth curve or an abelian variety, then $C_{\ell }(X,H^{i})$ holds; in fact we only showed there that $C_{\ell ,w}(X,H^{i})$ holds, that is, weak $\ell$ -independence. Given Lemma 3.7, the next result completes the proof of ‘strong’ $\ell$ -independence in these cases.

4 Unipotent fundamental groups over finite fields

In this section, we will prove the following result.

Let us first note the following consequence of the weak Lefschetz theorem.

Proof. If $\dim X>2$ then choose an iterated hyperplane section $Y{\hookrightarrow}X$ passing through $x$ with $\dim Y=2$ . The weak Lefschetz theorem [Reference GrothendieckSGA2, Exposé XII, Corollaire 3.5] says that the map $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(Y_{F^{\text{sep}}},\bar{x})\rightarrow \unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})$ is an isomorphism, hence the same is true of their $\mathbb{Q}_{\ell }$ -unipotent completions; in other words, $\unicode[STIX]{x1D70B}_{1}^{\ell }(Y,x)\rightarrow \unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)$ is an isomorphism for $\ell \neq p$ .

When $\ell =p$ we use the weak Lefschetz theorem slightly differently. We claim that for any unipotent isocrystal $E$ on $X/K$ , the induced map

$$\begin{eqnarray}H_{\text{rig}}^{i}(X/K,E)\rightarrow H_{\text{rig}}^{i}(Y/K,E|_{Y})\end{eqnarray}$$

is an isomorphism for $i=0,1$ and injective for $i=2$ ; this suffices to prove $\unicode[STIX]{x1D70B}_{1}^{p}(Y,x)\rightarrow \unicode[STIX]{x1D70B}_{1}^{p}(X,x)$ is an isomorphism by combining [Reference Chiarellotto and Le StumCL99, Propositions 1.2.2 and 1.3.1] with Tannakian duality. If $E$ is constant then this is simply the usual weak Lefschetz. In general one inducts on the degree of unipotence and uses the five lemma.◻

The key ingredient in the proof of Theorem 4.1 is the following concrete description of the Lie algebra $\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)$ , due to Pridham [Reference PridhamPri09].

Proposition 4.3. Let $L(H_{\ell }^{1}(X)^{\vee })$ denote the free Lie algebra on the dual of $H_{\ell }^{1}(X)$ , and $I_{\ell }$ the ideal generated by the image of the dual of the cup product

$$\begin{eqnarray}H_{\ell }^{2}(X)^{\vee }\overset{\cup _{\ell }^{\vee }}{\rightarrow }H_{\ell }^{1}(X)^{\vee }\otimes H_{\ell }^{1}(X)^{\vee }.\end{eqnarray}$$

Then there is a $W_{F}$ -equivariant isomorphism

$$\begin{eqnarray}L_{\ell ,X,x}=\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)\cong \frac{L(H_{\ell }^{1}(X)^{\vee })}{I_{\ell }}.\end{eqnarray}$$

Since Frobenius semisimplicity is known for $H_{\ell }^{1}(X)$ for $X$ smooth and projective of any dimension, and any $\ell$ , we deduce the following corollary.

Corollary 4.4. To prove Theorem 4.1, it suffices to show that the image of the cup product

$$\begin{eqnarray}H_{\ell }^{1}(X)\otimes H_{\ell }^{1}(X)\overset{\cup _{\ell }}{\rightarrow }H_{\ell }^{2}(X)\end{eqnarray}$$

satisfies weak $\ell$ -independence, that is, the family of representations $\{\operatorname{im}(\cup _{\ell })\}_{\ell }$ is weakly $\mathbb{Q}$ -compatible.

Clearly this holds when $X$ is a smooth, projective curve, since then the cup product map is surjective. When $X$ is a surface, we use the existence of Künneth projectors on $X$ .

We can now complete the proof of Theorem 4.1.

5 Cohomology over equicharacteristic local fields I

The main result of this section is the following theorem.

This is the main step in showing such $\ell$ -independence unconditionally for smooth and proper varieties over $F$ . The first key result we need in the proof of Theorem 5.1 essentially allows us to reduce to the ‘globally defined’ case, and in fact works in slightly larger generality than we will need. So for now let $S$ denote a Dedekind scheme, $s\in S$ a closed point and $\widehat{{\mathcal{O}}}_{S,s}$ the completed local ring at $s$ . By definition a semistable scheme over $\widehat{{\mathcal{O}}}_{S,s}$ is one that is étale locally smooth over $\operatorname{Spec}(\widehat{{\mathcal{O}}}_{S,s}[x_{1},\ldots ,x_{r}]/(x_{1}\cdots x_{r}))$ .

Proposition 5.2. Let ${\mathcal{X}}\rightarrow \operatorname{Spec}(\widehat{{\mathcal{O}}}_{S,s})$ be a semistable scheme of finite type and $n\geqslant 2$ an integer. Then there exist an étale neighbourhood $(U,u)\rightarrow (S,s)$ of $s$ and a flat scheme ${\mathcal{Y}}\rightarrow U$ , smooth away from $u$ and semistable at $u$ , such that

$$\begin{eqnarray}{\mathcal{Y}}\times _{U}\frac{{\mathcal{O}}_{U,u}}{\mathfrak{m}_{u}^{n}}\cong {\mathcal{X}}\times _{\widehat{{\mathcal{O}}}_{S,s}}\frac{\widehat{{\mathcal{O}}}_{S,s}}{\mathfrak{m}_{s}^{n}}\end{eqnarray}$$

as schemes over ${\mathcal{O}}_{U,u}/\mathfrak{m}_{u}^{n}\cong \widehat{{\mathcal{O}}}_{S,s}/\mathfrak{m}_{s}^{n}$ .

Proof. Choose a local parameter $t$ at $s$ . After shrinking $S$ we may assume that $S=\operatorname{Spec}(R)$ is affine and $t\in R$ . Then there exists an étale cover $\{U_{i}\rightarrow {\mathcal{X}}\}$ of ${\mathcal{X}}$ such that each $U_{i}$ is étale over some $\operatorname{Spec}(\widehat{{\mathcal{O}}}_{S,s}[x_{1},\ldots ,x_{d+1}]/(x_{1}\cdots x_{r}-t))$ (with $r$ and $d$ allowed to vary with $i$ .)

Choose some finitely generated, normal $R$ -algebra $A$ contained within $\widehat{{\mathcal{O}}}_{S,s}$ such that everything in sight is defined over $A$ , namely, ${\mathcal{X}}$ , the étale cover $\{U_{i}\rightarrow {\mathcal{X}}\}$ and the étale maps

$$\begin{eqnarray}U_{i}\rightarrow \operatorname{Spec}\biggl(\frac{A[x_{1},\ldots ,x_{d+1}]}{(x_{1}\cdots x_{r}-t)}\biggr).\end{eqnarray}$$

Now by Artin’s approximation theorem, the point $\unicode[STIX]{x1D6FC}:\operatorname{Spec}(\widehat{{\mathcal{O}}}_{S,s})\rightarrow \operatorname{Spec}(A)$ of the finite type $R$ -scheme $\operatorname{Spec}(A)$ , coming from the inclusion $A\subset \widehat{{\mathcal{O}}}_{S,s}$ , can be approximated modulo $t^{n}$ by a Henselian point; in other words, there exists a morphism $\unicode[STIX]{x1D6FC}^{h,n}:\operatorname{Spec}({\mathcal{O}}_{S,s}^{h})\rightarrow \operatorname{Spec}(A)$ which agrees with $\unicode[STIX]{x1D6FC}$ modulo $t^{n}$ . Then pulling back via $\unicode[STIX]{x1D6FC}^{h,n}$ and using the fact that $t$ is a local parameter at $s$ , we can see that there exists a semistable family ${\mathcal{Y}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{S,s}^{h})$ which agrees with ${\mathcal{X}}$ modulo  $t^{n}$ .

