2.1 General $(\varphi , \Gamma )$ -modules

Fix a compatible system of primitive p-power roots of unity $(\zeta _{p^n})_{n\in {\mathbf {Z}}_{\geq 0}}$ in $\overline {{\mathbf {Q}}}_p$ . Given a finite extension K of ${\mathbf {Q}}_p$ , consider $K(\zeta _{p^{\infty }}) = \bigcup _{n\in {\mathbf {Z}}_{\geq 0}} K(\zeta _{p^n})$ and denote by $\Gamma = \Gamma _K$ the Galois group $\operatorname {\mathrm {Gal}}(K(\zeta _{p^{\infty }})/K)$ . Moreover, for any $r\in [0, 1)$ , let

$$\begin{align*}{\mathcal{R}}_K^r := \left\{ f = \sum_{i\in {\mathbf{Z}}}a_i T^i: \begin{array}{l} a_i \in K^{\mathrm{unr}} \cap K(\zeta_{p^{\infty}}) \\ f\text{ is holomorphic on the annulus } r\leq |T| <1 \end{array}\!\right\} \end{align*}$$

and

$$\begin{align*}{\mathcal{R}}_K := \bigcup_{r\in [0, 1)} {\mathcal{R}}_K^r, \end{align*}$$

where $K^{\mathrm {unr}}$ is the maximal unramified extension of K in $\overline {{\mathbf {Q}}}_p$ and the infinite union is taken with respect to the inclusions ${\mathcal {R}}_K^{r} \hookrightarrow {\mathcal {R}}_K^{r'}$ for $r\leq r'<1$ . We call the ring ${\mathcal {R}}_K$ the Robba ring over K. It carries a $\varphi $ -action and a $\Gamma $ -action via the formula

$$\begin{align*}\varphi(T) = (1+T)^p -1 \quad \text{ and }\quad \gamma(T) = (1+T)^{\chi_{\operatorname{\mathrm{cyc}}}(\gamma)}-1 \text{ for any }\gamma \in \Gamma, \end{align*}$$

where $\chi _{\operatorname {\mathrm {cyc}}}$ is the cyclotomic character.

In what follows, we shall consider a more generalized version of ${\mathcal {R}}_K$ . Let E be a finite extension of ${\mathbf {Q}}_p$ . We denote by ${\mathcal {R}}_{K, E} := {\mathcal {R}}_K \otimes _{{\mathbf {Q}}_p}E$ and call it the Robba ring over K with coefficients in E. We linearize the actions of $\varphi $ and $\Gamma $ on ${\mathcal {R}}_{K, E}$ via $\varphi \otimes \operatorname {\mathrm {id}}$ and $\gamma \otimes \operatorname {\mathrm {id}}$ , respectively. In what follows, we often assume E is large enough so that $K \subset E$ .

By a $(\varphi , \Gamma )$ -module over ${\mathcal {R}}_{K, E}$ , we mean a finite free ${\mathcal {R}}_{K, E}$ -module D together with a $\varphi $ -semilinear endomorphism $\varphi _D$ and a semilinear action by $\Gamma $ , which commute with each other, such that the induced map

$$\begin{align*}\varphi_D: \varphi^*D = D \otimes_{\varphi}{\mathcal{R}}_{K, E} \rightarrow D \end{align*}$$

is an isomorphism. We shall denote by ${\mathbf{\mathsf{Mod}}}_{{\mathcal {R}}_{K, E}}^{(\varphi , \Gamma )}$ the category of $(\varphi , \Gamma )$ -modules over ${\mathcal {R}}_{K, E}$ .

Let ${\mathbf{\mathsf{Rep}}}_{K}(E)$ the category of Galois representations of $\operatorname {\mathrm {Gal}}_K$ with coefficients in E. Then, by [Reference Benois2, Proposition 1.1.4], there is a fully faithful functor

$$\begin{align*}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger} : {\mathbf{\mathsf{Rep}}}_{K}(E) \rightarrow {\mathbf{\mathsf{Mod}}}_{{\mathcal{R}}_{K, E}}^{(\varphi, \Gamma)}. \end{align*}$$

Moreover, by letting ${\mathbf{\mathsf{Mod}}}_{K, E}^{(\varphi , N)}$ (resp., ${\mathbf{\mathsf{Mod}}}_{K, E}^{\varphi }$ ) the category of $(\varphi , N)$ -modules (resp., $\varphi $ -modules) over $K_0 = K \cap {\mathbf {Q}}_p^{\mathrm {unr}}$ with coefficients in E, there is a functor (see, for example, [Reference Benois2, §1.2.3])

$$\begin{align*}{\mathcal{D}}_{{\mathrm{st}}}: {\mathbf{\mathsf{Mod}}}_{{\mathcal{R}}_{K, E}}^{(\varphi, \Gamma)} \rightarrow {\mathbf{\mathsf{Mod}}}_{K, E}^{(\varphi, N)} \quad (\text{resp., }{\mathcal{D}}_{\operatorname{\mathrm{cris}}}: {\mathbf{\mathsf{Mod}}}_{{\mathcal{R}}_{K, E}}^{(\varphi, \Gamma)} \rightarrow {\mathbf{\mathsf{Mod}}}_{K, E}^{\varphi}) \end{align*}$$

such that if $\rho \in {\mathbf{\mathsf{Rep}}}_{K}(E)$ is semistable (resp., crystalline), then ([Reference Berger3, Théorème 0.2])

$$\begin{align*}{\mathcal{D}}_{{\mathrm{st}}}({\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho)) = {\mathbf{D}}_{{\mathrm{st}}}(\rho) \quad (\text{resp., } {\mathcal{D}}_{\operatorname{\mathrm{cris}}}({\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho)) = {\mathbf{D}}_{\operatorname{\mathrm{cris}}}(\rho)). \end{align*}$$

Here, ${\mathbf {D}}_{{\mathrm {st}}}$ (resp., ${\mathbf {D}}_{\operatorname {\mathrm {cris}}}$ ) is Fontaine’s semistable (resp., crystalline) functor ([Reference Fontaine7, Reference Berger3]), assigning a Galois representation in ${\mathbf{\mathsf{Rep}}}_{K}(E)$ a $(\varphi , N)$ -module (resp., $\varphi $ -module) over $K_0$ with coefficients in E.

Now, let D be a $(\varphi , \Gamma )$ -module over ${\mathcal {R}}_{K, E}$ . Recall the cohomology of D is defined by the cohomology of the Herr complex

$$\begin{align*}0 \rightarrow D \xrightarrow{x \mapsto ((\varphi_D-1)x, (\gamma -1)x)} D \oplus D \xrightarrow{(x,y) \mapsto (\gamma -1)x - (\varphi_D -1)y} D \rightarrow 0, \end{align*}$$

where $\gamma $ is a (fixed) topological generator of $\Gamma $ . Note that, given $\alpha = (x, y)\in D \oplus D$ such that $(\gamma -1)x -(\varphi _D -1)y = 0$ , there is an extension

$$\begin{align*}0 \rightarrow D \rightarrow D_{\alpha} \rightarrow {\mathcal{R}}_{K, E} \rightarrow 0 \end{align*}$$

defined by

(2.1) $$ \begin{align} D_{\alpha} = D \oplus {\mathcal{R}}_{K, E}e, \quad (\varphi_{D_{\alpha}}-1)e = x, \quad (\gamma -1)e = y. \end{align} $$

It turns out that such an assignment gives rise to an isomorphism

$$\begin{align*}H^1(D) \cong \mathrm{Ext}_{(\varphi, \Gamma)}^1({\mathcal{R}}_{K, E}, D). \end{align*}$$