Now, finally, we note that there must be some finitely generated sub-algebra $B\subset {\mathcal{O}}_{S,s}^{h}$ , étale over $S$ such that ${\mathcal{Y}}$ is defined over $B$ and smooth away from $s$ . Setting $U=\operatorname{Spec}(B)$ then completes the proof.◻

To apply this result, we will specialise, and let $R\cong k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ be the ring of integers inside $F\cong k(\!(t)\!)$ . Suppose that we are given a semistable scheme ${\mathcal{X}}/R$ . We will let ${\mathcal{X}}^{\times }$ denote ${\mathcal{X}}$ endowed with the log structure given by the special fibre, and $X_{0}^{\times }$ the special fibre endowed with the inverse image log structure from ${\mathcal{X}}$ . Let $X$ denote the generic fibre of ${\mathcal{X}}$ . Then we may consider, as in [Reference NakayamaNak98], the log-étale cohomology $H_{\acute{\text{e}}\text{t}}^{i}(X_{0}^{\times ,\text{tame}},\mathbb{Q}_{\ell })$ of the log scheme $X_{0}^{\times }$ , as an $\ell$ -adic representation of $G_{F}$ (on which the wild inertia group $P_{F}\subset I_{F}$ acts trivially). We have the following logarithmic analogue of the smooth and proper base change theorem.

Proposition 5.3 [Reference NakayamaNak98, Proposition (4.2)].

There is an isomorphism

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(X_{0}^{\times ,\text{tame}},\mathbb{Q}_{\ell })\cong H_{\acute{\text{e}}\text{t}}^{i}(X_{F^{\text{sep}}},\mathbb{Q}_{\ell })\end{eqnarray}$$

of $G_{F}$ -representations.

Of course, there is a similar result for $p$ -adic cohomology, which goes as follows. Let $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}$ denote the category of $(\unicode[STIX]{x1D711},N)$ -modules over $K$ , that is, finite-dimensional vector spaces together with a semilinear, bijective Frobenius $\unicode[STIX]{x1D711}$ and a nilpotent endomorphism $N$ , such that $N\unicode[STIX]{x1D711}=q\unicode[STIX]{x1D711}N$ . Given such a $(\unicode[STIX]{x1D711},N)$ -module $V$ , we can put a connection on $V\otimes _{K}{\mathcal{R}}_{K}$ by setting $\unicode[STIX]{x1D6FB}(v\otimes 1)=N(v)\otimes t^{-1}$ , and this induces a fully faithful functor

$$\begin{eqnarray}-\otimes _{K}\,{\mathcal{R}}_{K}:\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}\rightarrow \text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{R}}_{K}}^{\unicode[STIX]{x1D6FB}}\end{eqnarray}$$

from $(\unicode[STIX]{x1D711},N)$ -modules over $K$ to $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{R}}_{K}$ , whose essential image is exactly the objects for which the connection acts unipotently. For ${\mathcal{X}}$ as above, we may consider the log-crystalline cohomology $H_{\log \text{-}\text{crys}}^{i}(X_{0}^{\times }/K^{\times }):=H_{\log \text{-}\text{crys}}^{i}(X_{0}^{\times }/W^{\times })\otimes K$ of its special fibre as a $(\unicode[STIX]{x1D711},N)$ -module over $K$ , as in [Reference Hyodo and KatoHK94] (the notation $W^{\times }$ , $K^{\times }$ refers to the fact that we are endowing $W$ with the Teichmüller lift of the log structure of the punctured point).

Proposition 5.4 [Reference Lazda and PálLP16, Theorem 5.46].

There is an isomorphism

$$\begin{eqnarray}H_{\log \text{-}\text{crys}}^{i}(X_{0}^{\times }/K^{\times })\otimes _{K}{\mathcal{R}}\cong H_{\text{rig}}^{i}(X/{\mathcal{R}}_{K})\end{eqnarray}$$

inside $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{R}}_{K}}^{\unicode[STIX]{x1D6FB}}$ .

We therefore obtain the following corollary.

Corollary 5.5. Let ${\mathcal{X}}\rightarrow \operatorname{Spec}(R)$ be a proper, semistable scheme with generic fibre $X$ . Then there exist a smooth, geometrically connected curve $C/k$ , a $k$ -rational point $c\in C(k)$ , a proper, semistable scheme ${\mathcal{Y}}\rightarrow C$ , smooth away from $c$ , and an isomorphism $R\cong \widehat{{\mathcal{O}}}_{C,c}$ (inducing $F\cong \widehat{k(C)}_{c}$ ) such that

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(X_{F^{\text{sep}}},\mathbb{Q}_{\ell })\cong H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}_{F^{\text{sep}}},\mathbb{Q}_{\ell })\end{eqnarray}$$

as $G_{F}$ representations for all $i,\ell \neq p$ , and

$$\begin{eqnarray}H_{\text{rig}}^{i}(X/{\mathcal{R}}_{K})\cong H_{\text{rig}}^{i}({\mathcal{Y}}_{F}/{\mathcal{R}}_{K})\end{eqnarray}$$

as $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules, for all $i$ . In particular, $C_{\ell ,w}(X,H^{i})$ is equivalent to $C_{\ell ,w}({\mathcal{Y}}_{F},H^{i})$ .

Our primary interest here is in $\ell$ -independence results; however, let us note in passing that Corollary 5.5 implies that the Clemens–Schmid exact sequence, obtained by the first author in collaboration with Tsuzuki in [Reference Chiarellotto and TsuzukiCT14], exists for ${\mathcal{X}}$ .

Corollary 5.6. Let $d=\dim ({\mathcal{X}}/R)=\dim {\mathcal{X}}-1$ . There is a Clemens–Schmid long exact sequence

$$\begin{eqnarray}\displaystyle \cdots \rightarrow H_{\text{rig}}^{m}(X_{0}/K)\rightarrow H_{\log \text{-}\text{crys}}^{m}(X_{0}^{\times }/K^{\times }) & \overset{N}{\rightarrow } & \displaystyle H_{\log \text{-}\text{crys}}^{m}(X_{0}^{\times }/K^{\times })(-1)\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{\text{rig}}^{2d-m}(X_{0}/K)^{\vee }(-d-1)\rightarrow \cdots \nonumber\end{eqnarray}$$

To return to our main focus, then, to prove Theorem 5.1 it suffices to do so in the ‘globally defined’ case, that is, we may assume that we have a smooth geometrically connected curve $C/k$ , $c\in C(k)$ such that $F=\widehat{k(C)}_{c}$ , and a semistable scheme ${\mathcal{X}}\rightarrow C$ , smooth away from $c$ , such that $X={\mathcal{X}}_{F}$ .

Let $U=C\setminus c$ , and consider a collection $\{{\mathcal{F}}_{\ell }\}_{\ell }$ of local systems on $U$ , where ${\mathcal{F}}_{\ell }$ is a lisse $\mathbb{Q}_{\ell }$ sheaf on $U$ for $\ell \neq p$ and ${\mathcal{F}}_{p}$ is an overconvergent $F$ -isocrystal on $U/K$ (here $F$ is the $q=\#k$ -power Frobenius). Then for any closed point $x\in U$ the action of geometric Frobenius on ${\mathcal{F}}_{\ell ,\bar{x}}$ for $\ell \neq p$ , plus the action of $F^{\deg (x)}$ on ${\mathcal{F}}_{p,x}$ , defines a collection $\{{\mathcal{F}}_{\ell ,x}\}_{\ell }$ of representations of the Weil group $W_{k(x)}$ .

Given such a system, we may also restrict to the Weil–Deligne group at $c$ . When $\ell \neq p$ , this is fairly standard, since we may consider the Galois group $G_{F}$ as a decomposition group at $c$ , and hence restrict the $\ell$ -adic representation $\unicode[STIX]{x1D70B}_{1}(U,\bar{\unicode[STIX]{x1D702}})\rightarrow {\mathcal{F}}_{\ell ,\bar{\unicode[STIX]{x1D702}}}$ of the fundamental group of $U$ (based at some geometric generic point) to get an $\ell$ -adic representation of $G_{F}$ , and hence a Weil–Deligne representation, which we shall call ${\mathcal{F}}_{\ell ,c}$ . When $\ell =p$ we use the construction of [Reference TsuzukiTsu98, § 6.1]. Pulling back to an appropriate tubular neighbourhood of $c$ gives a functor

$$\begin{eqnarray}F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\rightarrow \text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{R}}_{c}}^{\unicode[STIX]{x1D6FB}},\end{eqnarray}$$

where ${\mathcal{R}}_{c}$ is a copy of the Robba ring at $c$ . Then applying Marmora’s functor from $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over the Robba ring to Weil–Deligne representations [Reference MarmoraMar08] gives us a $K^{\text{un}}$ -valued representation of $W_{F}^{\prime }$ which we shall call ${\mathcal{F}}_{p,c}$ . The key result is then the following, whose proof we will defer to § 7 below.

We can now complete the proof of Theorem 5.1

Proof of Theorem 5.1.