Furthermore, we write $H^1_{{\mathrm {st}}}(D)$ (resp., $H^1_f(D)$ ) the subspace of $H^1(D)$ , consisting of those semistable (resp., crystalline) extensions $D_{\alpha }$ (i.e., those satisfy $\operatorname {\mathrm {rank}}_{K_0\otimes _{{\mathbf {Q}}_p}E}{\mathcal {D}}_{{\mathrm {st}}}(D_{\alpha }) = \operatorname {\mathrm {rank}}_{K_0 \otimes _{{\mathbf {Q}}_p} E} {\mathcal {D}}_{{\mathrm {st}}}(D) +1$ (resp., $\operatorname {\mathrm {rank}}_{K_0\otimes _{{\mathbf {Q}}_p}E}{\mathcal {D}}_{\operatorname {\mathrm {cris}}}(D_{\alpha }) = \operatorname {\mathrm {rank}}_{K_0 \otimes _{{\mathbf {Q}}_p} E} {\mathcal {D}}_{\operatorname {\mathrm {cris}}}(D) +1$ )). According to [Reference Benois2, Proposition 1.4.2], if $\rho \in {\mathbf{\mathsf{Rep}}}_{K}(E)$ , then

$$\begin{align*}H^1_{{\mathrm{st}}}({\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho)) \cong H^1_{{\mathrm{st}}}(K, \rho) \quad (\text{resp., } H^1_{f}({\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho)) \cong H_f^1(K, \rho)), \end{align*}$$

where

$$ \begin{align*} H_{{\mathrm{st}}}^1(K, \rho) &= \ker\left(H^1(K, \rho) \rightarrow H^1(K, \rho\otimes_{{\mathbf{Q}}_p}{\mathbf{B}}_{{\mathrm{st}}})\right) \\ (\text{resp., }H_{f}^1(K, \rho) &= \ker\left(H^1(K, \rho) \rightarrow H^1(K, \rho\otimes_{{\mathbf{Q}}_p}{\mathbf{B}}_{\operatorname{\mathrm{cris}}})\right)) \end{align*} $$

Footnote ¹ is the usual local Bloch–Kato Selmer group.

To conclude our discussion for general $(\varphi , \Gamma )$ -modules, we mention that if D is semistable,Footnote ² then $H_{{\mathrm {st}}}^1(D)$ and $H_f^1(D)$ can be computed by complexes $C_{{\mathrm {st}}}^{\bullet }$ and $C_{\operatorname {\mathrm {cris}}}^{\bullet }$ , respectively ([Reference Benois2, Proposition 1.4.4]). Here,

and

$$ \begin{align*} C_{\operatorname{\mathrm{cris}}}^{\bullet}(D) & = \left[{\mathcal{D}}_{\operatorname{\mathrm{cris}}}(D) \xrightarrow{a\mapsto (a, (\varphi-1)a)} \frac{{\mathcal{D}}_{\operatorname{\mathrm{cris}}}(D)}{\operatorname{\mathrm{Fil}}_{\operatorname{\mathrm{dR}}}^0{\mathcal{D}}_{\operatorname{\mathrm{cris}}}(D)} \oplus {\mathcal{D}}_{\operatorname{\mathrm{cris}}}(D) \right]. \end{align*} $$

2.2 $(\varphi , \Gamma )$ -modules of rank 1

Recall that $(\varphi , \Gamma )$ -modules of rank $1$ can be understood via continuous characters. More precisely, given a continuous character $\delta : K^\times \rightarrow E^\times $ and fixing a uniformizer $\varpi \in K$ , we can write $\delta = \delta ' \delta "$ with $\delta '|_{{\mathcal {O}}_{K}^{\times }} = \delta |_{{\mathcal {O}}_{K}^{\times }}$ , $\delta '(\varpi ) = 1$ and $\delta "(\varpi ) = \delta (\varpi )$ , $\delta "|_{{\mathcal {O}}_K^{\times }} = 1$ . By local class field theory, $\delta '$ defines a unique one-dimensional Galois representation $\chi _{\delta '}$ ; that is,

$$\begin{align*}\chi_{\delta'}: \operatorname{\mathrm{Gal}}_K \xrightarrow{\text{local Artin map}} \widehat{K^\times} \cong {\mathcal{O}}_K^\times \times \widehat{{\mathbf{Z}}} \xrightarrow{(a, b)\mapsto \delta'(a)} E^\times, \end{align*}$$

Footnote ³ which admits its associated $(\varphi , \Gamma )$ -module ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\chi _{\delta '})$ . However, we define ${\mathcal {R}}_{K, E}(\delta ") = {\mathcal {R}}_{K, E}e_{\delta "}$ such that $\varphi (e_{\delta "}) = \delta (\varpi )e_{\delta "}$ and $\gamma (e_{\delta "}) = e_{\delta "}$ . Then, the $(\varphi , \Gamma )$ -module associated with $\delta $ is defined to be

$$\begin{align*}{\mathcal{R}}_{K, E}(\delta) := {\mathcal{R}}_{K, E}(\delta") \otimes_{{\mathcal{R}}_{K, E}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\chi_{\delta'}). \end{align*}$$

In particular, the cyclotomic character $\operatorname {\mathrm {Gal}}_{K} \rightarrow {\mathcal {O}}_{E}^\times $ has the associated $(\varphi , \Gamma )$ -module ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\chi _{\operatorname {\mathrm {cyc}}})$ . By [Reference Nakamura16, Lemma 2.13], we know that

$$\begin{align*}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\chi_{\operatorname{\mathrm{cyc}}}) = {\mathcal{R}}_{K, E}(\mathrm{Nm}^{K}_{{\mathbf{Q}}_p}(z)|\mathrm{Nm}^{K}_{{\mathbf{Q}}_p}(z)|), \end{align*}$$

where $\mathrm {Nm}^K_{{\mathbf {Q}}_p}$ is the norm function from K to ${\mathbf {Q}}_p$ .

Lemma 2.1. Let $\delta : K^\times \rightarrow E^\times $ be the character

$$\begin{align*}\delta(z) = \left( \prod_{\sigma: K \hookrightarrow \overline{{\mathbf{Q}}}_p} \sigma(z)^{m_{\sigma}}\right) \left| \mathrm{Nm}_{{\mathbf{Q}}_p}^{K} (z)\right| \end{align*}$$

such that all $m_{\sigma } \geq 1$ .

• • If $\left ({\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\delta ))^{\vee }(\chi _{\operatorname {\mathrm {cyc}}}) \right )^{\varphi =1}$ is nonzero, then the inclusion $H_{{\mathrm {st}}}^1({\mathcal {R}}_{K, E}(\delta )) \hookrightarrow H^1({\mathcal {R}}_{K, E}(\delta ))$ is an isomorphism.

• • If $\left ({\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\delta ))^{\vee }(\chi _{\operatorname {\mathrm {cyc}}}) \right )^{\varphi =1} =0 $ , then the inclusion $H_f^1({\mathcal {R}}_{K, E}(\delta )) \kern1.4pt{\hookrightarrow}\kern1.4pt H_{{\mathrm {st}}}^1({\mathcal {R}}_{K, E}(\delta )\kern-0.7pt)$ is an isomorphism.