We may assume that $X$ is globally defined, that is, we have $C,c,U,{\mathcal{X}}$ as above such that $F=\widehat{k(C)}_{c}$ and ${\mathcal{X}}_{F}\cong X$ . We will apply Proposition 5.8, taking ${\mathcal{F}}_{\ell }$ to be the $i$ th higher direct image of the constant sheaf under the map $f:{\mathcal{X}}|_{U}\rightarrow U$ . Hence for $\ell \neq p$ we take ${\mathcal{F}}_{\ell }=\mathbf{R}^{i}f_{\ast }\mathbb{Q}_{\ell }$ , and for $\ell =p$ we take ${\mathcal{F}}_{p}$ to be the overconvergent $F$ -isocrystal $\mathbf{R}^{i}f_{\ast }{\mathcal{O}}_{X/K}^{\dagger }$ on $U/K$ constructed by Matsuda and Trihan in [Reference Matsuda and TrihanMT04]. By smooth and proper base change in étale cohomology we have, for any closed point $x\in U$ ,

$$\begin{eqnarray}{\mathcal{F}}_{\ell ,\overline{x}}=H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{X}}_{\bar{x}},\mathbb{Q}_{\ell }),\end{eqnarray}$$

and by the same result for Ogus’s convergent cohomology [Reference OgusOgu84, Proposition 3.5] we have

$$\begin{eqnarray}{\mathcal{F}}_{p,x}=H_{\text{rig}}^{i}({\mathcal{X}}_{x}/K(x)),\end{eqnarray}$$

where $K(x)$ is the fraction field of the Witt vectors of $k(x)$ . Hence by $C_{\ell ,w}({\mathcal{X}}_{x},H^{i})$ we know that $\{{\mathcal{F}}_{\ell }\}_{\ell }$ is a $\mathbb{Q}$ -compatible system on $U$ , and by Proposition 5.8 it follows that $\{{\mathcal{F}}_{\ell ,c}\}_{\ell }$ is a weakly $\mathbb{Q}$ -compatible system of Weil–Deligne representations. But again applying smooth and proper base change in étale cohomology tells us that ${\mathcal{F}}_{\ell ,c}\cong H_{\ell }^{i}(X)$ for $\ell \neq p$ , and the proof of [Reference Lazda and PálLP16, Proposition 5.52] tells us that ${\mathcal{F}}_{p,c}=H_{p}^{i}(X)$ . The result follows.◻

6 Cohomology over equicharacteristic local fields II

In this section we generalise Theorem 5.1 to apply to all smooth and proper varieties over an equicharacteristic local field, not just those with semistable reduction.

The proof will be via a rather delicate weight argument for Weil–Deligne representations. We will therefore let $\mathbb{Q}_{\ell }^{\prime }$ denote the field $\mathbb{Q}_{\ell }$ if $\ell \neq p$ , and $K^{\text{un}}$ if $\ell =p$ , and throughout consider $\mathbb{Q}_{\ell }^{\prime }$ -valued Weil–Deligne representations (i.e. $\mathbb{Q}_{\ell }^{\prime }$ -valued representations of $W_{F}^{\prime }$ ). If

$$\begin{eqnarray}(V,N,\unicode[STIX]{x1D70C}:W_{F}\rightarrow \operatorname{GL}(V))\end{eqnarray}$$

is such a representation then we have a canonical filtration

$$\begin{eqnarray}M_{k}:=\mathop{\sum }_{k=i-j}\ker \;N^{i+1}\cap \operatorname{im}N^{j}\end{eqnarray}$$

on $V$ , called the monodromy filtration. This is functorial in $V$ in the sense that if $f:V\rightarrow W$ is a morphism of Weil–Deligne representations, then $f(M_{k}V)\subset M_{k}W$ for all $k$ . Moreover, each $M_{k}$ is preserved by the $W_{F}$ -action, and hence there is an induced action of $W_{F}$ on $\operatorname{Gr}_{k}^{M}V$ .

Since the inertia $I_{F}$ acts through a finite quotient, this does not depend on the choice of Frobenius lift $\unicode[STIX]{x1D711}$ . We will need the following results concerning purity and Weil–Deligne representations.

We will also need the following result.

With these preliminaries in place, we can now prove Theorem 6.1.

Proof of Theorem 6.1.

Let $X/F$ be smooth and proper. Then by de Jong’s theorem on alterations [Reference de JongdeJ96, Theorem 8.2], we may choose a proper hypercover $X_{\bullet }\rightarrow X$ and finite extensions $F_{n}/F$ such that:

1. – the structure morphism $X_{n}\rightarrow \operatorname{Spec}(F)$ factors through $\operatorname{Spec}(F_{n})$ ;

2. – $X_{n}/F_{n}$ has strictly semistable reduction.

In particular, by applying Theorem 5.1, Proposition 6.3(iii) and Lemma 6.4, we can see that the cohomology groups $H_{\ell }^{i}(X_{n})$ , where $X_{n}$ is viewed as an $F$ -variety, are quasi-pure of weight $i$ , and moreover that they satisfy weak $\ell$ -independence. We now consider the spectral sequence

$$\begin{eqnarray}E_{1}^{n,i}=H_{\ell }^{i}(X_{n})\Rightarrow H_{\ell }^{i+n}(X).\end{eqnarray}$$

Since $E_{1}^{n,i}$ is quasi-pure of weight $i$ , it follows from Proposition 6.3(i) that $E_{2}^{i,n}$ is also quasi-pure of weight $n$ , and hence from Proposition 6.3(ii) that the spectral sequence degenerates at $E_{2}$ .

We will now consider the induced filtration $P_{\bullet }$ on $H_{\ell }^{i+n}(X)$ , whose $i$ th grade piece

$$\begin{eqnarray}\operatorname{Gr}_{i}^{P}H_{\ell }^{i+n}(X)=E_{2}^{n,i}\end{eqnarray}$$

is therefore quasi-pure of weight $i$ . If we look at the quotient map

$$\begin{eqnarray}H_{\ell }^{i+n}(X)\rightarrow \operatorname{Gr}_{i+n}^{P}=E_{2}^{0,i+n}\end{eqnarray}$$

then it follows from Proposition 6.3(i) that the kernel $P_{i+n-1}H_{\ell }^{i+n}(X)$ is quasi-pure of weight $i+n$ , and hence that the quotient map

$$\begin{eqnarray}P_{i+n-1}H_{\ell }^{i+n}(X){\twoheadrightarrow}\operatorname{Gr}_{i+n-1}^{P}=E_{2}^{1,i+n-1}\end{eqnarray}$$

must be zero by Proposition 6.3(ii). In particular, we have $P_{i+n-1}H_{\ell }^{i+n}(X)=P_{i+n-2}H_{\ell }^{i+n}(X)$ , and, again applying Proposition 6.3(ii), we can see that the quotient

$$\begin{eqnarray}P_{i+n-2}H_{\ell }^{i+n}(X){\twoheadrightarrow}\operatorname{Gr}_{i+n-2}^{P}=E_{2}^{2,i+n-2}\end{eqnarray}$$

must be zero. Continuing thus, we find that $P_{i+n-j}H_{\ell }^{i+n}(X)=P_{0}H_{\ell }^{i+n}(X)=0$ for all $j>0$ , from which we deduce that the $E_{2}$ -page of the spectral sequence only has non-zero terms in the first column. Put differently, we have a long exact sequence

$$\begin{eqnarray}0\rightarrow H_{\ell }^{i+n}(X)\rightarrow H_{\ell }^{i+n}(X_{0})\rightarrow H_{\ell }^{i+n}(X_{1})\rightarrow H_{\ell }^{i+n}(X_{2})\rightarrow \cdots\end{eqnarray}$$

for all $i,n,\ell$ . Since we know weak $\ell$ -independence for all the terms $H_{\ell }^{i+n}(X_{n})$ , we can therefore deduce weak $\ell$ -independence for $H_{\ell }^{i+n}(X)$ .◻

7 $L$ -functions and the proof of Proposition 5.8

The purpose of this section is to prove Proposition 5.8, as well as to give an application of Theorem 6.1 showing that the Hasse–Weil $L$ -functions of smooth and proper varieties over global function fields (including those defined using $p$ -adic cohomology) are independent of $\ell$ . The basic idea is that away from $\ell =p$ Proposition 5.8 follows from [Reference DeligneDel73, Théorème 9.8], and in fact Deligne’s proof also works at $\ell =p$ once certain facts about $p$ -adic cohomology of smooth curves are in place. The deduction of $\ell$ -independence for the Hasse–Weil $L$ -function is then entirely straightforward. In fact, it is not entirely accurate to describe this as an application of Theorem 6.1, since it is in fact a version of $\ell$ -independence for local $L$ -functions that forms a key component of the proof of Proposition 5.8, and therefore of Theorem 6.1. However, we thought it worth isolating this special case. All the results presented in this section are more or less special cases of more general results of Abe and Caro proved in [Reference Abe and CaroAC14]. However, in the simple case that we are interested in one can give proofs using simpler machinery from the theory of arithmetic ${\mathcal{D}}$ -modules on curves and formal curves, in particular that from [Reference CaroCar06] and [Reference CrewCre12].