Proof By [Reference Benois2, Corollary 1.4.5], we have the formula

$$\begin{align*}\dim_E H_{{\mathrm{st}}}^1({\mathcal{R}}_{K, E}(\delta)) - \dim_E H_f^1({\mathcal{R}}_{K, E}(\delta)) = \dim_{E} \left({\mathcal{D}}_{{\mathrm{st}}}({\mathcal{R}}_{K, E}(\delta))^{\vee}(\chi_{\operatorname{\mathrm{cyc}}}) \right)^{\varphi =1}. \end{align*}$$

Applying [Reference Rosso17, Proposition 2.1 & Lemma 2.3], we know that

$$\begin{align*}\dim_E H_{{\mathrm{st}}}^1({\mathcal{R}}_{K, E}(\delta)) \leq [K: {\mathbf{Q}}_p]+1 \quad \text{ and }\quad \dim_E H_f^1({\mathcal{R}}_{K, E}(\delta)) = [K: {\mathbf{Q}}_p]. \end{align*}$$

The lemma then follows easily.

Suppose $\delta : K^\times \rightarrow E^\times $ is a continuous character as in Lemma 2.1. Suppose ${\mathcal {R}}_{K, E}(\delta )$ is semistable and so $\operatorname {\mathrm {rank}}_{K_0 \otimes _{{\mathbf {Q}}_p}E}{\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\delta )) = 1$ . We fix a $K_0 \otimes _{{\mathbf {Q}}_p}E$ -basis $v_{\delta }$ for ${\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\delta ))$ and define

$$\begin{align*}\beta_{\delta}^* = -\operatorname{\mathrm{cl}}(0,0,v_{\delta}), \quad \alpha_{\delta}^* = \operatorname{\mathrm{cl}}(v_{\delta}, 0, 0) \in H^1(C_{{\mathrm{st}}}^{\bullet}({\mathcal{R}}_{K, E}(\delta))) = H_{{\mathrm{st}}}^1({\mathcal{R}}_{K, E}(\delta)). \end{align*}$$

Footnote ⁴

Lemma 2.2. Suppose $\eta : K^\times \rightarrow E^\times $ is a continuous character of the form $\eta (z) = \prod _{\sigma : K \hookrightarrow \overline {{\mathbf {Q}}}_p} \sigma (z)^{n_{\sigma }}$ with all $n_{\sigma } \leq 0$ . Suppose

$$\begin{align*}0 \rightarrow {\mathcal{R}}_{K, E}(\delta) \rightarrow D \rightarrow {\mathcal{R}}_{K, E}(\eta) \rightarrow 0 \end{align*}$$

is a semistable extension (in the sense of §2.1). Then,

$$\begin{align*}\operatorname{\mathrm{image}}\left( \partial: H^0({\mathcal{R}}_{K, E}(\eta)) \rightarrow H^1({\mathcal{R}}_{K, E}(\delta))\right) \subset H_{{\mathrm{st}}}^1({\mathcal{R}}_{K, E}(\delta)). \end{align*}$$

Moreover, there exists a unique ${\mathcal {L}}(D)\in E$ such that

$$\begin{align*}\beta_{\delta}^* + {\mathcal{L}}(D) \alpha_{\delta}^* \in \operatorname{\mathrm{image}} \partial. \end{align*}$$

Proof First of all, it follows from [Reference Benois2, Proposition 1.2.7] that ${\mathcal {R}}_{K, E}(\eta )$ is also semistable. Hence, by applying [op. cit., Proposition 1.4.4], we know that

$$\begin{align*}H^0({\mathcal{R}}_{K, E}(\eta)) = H_{{\mathrm{st}}}^0({\mathcal{R}}_{K, E}). \end{align*}$$

Taking the cohomology of the short exact sequence in the lemma, we have a commutative diagram

which shows the first claim.

Since ${\mathcal {R}}_{K, E}(\eta )$ is semistable and it is of rank $1$ over ${\mathcal {R}}_{K,E}$ , it is crystalline and ${\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\eta )) = {\mathcal {D}}_{\operatorname {\mathrm {cris}}}({\mathcal {R}}_{K, E}(\eta ))$ . This is because the monodromy operator is nilpotent. We consider the commutative diagram

induced by the short exact sequence in the lemma, where the rows are exact and the columns are the semistable complexes. Let $v_{\eta }$ be the element in ${\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\eta ))$ that gives rise to the basis of $H^0({\mathcal {R}}_{K, E}(\eta ))$ as in [Reference Rosso17, Proposition 2.1]. In particular, $v_{\eta } \in \operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^0 {\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{K, E}(\eta ))$ and $\varphi (v_{\eta }) = v_{\eta }$ . Using the relation $N \varphi = p \varphi N$ , one deduces that $1$ and $p^{-1}$ are Frobenius eigenvalues of ${\mathcal {D}}_{{\mathrm {st}}}(D)$ . We choose a lift $\widetilde {v}_{\eta }\in {\mathcal {D}}_{{\mathrm {st}}}(D)$ such that $\varphi (\widetilde {v}_{\eta }) = \widetilde {v}_{\eta }$ . This then implies that $N(\widetilde {v}_{\eta })$ has Frobenius eigenvalue $p^{-1}$ . The commutativity of the diagram then yields

Applying the exactness of the middle row, we see that

$$\begin{align*}(\widetilde{v}_{\eta}, 0, N(\widetilde{v}_{\eta})) = a(v_{\delta}, 0, 0) - b(0, 0, v_{\delta}). \end{align*}$$

Since $N(\widetilde {v}_{\eta })$ is a basis for the Frobenius eigensubspace of ${\mathcal {D}}_{{\mathrm {st}}}(D)$ on which $\varphi $ acts via $p^{-1}$ , we see that b is invertible. We then conclude that

$$\begin{align*}\partial: H^0({\mathcal{R}}_{K, E}(\eta)) \rightarrow H_{{\mathrm{st}}}^1({\mathcal{R}}_{K, E}(\delta)), \quad \operatorname{\mathrm{cl}}(v_{\eta}) \mapsto a \alpha_{\delta}^* + b\beta_{\delta}^*. \end{align*}$$

Therefore, ${\mathcal {L}}(D) := a/b$ .

3.1 Greenberg–Benois ${\mathcal {L}}$ -invariants over ${\mathbf {Q}}$

To define Greenberg–Benois ${\mathcal {L}}$ -invariants over ${\mathbf {Q}}$ , we start with a Galois representation

$$\begin{align*}\rho: \operatorname{\mathrm{Gal}}_{{\mathbf{Q}}} \rightarrow \operatorname{\mathrm{GL}}_n(E), \end{align*}$$

which is unramified outside a finite set of places. We denote by

$$\begin{align*}S = \{\ell: \rho|_{\operatorname{\mathrm{Gal}}_{{\mathbf{Q}}_{\ell}}} \text{ is ramified}\} \cup \{p, \infty\} \end{align*}$$

and let ${\mathbf {Q}}_{S}$ be the maximal extension of ${\mathbf {Q}}$ that is unramified outside S.

Recall the Bloch–Kato Selmer group associated with $\rho $ : Given $v\in S$ , define the local Selmer groups

$$\begin{align*}H_f^1({\mathbf{Q}}_{v}, \rho) = \left\{\begin{array}{@{}ll} \ker\left(H^1({\mathbf{Q}}_{\ell}, \rho) \rightarrow H^1(I_{\ell}, \rho)\right), & \text{ if }v=\ell \nmid p\infty, \\ \ker(H^1({\mathbf{Q}}_p, \rho) \rightarrow H^1({\mathbf{Q}}_p, \rho\otimes_{{\mathbf{Q}}_p}{\mathbf{B}}_{\operatorname{\mathrm{cris}}})), & \text{ if }v=p, \\ H^1({\mathbf{R}}, \rho), & \text{ if } v=\infty, \end{array}\right. \end{align*}$$

where $I_{\ell }$ stands for the inertia group at $\ell $ . Then, the Bloch-Kato Selmer group associated with $\rho $ is defined to be

$$\begin{align*}H_f^1({\mathbf{Q}}, \rho) := \ker\left( H^1(\operatorname{\mathrm{Gal}}_{{\mathbf{Q}}_S}, \rho) \rightarrow \bigoplus_{v\in S} \frac{H^1({\mathbf{Q}}_v, \rho)}{H_f^1({\mathbf{Q}}_v, \rho)}\right). \end{align*}$$

Let $\rho _p := \rho |_{\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_p}}$ . We follow [Reference Benois2, §2.1.2, 2.1.4] and proceed with the following conditions:

1. (B1) The local representation $\rho _p$ is semistable with Hodge–Tate weights $k_1 \leq k_2 \leq \cdots \leq k_n$ , giving rise to the de Rham filtration $\operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^{\bullet } {\mathbf {D}}_{{\mathrm {st}}}(\rho )$ .