Since the situation we are interested in in this section is somewhat different from that in the rest of the article, we will change notation slightly (only for this section; we will return to the usual notation in the next section). So we will let $F$ be a global function field with field of constants  $k$ , and let $X$ be a smooth, projective model for $F$ . We will let $U$ be an open subset of  $X$ , and let $S=X\setminus U$ . For any place $v\in |X|$ (with some fixed local parameter $t_{v}$ ) we will let $k_{v}$ denote the residue field at $v$ , $K_{v}$ the unique unramified extension of $K$ with residue field $k_{v}$ , $W_{v}$ its ring of integers and ${\mathcal{R}}_{v}$ a copy of the Robba ring at $v$ , considered as a subring of $K_{v}\unicode[STIX]{x27E6}t_{v},t_{v}^{-1}\unicode[STIX]{x27E7}$ . Let  $\unicode[STIX]{x1D711}_{v}$ denote a $\#k_{v}$ -power Frobenius on ${\mathcal{R}}_{v}$ . We will also denote $\#k_{v}$ -power Frobenius operators by  $F_{v}$ .

If we are given an overconvergent $F$ -isocrystal ${\mathcal{F}}\in F\text{-}\operatorname{Isoc}^{\dagger }(U/K)$ and a point $v\in X$ we can consider Tsuzuki’s functor

$$\begin{eqnarray}\mathbf{i}_{v}^{\ast }:F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\rightarrow \text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{R}}_{v}}^{\unicode[STIX]{x1D6FB}}\end{eqnarray}$$

to the category of $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{R}}_{v}$ , and therefore define the local $L$ -factor of ${\mathcal{F}}$ at $v$ by

$$\begin{eqnarray}L_{v}({\mathcal{F}},t):=\det (1-t^{\deg v}\unicode[STIX]{x1D711}_{v}\mid (\mathbf{i}_{v}^{\ast }{\mathcal{F}})^{\unicode[STIX]{x1D6FB}=0}).\end{eqnarray}$$

The global $L$ -function of ${\mathcal{F}}$ is then defined to be

$$\begin{eqnarray}\widehat{L}({\mathcal{F}},t):=\mathop{\prod }_{v\in |X|}L_{v}({\mathcal{F}},t);\end{eqnarray}$$

the notation $\widehat{L}$ used here is in order to distinguish it from the Etesse–Le Stum $L$ -function $L_{\text{EL}}$ from [Reference Etesse and Le StumEL93], which is defined by only considering local $L$ -factors at points of $U$ . Note that at places $v\in U$ the local $L$ -factor can also be described (by Dwork’s trick) as the inverse characteristic polynomial $\det (1-t^{\deg v}F^{\deg v}\mid {\mathcal{F}}_{v})$ of the linearised Frobenius of ${\mathcal{F}}$ acting on the stalk ${\mathcal{F}}_{v}$ of ${\mathcal{F}}$ at $v$ . Our first task will be to prove a function equation for $\widehat{L}({\mathcal{F}},t)$ , and to do so we will need to interpret the local factors $L_{v}({\mathcal{F}},t)$ in terms of arithmetic ${\mathcal{D}}$ -modules on $X$ .

So we will denote by ${\mathcal{X}}$ a lift of $X$ to a smooth projective formal curve over $W$ , by ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ Berthelot’s ring of arithmetic differential operators on ${\mathcal{X}}$ , and by $F\text{-}\operatorname{Mod}_{\text{hol}}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ (respectively, $F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ ) the category of holonomic $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ -modules (respectively, the bounded derived category of ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ -modules equipped with Frobenius structure, whose cohomology sheaves are holonomic). For any complex ${\mathcal{K}}\in F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ we will denote by $L({\mathcal{K}},t)$ the $L$ -function of ${\mathcal{K}}$ as defined in the introduction to [Reference CaroCar06]; let us quickly recall his definition. For $v\in |X|$ we may lift $v$ to a morphism $i_{v}:\operatorname{Spf}(W_{v})\rightarrow {\mathcal{X}}$ , and for any ${\mathcal{K}}\in F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ we may consider the pullback $i_{v}^{+}{\mathcal{K}}\in F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{\operatorname{Spf}(W_{v}),\mathbb{Q}}^{\dagger })$ . Its cohomology sheaves are therefore $F$ -isocrystals over $K_{v}$ ; we may therefore consider $F^{\deg v}$ as a $K$ -linear operator on these cohomology sheaves. Caro then defines

$$\begin{eqnarray}\displaystyle L_{v}({\mathcal{K}},t) & := & \displaystyle \mathop{\prod }_{i\in \mathbb{Z}}\textstyle \det _{K}(1-t^{\deg v}F^{\deg v}|H^{i}(i_{v}^{+}{\mathcal{K}}))^{(-1)^{r}/(\deg v)}\nonumber\\ \displaystyle L({\mathcal{K}},t) & := & \displaystyle \mathop{\prod }_{v\in |X|}L_{v}({\mathcal{K}},t).\nonumber\end{eqnarray}$$

The functional equation that we require will follow from the existence of the ‘intermediate extension’ functor

$$\begin{eqnarray}j_{!+}:F\text{-}\operatorname{Isoc}(U/K)\rightarrow F\text{-}\operatorname{Mod}_{\text{hol}}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })\end{eqnarray}$$

which will be such that:

1. (i) $\widehat{L}({\mathcal{F}},t)=L(j_{!+}{\mathcal{F}},t)$ ;

2. (ii) $\mathbf{D}(j_{!+}{\mathcal{F}})\cong j_{!+}({\mathcal{F}}^{\vee })$ , where $\mathbf{D}$ is the dual functor for ${\mathcal{D}}$ -modules, and $(-)^{\vee }$ that for isocrystals.

As noted at the beginning of this section, this is a special case of much more general results of [Reference Abe and CaroAC14], but we thought it worthwhile to go through the construction in detail in the simple case that we are interested in. To construct $j_{!+}$ , first let us consider Berthelot’s ring ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ of arithmetic differential operators on ${\mathcal{X}}$ with overconvergent singularities along $S$ , and the corresponding categories $F\text{-}\operatorname{Mod}_{\text{hol}}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S))$ and $F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S))$ of (complexes of) holonomic $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ -modules. Then, thanks to [Reference CaroCar06, Théorème 2.3.3], we may consider any ${\mathcal{F}}\in F\text{-}\operatorname{Isoc}^{\dagger }(U/K)$ as a holonomic $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ -module, which we shall do. We have canonical functors

$$\begin{eqnarray}\displaystyle j_{+}:F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)) & \rightarrow & \displaystyle F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }),\nonumber\\ \displaystyle j^{+}:F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }) & \rightarrow & \displaystyle F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S))\nonumber\end{eqnarray}$$

given by restriction and extension of scalars along ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }\rightarrow {\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ respectively; these are both exact for the natural $t$ -structures on both sides. Define $j_{!}$ to be $\mathbf{D}j_{+}\mathbf{D}$ where $\mathbf{D}$ is the duality functor; note that this is exact for holonomic $F\text{-}{\mathcal{D}}$ -modules by (the proof of) [Reference VirrionVir00, Proposition III.4.4]. By [Reference VirrionVir00, Proposition I.4.4] we have $j^{+}\mathbf{D}=\mathbf{D}j^{+}$ and hence we get adjoint pairs $(j^{+},j_{+})$ and $(j_{!},j^{+})$ . Also by [Reference CaroCar06, 1.1.8] we have $j^{+}j_{!}\cong j^{+}j_{+}\cong \operatorname{id}$ .