2. (B2) The Frobenius action on ${\mathbf {D}}_{{\mathrm {st}}}(\rho _p)$ is semisimple at $1$ and $p^{-1}$ .

3. (GB1) $H_f^1({\mathbf {Q}}, \rho ) = 0 = H_f^1({\mathbf {Q}}, \rho ^{\vee }(1))$ .Footnote ⁵

4. (GB2) $H^0(\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_S}, \rho ) = 0 = H^0(\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_S}, \rho ^{\vee }(1))$ .

5. (GB3) The associated $(\varphi , \Gamma )$ -module ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _p)$ has no saturated subquotientFootnote ⁶ isomorphic to $U_{k,m}$ with $k\geq 1$ and $m\geq 0$ ([Reference Benois2, §2.1.2]), where $U_{k,m}$ is the unique crystalline $(\varphi , \Gamma )$ -module sitting in a nonsplit short exact sequence

   $$\begin{align*}0 \rightarrow {\mathcal{R}}_{{\mathbf{Q}}_p, E}(|z|z^k) \rightarrow U_{k,m} \rightarrow {\mathcal{R}}_{{\mathbf{Q}}_p, E}(z^{-m}) \rightarrow 0. \end{align*}$$

Given a regular submodule $D\subset {\mathbf {D}}_{{\mathrm {st}}}(\rho _p)$ (i.e., a $(\varphi , N)$ -submodule such that ${\mathbf {D}}_{{\mathrm {st}}}(\rho _p) = D \oplus \operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^0{\mathbf {D}}_{{\mathrm {st}}}(\rho _p)$ ), Benois defines a five-step filtration

(3.1) $$ \begin{align} D^{\operatorname{\mathrm{GB}}}_i := \left\{ \begin{array}{cl} 0, & i=-2, \\ (1-p^{-1}\varphi^{-1})D + N(D^{\varphi = 1}), & i=-1,\\ D, & i=0,\\ D + {\mathbf{D}}_{{\mathrm{st}}}(\rho_p)^{\varphi=1}\cap N^{-1}(D^{\varphi=p^{-1}}), & i=1,\\ {\mathbf{D}}_{{\mathrm{st}}}(\rho_p), & i=2. \end{array}\right. \end{align} $$

Such a filtration then yields a filtration on ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _p)$ by

$$\begin{align*}\operatorname{\mathrm{Fil}}_i^{\operatorname{\mathrm{GB}}} {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p) = {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p) \cap \left( D_i^{\operatorname{\mathrm{GB}}} \otimes_{{\mathbf{Q}}_p} {\mathcal{R}}_{{\mathbf{Q}}_p, E}[1/t]\right), \end{align*}$$

where $t = \log (1+T) \in {\mathcal {R}}_{{\mathbf {Q}}_p, E}$ .

Using this filtration, we define the exceptional subquotient

$$\begin{align*}W := \operatorname{\mathrm{Fil}}_1^{\operatorname{\mathrm{GB}}} {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p) / \operatorname{\mathrm{Fil}}_{-1}^{\operatorname{\mathrm{GB}}} {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p). \end{align*}$$

By [Reference Benois2, Proposition 2.1.7], we have

$$\begin{align*}\begin{array}{rcl} W \cong W_0 \oplus W_1 \oplus M & & \operatorname{\mathrm{rank}} W_0 = \dim_E H^0(W^{\vee}(1)), \\ \operatorname{\mathrm{Gr}}_0^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p) \cong W_0 \oplus M_0 & \text{ with } & \operatorname{\mathrm{rank}} W_1 = \dim_E H^0(W),\\ \operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p) \cong W_1 \oplus M_1 & & \operatorname{\mathrm{rank}} M_0 = \operatorname{\mathrm{rank}} M_1, \end{array} \end{align*}$$

where M, $M_0$ , and $M_1$ sit inside a short exact sequence

$$\begin{align*}0 \rightarrow M_0 \rightarrow M \rightarrow M_1 \rightarrow 0. \end{align*}$$

Moreover, one has

$$ \begin{align*} H^1(W) & = \operatorname{\mathrm{coker}} \left(H^1(\operatorname{\mathrm{Fil}}_{-1}^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \rightarrow H^1(\operatorname{\mathrm{Fil}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \right),\\ H_f^1(W) & = \operatorname{\mathrm{coker}} \left(H_f^1(\operatorname{\mathrm{Fil}}_{-1}^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \rightarrow H_f^1(\operatorname{\mathrm{Fil}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \right), \end{align*} $$

and $\dim _E H^1(W)/H_f^1(W) = e_D = \operatorname {\mathrm {rank}} M_0 + \operatorname {\mathrm {rank}} W_0 +\operatorname {\mathrm {rank}} W_1$ ([Reference Benois2, §2.2.1]).

Under the assumption (GB1) and (GB2), one applies Poitou–Tate exact sequence and deduces an isomorphism

$$\begin{align*}H^1(\operatorname{\mathrm{Gal}}_{{\mathbf{Q}}_S}, \rho) \cong \bigoplus_{v\in S} \frac{H^1({\mathbf{Q}}_v, \rho)}{H_f^1({\mathbf{Q}}_v, \rho)}. \end{align*}$$

Note that the latter space contains an $e_D$ -dimensional subspace $\frac {H^1(W)}{H_f^1(W)} \cong \frac {H^1(\operatorname {\mathrm {Fil}}_1^{\operatorname {\mathrm {GB}}}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _p))}{H^1_f({\mathbf {Q}}_p, \rho )}$ . We then define $H^1(D, \rho )$ to be the image of $\frac {H^1(W)}{H_f^1(W)}$ in $H^1(\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_S}, \rho )$ .

To define the ${\mathcal {L}}$ -invariant, we further assume that

1. (GB4) $W_0 = 0$ and the Hodge–Tate weights for $\operatorname {\mathrm {Gr}}_1^{\operatorname {\mathrm {GB}}} {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _p)$ are positive (see [Reference Benois2, Proposition 1.5.9]).

Benois shows that there is a decomposition ([Reference Benois2, §2.1.9] (see also the discussion in [Reference Harron and Jorza11, §1.2]))

$$\begin{align*}H^1(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \cong H_f^1(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \oplus H_c^1(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \end{align*}$$

and isomorphisms

$$\begin{align*}H_f^1(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \cong {\mathcal{D}}_{\operatorname{\mathrm{cris}}}(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)) \cong H_c^1(\operatorname{\mathrm{Gr}}_1^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_p)). \end{align*}$$

There are natural morphisms $\varrho _{D, ?}: H^1(D, \rho ) \rightarrow {\mathcal {D}}_{\operatorname {\mathrm {cris}}}(\operatorname {\mathrm {Gr}}_1^{\operatorname {\mathrm {GB}}}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _p))$ (for $? \in \{f, c\}$ ) making the diagram

commutative. Under the assumption of (GB4), Benois shows that $\varrho _{D, c}$ is an isomorphism, and so one can define the Greenberg–Benois ${\mathcal {L}}$ - invariant attached to $\rho $ (with respect to D) as

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho) = {\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho, D) := \det\left( \varrho_{D, f} \circ \varrho_{D, c}^{-1}\right)\in E. \end{align*}$$

3.2 Greenberg–Benois ${\mathcal {L}}$ -invariants over general number fields

To define the Greenberg–Benois ${\mathcal {L}}$ -invariants over general number fields, we follow the idea in [Reference Rosso17] (see also [Reference Hida10]) and consider the induction of a Galois representation. More precisely, let F be a number field and suppose we are given a Galois representation

$$\begin{align*}\rho: \operatorname{\mathrm{Gal}}_F \rightarrow \operatorname{\mathrm{GL}}_n(E), \end{align*}$$

where E is (again) a finite extension of ${\mathbf {Q}}_p$ . We shall consider the induction $\operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_F \rho $ and define S similarly as before.

Assume the following conditions hold for $\rho $ :

1. (B1) For each place ${\mathfrak {p}} |p$ in F, $\rho _{{\mathfrak {p}}} = \rho |_{\operatorname {\mathrm {Gal}}_{F_{{\mathfrak {p}}}}}$ is semistable with Hodge–Tate weights $k_{{\mathfrak {p}}, \sigma , 1} \leq k_{{\mathfrak {p}}, \sigma , 2}\leq \cdots \leq k_{{\mathfrak {p}}, \sigma , n}$ , where $\sigma : F_{{\mathfrak {p}}} \hookrightarrow \overline {{\mathbf {Q}}}_p$ .

2. (B2) For each place ${\mathfrak {p}} |p$ in F, the Frobenius action on ${\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ is semistable at $1$ and $p^{-1}$ .

3. (GB1) $H_f^1({\mathbf {Q}}, \operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_{F} \rho ) = 0 = H_f^1({\mathbf {Q}}, \operatorname {\mathrm {Ind}}_F^{{\mathbf {Q}}} \rho ^{\vee }(1))$ .

4. (GB2) $H^0(\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_S}, \rho ) = 0 = H^0(\operatorname {\mathrm {Gal}}_{{\mathbf {Q}}_S}, \rho ^{\vee }(1))$ .

5. (GB3) The associated $(\varphi , \Gamma )$ -module ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}_F^{{\mathbf {Q}}}\rho )_p) = \bigoplus _{{\mathfrak {p}}|p} {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _{{\mathfrak {p}}})$ has no saturated subquotient isomorphic to $U_{k,m}$ with $k\geq 1$ and $m\geq 0$ ([Reference Benois2, §2.1.2]).

For every ${\mathfrak {p}}|p$ , choose a regular subomdule $D_{{\mathfrak {p}}} \subset {\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ . Then, $D := \bigoplus _{{\mathfrak {p}} |p} D_{{\mathfrak {p}} } \subset \bigoplus _{{\mathfrak {p}} | p}{\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}}) = {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_{F}\rho )_p)$ is a regular submodule.Footnote ⁷ Moreover, if $W_0$ , $M_0$ , $M_1$ (resp., $W_{{\mathfrak {p}}, 0}$ , $M_{{\mathfrak {p}}, 0}$ , $M_{{\mathfrak {p}}, 1}$ ) are the corresponding subquotients of ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_{F}\rho )_p)$ (resp., ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _{{\mathfrak {p}}})$ ) with respect to D (resp., $D_{{\mathfrak {p}}}$ ), then we have decompositions

$$\begin{align*}W_0 = \bigoplus_{{\mathfrak{p}} | p}W_{{\mathfrak{p}}, 0}, \quad M_0 = \bigoplus_{{\mathfrak{p}}| p} M_{{\mathfrak{p}}, 0}, \quad M_1 = \bigoplus_{{\mathfrak{p}}|p} M_{{\mathfrak{p}}, 1}. \end{align*}$$

Hence, by assuming

1. (GB4) $W_{{\mathfrak {p}}, 0} =0$ for every ${\mathfrak {p}} |p$ and the Hodge–Tate weights for $\operatorname {\mathrm {Gr}}_1^{\operatorname {\mathrm {GB}}} {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}_F^{{\mathbf {Q}}} \rho )_p)$ are all positive,

we may then follow the same recipe and define the Greenberg–Benois ${\mathcal {L}}$ - invariant attached to $\rho $ (with respect to $\{D_{{\mathfrak {p}}}\}_{{\mathfrak {p}}|p}$ )

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho) = {\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho, \{D_{{\mathfrak{p}}}\}_{{\mathfrak{p}}|p}) := {\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\operatorname{\mathrm{Ind}}_{F}^{{\mathbf{Q}}}\rho, D)\in E. \end{align*}$$

5.1 Assumptions on the Galois representation

Let F be a number field and let E be a finite extension of ${\mathbf {Q}}_p$ such that, for every prime ideal ${\mathfrak {p}}$ in ${\mathcal {O}}_F$ sitting above p, $F_{{\mathfrak {p}}} \subset E$ . Write $F_{{\mathfrak {p}}, 0}$ for the maximal unramified extension of ${\mathbf {Q}}_p$ in $F_{{\mathfrak {p}}}$ ; we further assume that $F_{{\mathfrak {p}}, 0} = {\mathbf {Q}}_p$ for every ${\mathfrak {p}}$ . Suppose we are given a Galois representation

$$\begin{align*}\rho: \operatorname{\mathrm{Gal}}_F \rightarrow \operatorname{\mathrm{GL}}_n(E) \end{align*}$$

that is unramified outside a finite set of places. Let S be the set of places in F such that $\rho $ ramifies. We make the following assumptions:

1. (I) Basic assumptions:

   1. (B1) For any prime ideal ${\mathfrak {p}} \subset {\mathcal {O}}_{F}$ sitting above p, $\rho _{{\mathfrak {p}}} := \rho |_{\operatorname {\mathrm {Gal}}_{F_{{\mathfrak {p}}}}}$ is semistable with Hodge–Tate weights $0\leq k_{{\mathfrak {p}}, \sigma , 1} \leq k_{{\mathfrak {p}}, \sigma , 2} \leq \cdots \leq k_{{\mathfrak {p}}, \sigma , n-1} \leq k_{{\mathfrak {p}}, n}$ , where $\sigma : F_{{\mathfrak {p}}} \hookrightarrow \overline {{\mathbf {Q}}}_p$ .