The closed immersion $i:S\rightarrow X$ lifts to a closed immersion $i:{\mathcal{S}}\rightarrow {\mathcal{X}}$ and hence we obtain Berthelot’s pullback and pushforward functors

$$\begin{eqnarray}\displaystyle i^{!}:F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }) & \rightarrow & \displaystyle F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{S}},\mathbb{Q}}^{\dagger }),\nonumber\\ \displaystyle i_{+}:F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{S}},\mathbb{Q}}^{\dagger }) & \rightarrow & \displaystyle F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }),\nonumber\end{eqnarray}$$

and we define $i^{+}=\mathbf{D}i^{!}\mathbf{D}$ . It follows from [Reference BerthelotBer02, Théorème 4.3.13] that $i_{+}\mathbf{D}=\mathbf{D}i_{+}$ and hence by [Reference CaroCar06, Théorème 1.2.21] we have two adjoint pairs $(i^{+},i_{+})$ and $(i_{+},i^{!})$ . Also by [Reference BerthelotBer02, Théorème 5.3.3] we get $i^{!}i_{+}\cong i^{+}i_{+}\cong \operatorname{id}$ . By [Reference CaroCar06, (1.1.6.5)] we have, for any ${\mathcal{E}}\in F\text{-}D_{\text{hol}}^{b}({\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ , exact triangles

$$\begin{eqnarray}\displaystyle j_{!}j^{+}{\mathcal{E}} & \rightarrow & \displaystyle {\mathcal{E}}\rightarrow i_{+}i^{+}{\mathcal{E}}\overset{+1}{\rightarrow },\nonumber\\ \displaystyle i_{+}i^{!}{\mathcal{E}} & \rightarrow & \displaystyle {\mathcal{E}}\rightarrow j_{+}j^{+}{\mathcal{E}}\overset{+1}{\rightarrow }.\nonumber\end{eqnarray}$$

In particular, taking ${\mathcal{E}}=j_{+}{\mathcal{F}}$ , we get an exact triangle $j_{!}{\mathcal{F}}\rightarrow j_{+}{\mathcal{F}}\rightarrow i_{+}i^{+}j_{+}{\mathcal{F}}\overset{+1}{\rightarrow }$ , and we define $j_{!+}{\mathcal{F}}:=\operatorname{im}(j_{!}{\mathcal{F}}\rightarrow j_{+}{\mathcal{F}})$ . Note that this makes sense as a ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ -module because $j_{+}$ and  $j_{!}$ are exact. By exactness of $\mathbf{D}$ for holonomic $F\text{-}{\mathcal{D}}^{\dagger }$ -modules we have $\mathbf{D}j_{!+}{\mathcal{F}}\cong j_{!+}\mathbf{D}{\mathcal{F}}$ , and hence, using the main result of [Reference CaroCar05], we deduce that $\mathbf{D}(j_{!+}{\mathcal{F}})\cong j_{!+}({\mathcal{F}}^{\vee })$ . To compare the $L$ -functions of ${\mathcal{F}}$ and $j_{!+}{\mathcal{F}}$ essentially boils down to the following local calculation from [Reference CrewCre12].

Proof. The first case is noted in the proof of [Reference CaroCar06, Proposition 3.3.1] (in combination with the main result from [Reference CaroCar05]); we will consider the second. The key point is to show that we may calculate $j_{!}$ and $j_{+}$ entirely locally. So let $\mathbf{i}_{v}:\mathfrak{X}_{v}:=\operatorname{Spf}(W_{v}\unicode[STIX]{x27E6}t_{v}\unicode[STIX]{x27E7})\rightarrow {\mathcal{X}}$ be a lifting of the map $\operatorname{Spf}(k_{v}\unicode[STIX]{x27E6}t_{v}\unicode[STIX]{x27E7})\rightarrow X$ of the completed local ring at $v$ into $X$ (previously, $\mathbf{i}_{v}^{\ast }$ was simply notation, but shortly we shall see how to interpret it as a genuine pullback via $\mathbf{i}_{v}$ ). Let ${\mathcal{D}}_{v}^{\dagger }$ denote the ring of arithmetic differential operators on $\mathfrak{X}_{v}$ as constructed by Crew in [Reference CrewCre12], ${\mathcal{D}}_{v}^{\dagger }(v)$ the version with overconvergent singularities along the closed point of $\mathfrak{X}_{v}$ . Then (unlike the usual case in ${\mathcal{D}}$ -module theory), the map $\mathbf{i}_{v}$ is actually a map of ringed spaces $(\mathfrak{X}_{v},{\mathcal{D}}_{v}^{\dagger })\rightarrow ({\mathcal{X}},{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger })$ (respectively, $(\mathfrak{X}_{v},{\mathcal{D}}_{v}^{\dagger }(v))\rightarrow ({\mathcal{X}},{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S))$ ) and hence we have natural pullback functors

$$\begin{eqnarray}\displaystyle {\mathcal{E}} & \mapsto & \displaystyle {\mathcal{E}}\otimes _{{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }}{\mathcal{D}}_{v}^{\dagger }=:\mathbf{i}_{v}^{\ast }{\mathcal{E}},\nonumber\\ \displaystyle {\mathcal{E}} & \mapsto & \displaystyle {\mathcal{E}}\otimes _{{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)}{\mathcal{D}}_{v}^{\dagger }(v)=:\mathbf{i}_{v}^{\ast }{\mathcal{E}}\nonumber\end{eqnarray}$$

from holonomic $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ -modules to holonomic $F\text{-}{\mathcal{D}}_{v}^{\dagger }$ -modules (respectively, $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ -modules to $F\text{-}{\mathcal{D}}_{v}^{\dagger }(v)$ -modules). Note that we now have $\mathbf{i}_{v}^{\ast }$ denoting three distinct functors; however, we shall shortly see that they are all compatible, so no essential confusion should arise.

As in the global case, we may define functors $j_{v+}$ and $j_{v}^{+}$ to be restriction and extension of scalars respectively along the natural map ${\mathcal{D}}_{v}^{\dagger }\rightarrow {\mathcal{D}}_{v}^{\dagger }(v)$ , as well as a duality functor $\mathbf{D}$ using the machinery of [Reference VirrionVir00]. Defining $j_{v!}=\mathbf{D}j_{v+}\mathbf{D}$ , we then get adjoint pairs $(j_{v}^{+},j_{v+})$ and $(j_{v!},j_{v}^{+})$ as before, and hence we can define $j_{v!+}{\mathcal{E}}=\operatorname{im}(j_{v!}{\mathcal{E}}\rightarrow j_{v+}{\mathcal{E}})$ for holonomic $F\text{-}{\mathcal{D}}_{v}^{\dagger }(v)$ -module ${\mathcal{E}}$ . Applying [Reference VirrionVir00, Proposition I.4.4] to the map of rings $\mathbf{i}_{v}^{-1}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }\rightarrow {\mathcal{D}}_{v}^{\dagger }$ (respectively, $\mathbf{i}_{v}^{-1}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)\rightarrow {\mathcal{D}}_{v}^{\dagger }(v)$ ), we get $\mathbf{D}\mathbf{i}_{v}^{\ast }\cong \mathbf{i}_{v}^{\ast }\mathbf{D}$ .

Claim. For any coherent ${\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ -module ${\mathcal{E}}$ we have

$$\begin{eqnarray}\mathbf{i}_{v}^{\ast }j_{+}{\mathcal{E}}\cong j_{v+}\mathbf{i}_{v}^{\ast }{\mathcal{E}}.\end{eqnarray}$$

Proof of Claim.

In the usual manner we may reduce to the case of coherent ${\mathcal{D}}_{X_{n}}^{(m)}(S)$ -modules ${\mathcal{E}}$ , where $X_{n}={\mathcal{X}}\otimes _{W}W/p^{n+1}$ and

$$\begin{eqnarray}{\mathcal{D}}_{X_{n}}^{(m)}(S):={\mathcal{B}}(t_{S},p^{m+1})\otimes {\mathcal{D}}_{X_{n}}^{(m)}\end{eqnarray}$$

is the ring of differential operators constructed in [Reference BerthelotBer96, § 4.2.5] (here $t_{S}$ is a section of ${\mathcal{O}}_{{\mathcal{X}}}$ cutting out $S$ ). Defining the local version

$$\begin{eqnarray}{\mathcal{D}}_{v,n}^{(m)}(v):={\mathcal{B}}(t_{v},p^{m+1})\otimes {\mathcal{D}}_{v,n}^{(m)}\end{eqnarray}$$

entirely similarly, one can easily verify the isomorphism

$$\begin{eqnarray}{\mathcal{D}}_{v,n}^{(m)}(v)\cong {\mathcal{D}}_{v,n}^{(m)}\otimes _{\boldsymbol{ i}_{v}^{-1}{\mathcal{D}}_{X_{n}}^{(m)}}\mathbf{i}_{v}^{-1}{\mathcal{D}}_{X_{n}}^{(m)}(S)\end{eqnarray}$$