   2. (B2) For any prime ideal ${\mathfrak {p}} \subset {\mathcal {O}}_{F}$ sitting above p, the Frobenius eigenvalues on ${\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ are $p^m$ , …, $p^{m-n+1}$ such that $k_{{\mathfrak {p}}, \sigma , n}> m > k_{{\mathfrak {p}}, \sigma , n-1}$ , where the first inequality is always guaranteed by Lemma 4.2.Footnote ⁸

2. (II) Fontaine–Mazur assumptions:

   1. (FM1) For any ${\mathfrak {p}}|p$ , let $D_{{\mathfrak {p}}, (\varphi , N)}^{(i)}$ be the eigenspace in ${\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ on which the Frobenius acts via $p^{m-i}$ . We assume that the induced monodromy operator N on $D_{{\mathfrak {p}}, (\varphi , N)}^{(i)}$ gives an isomorphism

      $$\begin{align*}N: D_{{\mathfrak{p}}, (\varphi, N)}^{(i)} \rightarrow D_{{\mathfrak{p}}, (\varphi, N)}^{(i+1)}. \end{align*}$$

   2. (FM2) Define $\operatorname {\mathrm {Fil}}_j^{\varphi } {\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}}) := \sum _{i>n-1-j} D_{{\mathfrak {p}}, (\varphi , N)}^{(i)}$ , and we call the ascending filtration $\operatorname {\mathrm {Fil}}_{\bullet }^{\varphi } {\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ the Frobenius filtration on ${\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ . We assume the orthogonality

      $$\begin{align*}{\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}) = \operatorname{\mathrm{Fil}}_{\operatorname{\mathrm{dR}}}^{k_{{\mathfrak{p}}, \bullet, i}}{\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}) \oplus \operatorname{\mathrm{Fil}}_i^{\varphi} {\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}). \end{align*}$$

3. (III) Greenberg–Benois assumptions:

   1. (GB1) Vanishing of the Bloch–Kato Selmer groups

      $$\begin{align*}H_f^1({\mathbf{Q}}, \operatorname{\mathrm{Ind}}^{{\mathbf{Q}}}_{F} \rho(m)) = H^1_f({\mathbf{Q}}, \operatorname{\mathrm{Ind}}^{{\mathbf{Q}}}_{F}\rho^{\vee}(1-m)) = 0. \end{align*}$$

   2. (GB2) Vanishing of the zero-degree Galois cohomology

      $$\begin{align*}H^0(\operatorname{\mathrm{Gal}}_{{\mathbf{Q}}_S}, \operatorname{\mathrm{Ind}}^{{\mathbf{Q}}}_{F} \rho(m)) = H^0(\operatorname{\mathrm{Gal}}_{{\mathbf{Q}}_S},\operatorname{\mathrm{Ind}}^{{\mathbf{Q}}}_F\rho^{\vee}(1-m)) =0. \end{align*}$$

   3. (GB3) The associated $(\varphi , \Gamma )$ -module ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}_F^{{\mathbf {Q}}}\rho (m))_p) = \bigoplus _{{\mathfrak {p}}|p} {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _{{\mathfrak {p}}} (m))$ does not admit a subquotient of the form $U_{k,r}$ with $k\geq 1$ and $r\geq 0$ ([Reference Benois2, §2.1.2]).

   4. (GB4) For any ${\mathfrak {p}}|p$ , the space $W_{{\mathfrak {p}}, 0}$ for $\rho _{{\mathfrak {p}}}(m)$ vanishes (see [Reference Benois2, Proposition 2.1.7] or [Reference Rosso17, pp. 1238]).

Remark 5.1. For every ${\mathfrak {p}} | p$ , the Frobenius filtration $\operatorname {\mathrm {Fil}}_{\bullet }^{\varphi }{\mathbf {D}}_{{\mathrm {st}}}(\rho _{{\mathfrak {p}}})$ defines a filtration $\operatorname {\mathrm {Fil}}_{\bullet } {\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho )$ (similar as in the proof of Lemma 4.2). One observes that the graded pieces $\operatorname {\mathrm {Gr}}_{i}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho )$ of this filtration are all of rank $1$ over ${\mathcal {R}}_{F_{{\mathfrak {p}}}, E}$ (by [Reference Benois2, Proposition 1.2.7 (ii)]). In particular, $\operatorname {\mathrm {Fil}}_{\bullet }{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho )$ is a triangulation of ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho )$ . In fact, we have the following description for the graded pieces:

$$\begin{align*}\operatorname{\mathrm{Gr}}_i {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho) = {\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{n-i}), \end{align*}$$

where

$$\begin{align*}\delta_{n-i}(z) = \left(\prod_{\sigma: F_{{\mathfrak{p}}} \hookrightarrow \overline{{\mathbf{Q}}}_p} \sigma(z)^{-k_{{\mathfrak{p}}, \sigma, i}} \right)|\mathrm{Nm}^{F_{{\mathfrak{p}}}}_{{\mathbf{Q}}_p}z|^{-(m-n+i)}. \end{align*}$$

5.2 The main theorem

Theorem 5.4. Keep the notations and assumptions as above. We have an equality

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho(m)) = \prod_{{\mathfrak{p}} | p} - {\mathcal{L}}_{\operatorname{\mathrm{FM}}}(\rho_{{\mathfrak{p}}}). \end{align*}$$

Proof The proof of the theorem is similar to the proof of [Reference Benois2, Proposition 2.3.7], which relies on the following three steps:

Step 1. Fontaine–Mazur ${\mathcal {L}}$ -invariants and cohomology of $(\varphi , \Gamma )$ -modules.

Consider the triangulation $\operatorname {\mathrm {Fil}}_{\bullet }{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho )$ in Remark 5.1. We define

$$\begin{align*}\widetilde{W}_{{\mathfrak{p}}} := \operatorname{\mathrm{Fil}}_{n}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho)/\operatorname{\mathrm{Fil}}_{n-2}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho). \end{align*}$$

Hence, $\widetilde {W}_{{\mathfrak {p}}}$ sits inside the short exact sequence

for $\delta _{{\mathfrak {p}}, i} : F_{{\mathfrak {p}}}^\times \rightarrow E^\times $ described as in Remark 5.1.

As a result, $\widetilde {W}_{{\mathfrak {p}}}$ defines a class

$$ \begin{align*} \operatorname{\mathrm{cl}}(\widetilde{W}_{{\mathfrak{p}}}) & \in \mathrm{Ext}_{(\varphi, \Gamma)}({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 0}), {\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1})) \\ & \cong \mathrm{Ext}_{(\varphi, \Gamma)}({\mathcal{R}}, {\mathcal{R}}(\delta_{{\mathfrak{p}}, 1}\delta_{{\mathfrak{p}}, 0}^{-1})) \\ & \cong H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}\delta_{{\mathfrak{p}}, 0}^{-1})). \end{align*} $$

However, by construction, we know that $\widetilde {W}_{{\mathfrak {p}}}$ is semistable (since ${\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}}) = D_{{\mathfrak {p}}, (\varphi , N)}^{(0)} \oplus D_{{\mathfrak {p}}, (\varphi , N)}^{(1)}$ ), and so $\operatorname {\mathrm {cl}}(\widetilde {W}_{{\mathfrak {p}}}) \in H_{{\mathrm {st}}}^1({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1}))$ . Recall that $H_{{\mathrm {st}}}^1({\mathcal {R}}_{F_{{\mathfrak {p}}}, E} (\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1}))$ can be computed via the complex $C_{{\mathrm {st}}}^{\bullet }({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1}))$ with

where the first map is given by $a \mapsto \big (a\ \mod \operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^0 {\mathcal {D}}_{{\mathrm {st}}}({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1})), (\varphi -1)a, Na \big )$ , while the second arrow is defined by $(a, b, c)\mapsto Nb-(p\varphi -1)c$ .