and the claim follows. ◻

Combining this with flatness of $\mathbf{i}_{v}^{-1}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }\rightarrow {\mathcal{D}}_{v}^{\dagger }$ , we can see that we get an isomorphism

$$\begin{eqnarray}j_{v!+}(\mathbf{i}_{v}^{\ast }{\mathcal{E}})\cong \mathbf{i}_{v}^{\ast }(j_{!+}{\mathcal{E}})\end{eqnarray}$$

for any holonomic $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }(\,^{\dagger }S)$ -module ${\mathcal{E}}$ . Let $i_{v}:\operatorname{Spf}(W_{v})\rightarrow {\mathcal{X}}$ denote the composition of $\mathbf{i}_{v}$ with the inclusion of the zero section $\operatorname{Spf}(W_{v})\rightarrow \mathfrak{X}_{v}$ , so that $i_{v}^{+}$ factors through $\mathbf{i}_{v}^{\ast }$ . If we let ${\mathcal{E}}_{v}^{\dagger }\subset {\mathcal{R}}_{v}$ denote a copy of the bounded Robba ring at $v$ , then the pullback functor

$$\begin{eqnarray}\mathbf{i}_{v}^{\ast }:F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\rightarrow \text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{R}}_{v}}^{\unicode[STIX]{x1D6FB}}\end{eqnarray}$$

factors through the category $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{{\mathcal{E}}_{v}^{\dagger }}^{\unicode[STIX]{x1D6FB}}$ of $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{E}}_{v}^{\dagger }$ , and the diagram

commutes, where the right-hand vertical arrow is that viewing a $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D6FB})$ -module as a holonomic $F_{v}\text{-}{\mathcal{D}}_{v}^{\dagger }(v)$ -module. The upshot of all these complicated trivialities is that to prove (2), we may reduce to the local case; in other words, we may replace ${\mathcal{X}}$ by $\mathfrak{X}_{v}$ .

We may therefore suppose that we are given a $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D6FB})$ -module $M$ over ${\mathcal{E}}_{v}^{\dagger }$ , and we must show that

$$\begin{eqnarray}(i_{v}^{+}j_{v!+}M)[-1]\cong (M\otimes {\mathcal{R}}_{v})^{\unicode[STIX]{x1D6FB}=0}\end{eqnarray}$$

as $F_{v}$ -isocrystals over $K_{v}$ . Note that by [Reference CrewCre12, Proposition 5.1.3] the complex $i_{v}^{+}j_{+}M$ is concentrated in degrees 0 and $-1$ . From the exact triangle $j_{v!}M\rightarrow j_{v+}M\rightarrow i_{v+}i_{v}^{+}j_{v+}M\overset{+1}{\rightarrow }$ we therefore deduce the exact sequence

$$\begin{eqnarray}0\rightarrow j_{v!+}M\rightarrow j_{v+}M\rightarrow i_{v+}H^{0}(i_{v}^{+}j_{v+}M)\rightarrow 0\end{eqnarray}$$

of $F\text{-}{\mathcal{D}}_{v}^{\dagger }$ -modules. Now applying $i_{v}^{+}$ and looking at the long exact sequence in cohomology, we can see that $i_{v}^{+}j_{v!+}M$ is concentrated in degree $-1$ and that $i_{v}^{+}j_{!+}M[-1]\cong H^{-1}(i_{v}^{+}j_{v+}M)$ . Now the claim follows from again applying [Reference CrewCre12, Proposition 5.1.3].◻

Hence applying Poincaré duality for $F\text{-}{\mathcal{D}}_{{\mathcal{X}},\mathbb{Q}}^{\dagger }$ -modules [Reference CaroCar06, Theorem 1.3.5] together with the cohomological formula for $L$ -functions [Reference CaroCar06, Théorème 3.3.4] gives the following corollary.

Corollary 7.3. For any ${\mathcal{F}}\in F\text{-}\operatorname{Isoc}^{\dagger }(U/K)$ we have

$$\begin{eqnarray}\widehat{L}({\mathcal{F}},t)=\unicode[STIX]{x1D716}({\mathcal{F}},t)\widehat{L}({\mathcal{F}}^{\vee }(1),t^{-1})\end{eqnarray}$$

where $\unicode[STIX]{x1D716}({\mathcal{F}},t)$ is a monomial in $t$ .

The next key ingredient in the proof of Proposition 5.8 is the ability to twist overconvergent $F$ -isocrystals by characters of the idèle class group. So let $\unicode[STIX]{x1D712}:\mathbb{A}_{F}^{\ast }/F^{\ast }\rightarrow \mathbb{C}^{\ast }$ be a finite-order character, unramified at all places of $U$ . Via the isomorphism $\mathbb{A}_{F}^{\ast }/F^{\ast }\cong W_{F}^{\text{ab}}$ of global class field theory, we may view $\unicode[STIX]{x1D712}$ as a character of the Weil group $W_{F}$ of $F$ , and, since the image of $\unicode[STIX]{x1D712}$ consists of roots of unity, we may choose a finite extension $L/K$ and view $\unicode[STIX]{x1D712}$ as a continuous character $\unicode[STIX]{x1D712}:G_{F}\rightarrow L^{\ast }$ , unramified at all places of $U$ . Hence we obtain a character $\unicode[STIX]{x1D712}:\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U,\bar{\unicode[STIX]{x1D702}})\rightarrow L^{\ast }$ where $\bar{\unicode[STIX]{x1D702}}$ is some choice of geometric generic point.

Now, by [Reference CrewCre87, Theorem 2.1] there is an equivalence of categories between continuous $L$ -valued representations of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U,\bar{\unicode[STIX]{x1D702}})$ which have finite local monodromy at all places of $S$ and the extension of scalars $F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\otimes _{K}L$ of the $K$ -linear category $F\text{-}\operatorname{Isoc}^{\dagger }(U/K)$ to $L$ . Hence we may associate to $\unicode[STIX]{x1D712}$ a rank-one isocrystal ${\mathcal{E}}_{\unicode[STIX]{x1D712}}\in F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\otimes _{K}L$ . While we defined $L$ -functions for objects in $F\text{-}\operatorname{Isoc}^{\dagger }(U/K)$ , essentially the same definition works for $F\text{-}\operatorname{Isoc}^{\dagger }(U/K)\otimes _{K}L$ , and hence it makes sense to speak of the twisted $L$ -function $L({\mathcal{F}}\unicode[STIX]{x1D712},t):=L({\mathcal{F}}\otimes {\mathcal{E}}_{\unicode[STIX]{x1D712}},t)$ .

To prove Proposition 5.8 we now proceed exactly as in the proof of [Reference DeligneDel73, Théorème 9.8], which shows that the semisimplifications $\{{\mathcal{F}}_{\ell ,c}^{\text{ss}}\}$ (as Weil–Deligne representations, i.e. not just their Frobenius semisimplifications) are weakly $\mathbb{Q}$ -compatible. Hence the traces of $W_{F}$ of the graded pieces of the weight filtration have values in $\mathbb{Q}$ independence of $\ell$ , and we can then conclude using the weight monodromy theorem ([Reference DeligneDel80, Théorème 1.8.4] for $\ell \neq p$ , [Reference CrewCre98, Theorem 10.8] for $\ell =p$ ), since this states that the weight filtration coincides with the monodromy filtration.

Before moving on to unipotent fundamental groups over local fields, let us give an ‘application’ of Theorem 6.1 to give a new proof of a result of Abe and Caro [Reference Abe and CaroAC14, Corollary 4.3.13] concerning Hasse–Weil $L$ -functions of varieties over global function fields (at least in the semistable case). Of course, as remarked earlier, this ‘application’ is more of a special case of the proof of Proposition 5.8; however, we thought it was worth isolating as a separate result in its own right. So suppose that we have a smooth and proper variety $Y/F$ , then for any place $v$ of $F$ , we can base change $Y$ to the completion $F_{v}$ , and consider the rigid cohomology groups $H_{\text{rig}}^{i}(Y_{F_{v}}/{\mathcal{R}}_{v})$ as $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{R}}_{v}$ .