Now, choose a basis $v_{{\mathfrak {p}}, 0}\in D_{{\mathfrak {p}}, (\varphi , N)}^{(0)}$ over $F_{{\mathfrak {p}}, 0}\otimes _{{\mathbf {Q}}_p} E$ and let $v_{{\mathfrak {p}}, 1} := Nv_{{\mathfrak {p}}, 0}$ , which is a $F_{{\mathfrak {p}}, 0}\otimes _{{\mathbf {Q}}_p} E$ -basis for $D_{{\mathfrak {p}}, (\varphi , N)}^{(1)}$ . We again denote by $v_{{\mathfrak {p}}, i}$ for the image of $v_{{\mathfrak {p}}, i}$ in ${\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}}(\delta _{{\mathfrak {p}}, 0}^{-1}))$ . By the proof of [Reference Benois2, Proposition 1.4.4 (ii)], we know that the class $\operatorname {\mathrm {cl}}(\widetilde {W}_{{\mathfrak {p}}})$ in $H^1(C_{{\mathrm {st}}}^{\bullet }({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1})))$ is given by

$$\begin{align*}\operatorname{\mathrm{cl}}(a, b, c) = \operatorname{\mathrm{cl}}(a, (\varphi-1)v_{{\mathfrak{p}}, 0}, Nv_{{\mathfrak{p}}, 0}) = \operatorname{\mathrm{cl}}(a, 0 , v_{{\mathfrak{p}}, 1}), \end{align*}$$

where $a\in {\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}}(\delta _{{\mathfrak {p}}, 0}^{-1}))$ such that $v_{{\mathfrak {p}}, 0} + a \in \operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^0{\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}}(\delta _{{\mathfrak {p}}, 0}^{-1}))$ . After untwisting, a defines an element, still denoted by $a\in {\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}})$ such that $v_{{\mathfrak {p}}, 0} +a \in \operatorname {\mathrm {Fil}}_{\operatorname {\mathrm {dR}}}^{k_{{\mathfrak {p}}, \bullet , n}}{\mathcal {D}}_{{\mathrm {st}}}(\widetilde {W}_{{\mathfrak {p}}})$ . However, by construction,

$$\begin{align*}\operatorname{\mathrm{Fil}}_{\operatorname{\mathrm{dR}}}^{k_{{\mathfrak{p}}, \bullet, n}}{\mathcal{D}}_{{\mathrm{st}}}(\widetilde{W}_{{\mathfrak{p}}}) = \operatorname{\mathrm{Gr}}_{\operatorname{\mathrm{dR}}}^{n-1}{\mathbf{D}}_{{\mathrm{st}}}(\rho) \text{ (notation as in Lemma 4.1)}. \end{align*}$$

Hence, we conclude that

(5.1) $$ \begin{align} \operatorname{\mathrm{cl}}(\widetilde{W}_{{\mathfrak{p}}}) = \operatorname{\mathrm{cl}}(-{\mathcal{L}}_{\operatorname{\mathrm{FM}}}(\rho_{{\mathfrak{p}}})v_{{\mathfrak{p}}, 1}, 0, v_{{\mathfrak{p}}, 1})\in H^1(C_{{\mathrm{st}}}^{\bullet}({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}\delta_{{\mathfrak{p}}, 0}^{-1}))). \end{align} $$

Step 2. Computing ${\mathcal {L}}_{\operatorname {\mathrm {GB}}}(\rho )$ .

Next, we would also like to compute the Greenberg–Benois ${\mathcal {L}}$ -invariant ${\mathcal {L}}_{\operatorname {\mathrm {GB}}}(\rho )$ via cohomology of $(\varphi , \Gamma )$ -modules. As before, because of the decomposition ${\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }((\operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_{F}\rho )_p) = \bigoplus _{{\mathfrak {p}}|p}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\operatorname {\mathrm {Ind}}_{F_{{\mathfrak {p}}}}^{{\mathbf {Q}}_p}\rho _{{\mathfrak {p}}})$ , we can study each ${\mathfrak {p}}$ individually. Hence, fix ${\mathfrak {p}} | p$ . Computing the five-step filtration (3.1) explicitly, we have

$$\begin{align*}D_{{\mathfrak{p}}, i}^{\operatorname{\mathrm{GB}}} = \left\{ \begin{array}{@{}ll} 0, & i=-2, \\ \operatorname{\mathrm{Fil}}_{n-2}^{\varphi}{\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}(m)), & i=-1,\\ D_{{\mathfrak{p}}}, & i=0\\ {\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}(m)), & i=1,\\ {\mathbf{D}}_{{\mathrm{st}}}(\rho_{{\mathfrak{p}}}(m)), & i=2, \end{array}\right. \end{align*}$$

which gives rise to a five-step filtration $\operatorname {\mathrm {Fil}}_{\bullet }^{\operatorname {\mathrm {GB}}}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _{{\mathfrak {p}}}(m))$ .

Let us simplify the notation and write

$$\begin{align*}W_{{\mathfrak{p}}} = \operatorname{\mathrm{Fil}}_1^{\operatorname{\mathrm{GB}}} {\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_{{\mathfrak{p}}}(m))/\operatorname{\mathrm{Fil}}_{-1}^{\operatorname{\mathrm{GB}}}{\mathbf{D}}_{\operatorname{\mathrm{rig}}}^{\dagger}(\rho_{{\mathfrak{p}}}(m)). \end{align*}$$

Similar as before, we see that $W_{{\mathfrak {p}}}$ sits inside the short exact sequence

(5.2)

where

$$\begin{align*}\delta_{{\mathfrak{p}}, n-i}' = \delta_{{\mathfrak{p}}, n-i}\left(\mathrm{Nm}^{F_{{\mathfrak{p}}}}_{{\mathbf{Q}}_p}z\right)^m |\mathrm{Nm}^{F_{{\mathfrak{p}}}}_{{\mathbf{Q}}_p}z|^m = \left(\prod_{\sigma: F_{{\mathfrak{p}}} \hookrightarrow \overline{{\mathbf{Q}}}_p} \sigma(z)^{-k_{{\mathfrak{p}}, \sigma, i}+m} \right)|\mathrm{Nm}^{F_{{\mathfrak{p}}}}_{{\mathbf{Q}}_p}z|^{n-i}. \end{align*}$$

By taking cohomology, we have the connecting homomorphism

$$\begin{align*}\partial: H^0({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 0}')) \rightarrow H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}')) = H_{{\mathrm{st}}}^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}')), \end{align*}$$

where the equation follows from Lemma 2.1. Denoted by $\alpha _{\delta _{{\mathfrak {p}}, 1}'}^*$ and $\beta _{\delta _{{\mathfrak {p}}, 1}'}^*$ the two classes in $H^1({\mathcal {R}}_{F_{{\mathfrak {p}}, E}}(\delta _{{\mathfrak {p}}, 1}'))$ in Lemma 2.2. We know from loc. cit. that $\partial $ gives rise to a unique number ${\mathcal {L}}(W_{{\mathfrak {p}}})\in E$ such that

$$\begin{align*}\beta_{\delta_{{\mathfrak{p}}, 1}'}^* + {\mathcal{L}}(W_{{\mathfrak{p}}}) \alpha_{\delta_{{\mathfrak{p}}, 1}'}^* \in \operatorname{\mathrm{image}} \partial. \end{align*}$$

We claim that

(5.3) $$ \begin{align} {\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho(m)) = \prod_{{\mathfrak{p}}|p} {\mathcal{L}}(W_{{\mathfrak{p}}}). \end{align} $$

Note that, in the definition of ${\mathcal {L}}_{\operatorname {\mathrm {GB}}}(\rho (m))$ , one studies the cohomology of $\operatorname {\mathrm {Gr}}_1^{\operatorname {\mathrm {GB}}}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger }(\rho _{{\mathfrak {p}}}(m))$ . However, we are now having cohomology classes in $H^1(\operatorname {\mathrm {Gr}}_0^{\operatorname {\mathrm {GB}}}{\mathbf {D}}_{\operatorname {\mathrm {rig}}}^{\dagger } (\rho _{{\mathfrak {p}}}(m)))$ . To resolve this, we look at the short exact sequence

By [Reference Benois2, Proposition 2.2.4], the Greenberg–Benois ${\mathcal {L}}$ -invariant computed by this exact sequence (at each ${\mathfrak {p}}$ ) is the same as ${\mathcal {L}}_{\operatorname {\mathrm {GB}}}(\rho (m))$ . Here,

$$\begin{align*}\kappa_{{\mathfrak{p}}, i}(z) = \left(\prod_{\sigma: F_{{\mathfrak{p}}} \hookrightarrow \overline{{\mathbf{Q}}}_p} \sigma(z)^{k_{{\mathfrak{p}}, \sigma, n-i}-m+1} \right) |\mathrm{Nm}^{F_{{\mathfrak{p}}}}_{{\mathbf{Q}}_p}z|^{1-i}, \end{align*}$$

and we want to compute ${\mathcal {L}}_{\operatorname {\mathrm {GB}}}(\rho (m))$ using the cohomology of ${\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\kappa _{{\mathfrak {p}}, 1})$ .

By (B2), we have $k_{{\mathfrak {p}}, \sigma , n-1}-m+1\leq 0$ and we let $u_{{\mathfrak {p}}} := \min _{\sigma }\{ k_{{\mathfrak {p}}, \sigma , n-1}-m+1\}$ . By [Reference Rosso17, (2.8)], there is an injection

$$\begin{align*}H^1({\mathcal{R}}_{{\mathbf{Q}}_p, E}(z^{u_{{\mathfrak{p}}}})) \hookrightarrow H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\kappa_{{\mathfrak{p}}, 1})), \quad \begin{array}{l} x_{u_{{\mathfrak{p}}}} \mapsto x_{k_{{\mathfrak{p}}, \bullet, n-1}-m+1} \\ y_{u_{{\mathfrak{p}}}} \mapsto y_{k_{{\mathfrak{p}}, \bullet, n-1}-m+1} \end{array}, \end{align*}$$

where $x_{u_{{\mathfrak {p}}}}$ , $x_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1}$ , $y_{u_{{\mathfrak {p}}}}$ , and $ y_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1}$ are as defined in loc. cit. Footnote ⁹ ,Footnote ¹⁰ By the discussion on [Reference Rosso17, pp. 1238], we have a commutative diagram

(5.4)

where $\iota _c$ is an isomorphism. Moreover, [op. cit., Corollary 3.9] yields that

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho(m)) = \det(\iota_f \circ \iota_c^{-1}). \end{align*}$$

In particular, if ${\mathcal {L}}_{{\mathfrak {p}}}\in E$ such that

then

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho(m)) = \prod_{{\mathfrak{p}}|p} {\mathcal{L}}_{{\mathfrak{p}}}. \end{align*}$$

By definition, $H^1\left (\bigoplus _{{\mathfrak {p}}|p}D_{{\mathfrak {p}}}^{\vee }(1-m), \operatorname {\mathrm {Ind}}^{{\mathbf {Q}}}_{F}\rho ^{\vee }(1-m)\right ) \cong \bigoplus _{{\mathfrak {p}}|p}\frac {H^1(W_{{\mathfrak {p}}}^{\vee }(\chi _{\operatorname {\mathrm {cyc}}}))}{H_f^1(W_{{\mathfrak {p}}}^{\vee }(\chi _{\operatorname {\mathrm {cyc}}}))}$ . The vertical morphism in (5.4) is compatible with the natural morphism

$$\begin{align*}H^1(W_{{\mathfrak{p}}}^{\vee}(\chi_{\operatorname{\mathrm{cyc}}})) \rightarrow H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\kappa_{{\mathfrak{p}}, 1})), \end{align*}$$

induced from the short exact sequence (5.2). Note that the exact sequence

$$\begin{align*}H^1(W_{{\mathfrak{p}}}^{\vee}(\chi_{\operatorname{\mathrm{cyc}}})) \rightarrow H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\kappa_{{\mathfrak{p}}, 1})) \xrightarrow{\partial} H^2({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\kappa_{{\mathfrak{p}}, 0})) \end{align*}$$

is dual to the exact sequence

$$\begin{align*}H^0({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 0}')) \xrightarrow{\partial} H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}')) \rightarrow H^1(W_{{\mathfrak{p}}}). \end{align*}$$

We have $\partial ({\mathcal {L}}_{{\mathfrak {p}}} x_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1} + y_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1}) = 0 \in H^2({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\kappa _{{\mathfrak {p}}, 0}))$ and $\beta _{\delta _{{\mathfrak {p}}, 1}'}^* + {\mathcal {L}}(W_{{\mathfrak {p}}}) \alpha _{\delta _{{\mathfrak {p}}, 1}'}^* = 0 \in H^1(W_{{\mathfrak {p}}})$ . Moreover, using the relation between $x_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1}$ (resp., $y_{k_{{\mathfrak {p}}, \bullet , n-1}-m+1}$ ) and $\beta _{\delta _{{\mathfrak {p}}, 1}'}^*$ (resp., $-\alpha _{\delta _{{\mathfrak {p}}, 1}'}^*$ ), one sees that

$$\begin{align*}{\mathcal{L}}_{{\mathfrak{p}}} = {\mathcal{L}}(W_{{\mathfrak{p}}}), \end{align*}$$

which concludes our claim.

Step 3. Conclusion.

By construction, $W_{{\mathfrak {p}}}$ defines a class (see (5.2))

$$ \begin{align*} \mathrm{cl}(W_{{\mathfrak{p}}}) & \in \mathrm{Ext}_{(\varphi, \Gamma)}^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 0}'), {\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}')) \\ & \cong \mathrm{Ext}_{(\varphi, \Gamma)}^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}, {\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}'\delta_{{\mathfrak{p}}, 0}'^{-1})) \\ & \cong H^1({\mathcal{R}}_{F_{{\mathfrak{p}}}, E}(\delta_{{\mathfrak{p}}, 1}\delta_{{\mathfrak{p}}, 0}^{-1})). \end{align*} $$

Note that, as classes in $H^1({\mathcal {R}}_{F_{{\mathfrak {p}}}, E}(\delta _{{\mathfrak {p}}, 1}\delta _{{\mathfrak {p}}, 0}^{-1}))$ , we have

$$\begin{align*}\operatorname{\mathrm{cl}}(W_{{\mathfrak{p}}}) = \operatorname{\mathrm{cl}}(\widetilde{W}_{{\mathfrak{p}}}). \end{align*}$$

Unwinding everything, we have

$$\begin{align*}c\operatorname{\mathrm{cl}}({\mathcal{L}}(W_{{\mathfrak{p}}})v_{{\mathfrak{p}}, 1}, 0, -v_{{\mathfrak{p}}, 1}) = \operatorname{\mathrm{cl}}(W_{{\mathfrak{p}}}) = \operatorname{\mathrm{cl}}(\widetilde{W}_{{\mathfrak{p}}}) \stackrel{(5.1)}{=} \operatorname{\mathrm{cl}}(-{\mathcal{L}}_{\operatorname{\mathrm{FM}}}(\rho_{{\mathfrak{p}}})v_{{\mathfrak{p}}, 1}, 0, v_{{\mathfrak{p}}, 1}) \end{align*}$$

for some $c\in E$ . In particular, we conclude that

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{FM}}}(\rho_{{\mathfrak{p}}}) = -{\mathcal{L}}(W_{{\mathfrak{p}}}) \end{align*}$$

and so,

$$\begin{align*}{\mathcal{L}}_{\operatorname{\mathrm{GB}}}(\rho(m)) = \prod_{{\mathfrak{p}}|p} -{\mathcal{L}}_{\operatorname{\mathrm{FM}}}(\rho_{{\mathfrak{p}}}) \end{align*}$$

by (5.3).