Definition 7.4. Define

$$\begin{eqnarray}\displaystyle L_{\text{HW},v}(F,H_{\text{rig}}^{i}(Y),t) & := & \displaystyle \det (1-\unicode[STIX]{x1D711}_{v}t^{\deg v}\mid H_{\text{rig}}^{i}(Y_{F_{v}}/{\mathcal{R}}_{v})^{\unicode[STIX]{x1D6FB}=0}),\nonumber\\ \displaystyle L_{\text{HW}}(F,H_{\text{rig}}^{i}(Y),t) & = & \displaystyle \mathop{\prod }_{v}L_{\text{HW},v}(F,H_{\text{rig}}^{i}(Y),t).\nonumber\end{eqnarray}$$

There is another $p$ -adic definition of the Hasse–Weil $L$ -function given by Abe and Caro in [Reference Abe and CaroAC14, § 4.3], which they denote by $L_{\text{HW}}(F,{\mathcal{H}}^{i}(Y/K),t)$ . To see that these two definitions coincide, given Proposition 7.1 above, it suffices to note that if we have a smooth projective model ${\mathcal{Y}}\rightarrow U$ over some open subset of $U$ , and let ${\mathcal{H}}^{i}$ denote Matsuda and Trihan’s overconvergent pushforward $\mathbf{R}^{i}f_{\ast }{\mathcal{O}}_{Y/K}^{\dagger }$ , then for any place $v$ of $F$ , we have $\mathbf{i}_{v}^{\ast }{\mathcal{H}}^{i}\cong H_{\text{rig}}^{i}(Y_{F_{v}}/{\mathcal{R}}_{v})$ . This was noted in the proof of [Reference Lazda and PálLP16, Proposition 5.52]. We have the following ‘corollary’ of Theorem 6.1.

8 Unipotent fundamental groups over equicharacteristic local fields

We will now suppose that we have $F\cong k(\!(t)\!)$ equicharacteristic local. The main result of this section is the following theorem.

Of course, the basic strategy of proof will be identical to that in § 5. The first step will be in showing that an appropriate analogue of Propositions 5.3 and 5.4 holds for unipotent fundamental groups. So let ${\mathcal{X}}$ , ${\mathcal{X}}^{\times }$ , $X_{0}^{\times }$ be as in the paragraph before Proposition 5.3, let $x_{0}\in X_{0}^{\text{sm}}(k)$ denote the specialisation of our given point $x\in X(F)$ , and choose a geometric point $\bar{x}_{0}$ above it. For $\ell \neq p$ we will consider $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},\bar{x}_{0})$ , the log-étale fundamental group of $X_{0}^{\times ,\text{tame}}$ as defined in [Reference LepageLep13, § 2]. This has a natural $G_{F}$ -action (trivial on wild inertia $P_{F}$ ), and hence so does its $\mathbb{Q}_{\ell }$ -pro-unipotent completion $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},\bar{x}_{0})_{\mathbb{Q}_{\ell }}$ .

Proposition 8.2 [Reference LepageLep13, Theorems 2.5, 2.7].

There is an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},\bar{x}_{0})_{\mathbb{Q}_{\ell }}\cong \unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},\bar{x})_{\mathbb{Q}_{\ell }}\end{eqnarray}$$

of pro-unipotent groups with $G_{F}$ -action.

For $\ell =p$ we will consider the category $\operatorname{Uni}_{K}(X_{0}^{\times }/K^{\times })$ of unipotent log-isocrystals on $X_{0}^{\times }$ , relative to $W$ equipped with the Teichmüller lift of the log structure on $k$ defined by $\mathbb{N}\rightarrow k$ , $1\mapsto 0$ , as defined in [Reference ShihoShi00, § 4]. This is a Tannakian category by [Reference ShihoShi00, Proposition 4.14], and $x_{0}$ defines a fibre functor

$$\begin{eqnarray}x_{0}^{\ast }:\operatorname{Uni}_{K}(X_{0}^{\times }/K^{\times })\rightarrow \operatorname{Vec}_{K}.\end{eqnarray}$$

We let $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})$ denote the corresponding pro-unipotent group. To put a Frobenius and monodromy operator on this pro-unipotent group, that is, to turn this into a non-abelian $(\unicode[STIX]{x1D711},N)$ -module, we follow the approach of [Reference LazdaLaz16]. So let $k^{\times }$ denote $k$ considered with the log structure of the punctured point, and $f_{0}:X_{0}^{\times }\rightarrow \operatorname{Spec}(k^{\times })$ the structure morphism.

Consider the Tannakian category

$$\begin{eqnarray}{\mathcal{N}}F\text{-Isoc}(X_{0}^{\times }/K)\end{eqnarray}$$

of relatively unipotent log- $F$ -isocrystals on $X_{0}$ , relative to $W$ equipped with the trivial log structure. In other words,

$$\begin{eqnarray}{\mathcal{N}}F\text{-Isoc}(X_{0}^{\times }/K)\subset F\text{-Crys}(X_{0}^{\times }/W)_{\mathbb{ Q}}\end{eqnarray}$$

is the full sub-category of objects which are iterated extensions of those pulled back from the category $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}$ of $(\unicode[STIX]{x1D711},N)$ -modules via

$$\begin{eqnarray}f_{0}^{\ast }:\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}=F\text{-Isoc}(\operatorname{Spec}(k^{\times })/K)\rightarrow F\text{-Isoc}^{\dagger }(X_{0}^{\times }/K).\end{eqnarray}$$

We therefore have a pair of functors

$$\begin{eqnarray}x_{0}^{\ast }:{\mathcal{N}}F\text{-Isoc}(X_{0}^{\times }/K)\leftrightarrows \text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}:f_{0}^{\ast },\end{eqnarray}$$

and using the formalism of [Reference LazdaLaz15, § 2.1], we may attach to this situation a ‘relative fundamental group’

$$\begin{eqnarray}G_{X_{0}^{\times },x_{0}}^{\log \text{-}\text{crys}}:=G({\mathcal{N}}F\text{-Isoc}(X_{0}^{\times }/K),x_{0}^{\ast })\end{eqnarray}$$

in the terminology of [Reference LazdaLaz15]; this is a pro-unipotent group scheme over the category $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}$ . Concretely, then, such an object is something of the form $\operatorname{Spec}(A)$ , where $A$ is a Hopf $K$ -algebra equipped with the structure of an ind- $(\unicode[STIX]{x1D711},N)$ -module. Moreover, the Hopf-algebra and $(\unicode[STIX]{x1D711},N)$ -module structures are compatible in the sense that the maps

$$\begin{eqnarray}\displaystyle A\otimes _{K}A\overset{m}{\rightarrow }A,\quad A\overset{c}{\rightarrow }A\otimes _{K}A, & & \displaystyle \nonumber\\ \displaystyle K\overset{u}{\rightarrow }A,\quad A\overset{\unicode[STIX]{x1D700}}{\rightarrow }K,\quad A\overset{\unicode[STIX]{x1D704}}{\rightarrow }A, & & \displaystyle \nonumber\end{eqnarray}$$

expressing the Hopf algebra structure of $A$ , are in fact morphisms of $(\unicode[STIX]{x1D711},N)$ -modules. By [Reference LazdaLaz15, Proposition 2.3] this group scheme $G_{X_{0}^{\times },x_{0}}^{\log \text{-}\text{crys}}$ is uniquely characterised by the fact that its category of enriched representations, that is, representations in objects of the category $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{K}^{N}$ , is equivalent to ${\mathcal{N}}F\text{-}\operatorname{Isoc}(X_{0}^{\times }/K)$ in such a way that $x_{0}^{\ast }$ gets identified with the forgetful functor.

By [Reference LazdaLaz16, Proposition 9.9] the affine group scheme over $K$ underlying $G_{X_{0},x_{0}}^{\log \text{-}\text{crys}}$ , that is, the affine group scheme obtained by forgetting $(\unicode[STIX]{x1D711},N)$ -module structures, is simply $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})$ . Hence we may view this latter group scheme canonically as having the structure of a non-abelian $(\unicode[STIX]{x1D711},N)$ -module. Hence the base change $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})\otimes _{K}{\mathcal{R}}_{K}$ can be viewed as a non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module over ${\mathcal{R}}_{K}$ . Alternatively, we may consider the non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{R}}_{K},x)$ defined in § 2. The following result is then the non-abelian analogue of Proposition 5.4.

Proposition 8.3. There is a natural isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})\otimes _{K}{\mathcal{R}}_{K}\cong \unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{R}}_{K},x)\end{eqnarray}$$

of non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules.

Proof. By definition there exists a non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)\otimes _{{\mathcal{E}}_{K}^{\dagger }}{\mathcal{R}}_{K}\cong \unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{R}}_{K},x).\end{eqnarray}$$

Let ${\mathcal{E}}_{K}^{+}=W\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}\otimes _{W}K$ be the ring of bounded power series over $K$ ; thus we have ${\mathcal{E}}_{K}^{+}\subset {\mathcal{E}}_{K}^{\dagger }$ as the sub-ring of series with no terms in negative powers of $t$ , and we also have a natural ‘evaluate at $t=0$ map’ ${\mathcal{E}}_{K}^{+}\rightarrow K$ . Thanks to [Reference TsuzukiTsu98, Proposition 3.1.4], the category $\text{}\underline{\mathbf{M}\unicode[STIX]{x1D6F7}}_{S_{K}}^{\unicode[STIX]{x1D6FB},\log }$ of log- $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{E}}_{K}^{+}$ is Tannakian, and it was shown in [Reference LazdaLaz16, § 9] (see, in particular, Propositions 9.6 and 9.9) that there exists a non-abelian log- $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -module $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}({\mathcal{X}}^{\times }/{\mathcal{E}}_{K}^{+},x)$ over ${\mathcal{E}}_{K}^{+}$ such that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x) & \cong & \displaystyle \unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}({\mathcal{X}}^{\times }/{\mathcal{E}}_{K}^{+},x)\otimes _{{\mathcal{E}}_{K}^{+}}{\mathcal{E}}_{K}^{\dagger },\nonumber\\ \displaystyle \unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0}) & \cong & \displaystyle \unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}({\mathcal{X}}^{\times }/{\mathcal{E}}_{K}^{+},x)\otimes _{{\mathcal{E}}_{K}^{+}}K.\nonumber\end{eqnarray}$$

We can thus conclude by applying [Reference Lazda and PálLP16, Proposition 5.34]. ◻

Now following the argument of the previous section, and given Theorem 4.1, to prove Theorem 8.1 it suffices to show the following. Suppose that we are given a smooth, geometrically connected curve $C/k$ , $c\in C(k)$ such that $F\cong \widehat{k(C)}_{c}$ , and a semistable scheme ${\mathcal{X}}\rightarrow C$ , smooth away from $c$ , such that $X={\mathcal{X}}_{F}$ . Suppose further that $x$ comes from a section $s:C\rightarrow {\mathcal{X}}$ . Let $U=C\setminus c$ . Then we want to show that there exist local systems ${\mathcal{F}}_{\ell ,k}$ on $U$ for all $\ell$ prime and $k\geqslant 1$ such that:

1. (i) for any closed point $y\in U$ the fibre of ${\mathcal{F}}_{\ell ,k}$ at $y$ is isomorphic (as a $W_{k(y)}$ -representation) to the quotient of the universal enveloping algebra of $\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\ell }({\mathcal{X}}_{y},s(y))$ by the $k$ th power of its augmentation ideal;

2. (ii) the fibre of ${\mathcal{F}}_{\ell ,k}$ at $c$ is isomorphic (as a $W_{F}^{\prime }$ -representation) to the quotient of the universal enveloping algebra of $\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\ell }(X,x)$ by the $k$ th power of its augmentation ideal.

For $\ell =p$ these follow from the results of [Reference LazdaLaz15, § 3] and [Reference LazdaLaz16, § 6]. Specifically, in [Reference LazdaLaz15, § 3] it was shown how to define, in the above situation, a pro-unipotent affine group scheme $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{U}/U,s)$ over the category $F\text{-}\operatorname{Isoc}(U/K)$ , such that for any closed point $y\in U$ the fibre $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{U}/U,s)_{y}$ can be identified with the unipotent rigid fundamental group $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{y}/K,s(y))$ of the fibre ${\mathcal{X}}_{y}$ . It was then proved in [Reference LazdaLaz16, Theorem 6.1] that we have an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{U}/U,s)_{c}\cong \unicode[STIX]{x1D70B}_{1}^{\text{rig}}(X/{\mathcal{E}}_{K}^{\dagger },x)\end{eqnarray}$$

as non-abelian $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$ -modules over ${\mathcal{E}}_{K}^{\dagger }$ , where $\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{U}/U,s)_{c}$ denotes the ‘fibre’ at $c$ . Taking the quotients of the universal enveloping algebra of $\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\text{rig}}({\mathcal{X}}_{U}/U,s)$ by the powers of the augmentation ideal then gives us the required $F$ -isocrystals on $U/K$ .

For $\ell \neq p$ these two follow from Grothendieck’s homotopy exact sequence: we know that $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U,\bar{\unicode[STIX]{x1D702}})$ acts on $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\bar{\unicode[STIX]{x1D702}}},s(\bar{\unicode[STIX]{x1D702}}))$ , hence on its pro-unipotent completion $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\bar{\unicode[STIX]{x1D702}}},s(\bar{\unicode[STIX]{x1D702}}))_{\mathbb{Q}_{\ell }}$ and therefore on the universal enveloping algebra $\hat{{\mathcal{U}}}(\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\bar{\unicode[STIX]{x1D702}}},s(\bar{\unicode[STIX]{x1D702}}))_{\mathbb{Q}_{\ell }})$ of the associated Lie algebra (again, here $\bar{\unicode[STIX]{x1D702}}$ is any geometric generic point). Hence there is a natural action of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U,\bar{\unicode[STIX]{x1D702}})$ on $\hat{{\mathcal{U}}}(\operatorname{Lie}\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{\bar{\unicode[STIX]{x1D702}}},s(\bar{\unicode[STIX]{x1D702}}))_{\mathbb{Q}_{\ell }})/\mathfrak{a}^{k}$ for any $k$ ( $\mathfrak{a}$ being the augmentation ideal), and this gives the required lisse $\mathbb{Q}_{\ell }$ -sheaf on $U$ .

The astute reader will object that there is a slight gap in the argument outlined above, in that while we can certainly assume that our variety is globally defined, we have not explained why we can assume the existence of a global section. Examining the proof of Proposition 5.2, it suffices to observe that again the given formal section can be arbitrarily well approximated by a Henselian section, which must descend to some étale neighbourhood of $s$ . While this will not agree with our given formal section, it will do so on the special fibre, which suffices by Propositions 8.2 and 8.3.

9 Curves in mixed characteristic

Finally, we will use the results of the previous section to deduce an $\ell$ -independence result for the unipotent fundamental group of a semistable curve in mixed characteristic. So for this section $F$ will be assumed to be a mixed characteristic local field, with ring of integers $R$ , and our main result is as follows.

Proof. The basic idea is that $(\hat{{\mathcal{U}}}/\mathfrak{a}^{k})_{\ell }$ (as a Weil–Deligne representation) only depends on the special fibre $X_{0}^{\times }$ of ${\mathcal{X}}$ as a log scheme, and since $X_{0}$ is a curve, this deforms to equicharacteristic. More concretely, when $\ell \neq p$ we consider the log-unipotent fundamental group $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},x_{0})_{\mathbb{Q}_{\ell }}$ of the log scheme $X_{0}^{\times ,\text{tame}}$ as a $G_{F}$ -representation; then again applying [Reference LepageLep13, Theorems 2.3, 2.5], we have

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},x_{0})_{\mathbb{Q}_{\ell }}\cong \unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},x)_{\mathbb{Q}_{\ell }}\end{eqnarray}$$

as $G_{F}$ -representations. When $\ell =p$ we instead consider the log-crystalline fundamental group $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})$ defined in § 8 above. Thus by [Reference Andreatta, Iovita and KimAIK15, Theorem 1.4] or [Reference LazdaLaz16, Theorem 10.2] we have an isomorphism

$$\begin{eqnarray}\mathbf{D}_{\text{st}}(\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{F^{\text{sep}}},x)_{\mathbb{Q}_{p}})\cong \unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})\end{eqnarray}$$

of $(\unicode[STIX]{x1D711},N)$ -modules over $K$ (after forgetting the Hodge filtration on the former).

But now since the log-smooth deformation theory of curves is unobstructed by [Reference KatoKat96, Proposition 8.6], we may deform $X_{0}^{\times }$ to characteristic $p$ ; in other words, there exists a semistable family ${\mathcal{Y}}\rightarrow \operatorname{Spec}(k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7})$ whose special fibre is isomorphic to $X_{0}^{\times }$ as a log scheme. Hence by applying Theorem 8.1 we know that the Weil–Deligne representations associated to the quotients of the universal enveloping algebras of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X_{0}^{\times ,\text{tame}},x_{0})_{\mathbb{Q}_{\ell }}$ and $\unicode[STIX]{x1D70B}_{1}^{\log \text{-}\text{crys}}(X_{0}^{\times }/K^{\times },x_{0})$ are weakly independent of $\ell$ , and we may conclude.◻