text:
\section{Introduction}

In their remarkable paper \cite{MTT}, Mazur, Tate and Teitelbaum
proposed a conjectural formula for the derivative at $s=1$ of the
$p$-adic $L$-function of an elliptic curve $E$ over $\BQ$ when $p$
is a prime of split multiplicative reduction. An important quantity
in this formula is the so called $\CL$-invariant, namely
$\CL(E)=\log_p(q_E)/v_p(q_E)$ where $q_E\in  \BQ_p^\times$ is the
Tate period for $E$. This conjectural formula was proved by
Greenberg and Stevens \cite{GS} using Hida's families. Indeed, for
the weight $2$ newform $f$ attached to $E$, there exists a family of
$p$-adic ordinary Hecke eigenforms containing $f$. A key formula
they proved is \begin{equation} \CL(E)= -2
\frac{\alpha'(f)}{\alpha(f)} \label{eq:L-elliptic}
\end{equation} where $\alpha$ is the function of $U_p$-eigenvalues of
the eigenforms in the Hida family. On the other hand, they showed
that $-2\frac{\alpha'(f)}{\alpha(f)}$ is equal to $\frac{L'_p(f,
1)}{L(f,1)}$. Combining these two facts they obtained the
conjectural formula.

In this paper we will focus on (\ref{eq:L-elliptic}) which was later
generalized by Colmez \cite{Cz2010} to the non-ordinary setting. We
state Colmez's result below.

\begin{thm}\label{thm:colmez} $($\cite{Cz2010}$)$ Suppose that, at each closed point $z$ of $\mathrm{Max}(S)$
one of the Hodge-Tate weight of $\CV_z$ is $0$, and there exists
$\alpha\in S$ such that
$(\BB_{\cris,S}^{\varphi=\alpha}\widehat{\otimes}_S \CV
)^{G_{\BQ_p}}$ is locally free of rank $1$ over $S$. Suppose $z_0$
is a closed point of $\mathrm{Max}(S)$ such that $\CV_{z_0}$ is
semistable with Hodge-Tate weights \footnote{In this paper, the
Hodge-Tate weights are defined to be minus the generalized
eigenvalues of Sen's operators. In particular the Hodge-Tate weight
of the cyclotomic character $\chi_\cyc$ is $-1$.} $0$ and $k\geq 1$.
Then the differential
$$\frac{\mathrm{d}\alpha}{\alpha} - \frac{1}{2} \CL \mathrm{d}\kappa+
\frac{1}{2}\mathrm{d}\delta $$ is zero at $z_0$, where $\CL$ is the
Fontaine-Mazur $\CL$-invariant of $\CV_{z_0}$.
\end{thm}

See \cite{Cz2010} for the precise meanings of $\kappa$ and $\delta$.
Roughly speaking, $\mathrm{d}\delta$ is the derivative of Frobenius,
and $\mathrm{d}\kappa$ is the derivative of Hodge-Tate weights.

The condition that
``$(\BB_{\cris,S}^{\varphi=\alpha}\widehat{\otimes}_S \CV
)^{G_{\BQ_p}}$ is locally free of rank $1$ over $S$'' in Theorem
\ref{thm:colmez} is equivalent to that $\CV$ admits a triangulation
\cite{Cz2008}.
 So, Theorem \ref{thm:colmez} means that the derivatives of
Frobenius and the derivatives of Hodge-Tate weights of a family of
$2$-dimensional representations of $G_{\BQ_p}$ with a triangulation
satisfy a non-trivial relation at each semistable (but
non-crystalline) point.

Colmez's theorem was generalized by Zhang \cite{Zhang} for families
of $2$-dimensional Galois representations of $G_K$ ($K$ a finite
extension of $\BQ_p$) and Pottharst \cite{Pott} who considered
families of (not necessarily \'etale) $(\varphi,\Gamma)$-modules of
rank $2$ instead of families of $2$-dimensional Galois
representations.

In this paper we give a generalization of Colmez's theorem which
includes the above generalizations as special cases.

Fix a finite extension $K$ of $\BQ_p$. What we work with is a family
of $K$-$B$-pair (called $S$-$B$-pair in our context) that is locally
triangulable. We will provide conditions for Fontaine-Mazur
$\CL$-invariant to be defined. Note that, the $\CL$-invariant is now
a vector with component number equal to $[K:\BQ_p]$.

\begin{thm}\label{thm:main}
Let $W$ be an $S$-$B$-pair that is semistable at a point $z\in
\Max(S)$. Suppose that $W$ is locally triangulable at $z$ with the
local triangulation parameters $(\delta_1,\cdots,\delta_n)$. Assume
that for $D_z$, the filtered $E$-$(\varphi,N)$-module attached to
$W_z$, the Fontaine-Mazur $\CL$-invariant $\vec{\mathcal{L}}_{s,t}$
$($see Definition \ref{defn:Fontaine-Mazur}$)$ can be defined for
 $ s, t \in \{1,2,\cdots, n\}$.  Then
$$ \frac{1}{[K:\BQ_p]}\left(\frac{\mathrm{d} \delta_t(p)}{\delta_t(p)} - \frac{\mathrm{d}
\delta_s(p)}{\delta_s(p)}\right)
 + \vec{\CL}_{s,t} \cdot (\mathrm{d}\vec{w}(\delta_t)-\mathrm{d}\vec{w}(\delta_s)) = 0 . $$
Here, $\vec{w}(\delta_i)$ is the Hodge-Tate weight of the character
$\delta_i$.
\end{thm}

In \cite{Xie2015} we proved Theorem \ref{thm:main} for a special
case, where we consider the case of $K=\BQ_p$ and demand that the
Frobenius is simisimple at $z$. The motivation and some potential
applications of our theorem was also discussed in \cite{Xie2015}.

Our paper is orginized as follows. In Section \ref{sec:B-pairs} we
recall the theory of $B$-pairs built by Berger. Then in Section
\ref{sec:family-B-pairs} we extend a part of this theory to families
of $B$-pairs, and discuss the relation between triangulations of
semistable $B$-pairs and refinements of their associated filtered
$(\varphi,N)$-modules. In Section \ref{sec:coh} we compare
cohomology groups of $(\varphi,\Gamma)$-modules and those of
$B$-pairs, and then attach a $1$-cocycle to each infinitesimal
deformation of a $B$-pair. In Section \ref{sec:aux-for} we use the
reciprocity law to build an auxiliary formula for $L$-invariants.
The $L$-invariant is defined in Section \ref{sec:L-inv}. In Section
\ref{sec:proj-van-prop} we prove a formula called ``projection
vanishing property'' for the above $1$-cocycle. Finally in Section
\ref{sec:main} we use the auxiliary formula in Section
\ref{sec:aux-for} and the projection vanishing property to deduce
Theorem \ref{thm:main}.

\section*{Notations}

Let $K$ be a finite extension of $\BQ_p$, $G_K$ the absolute Galois
group $\mathrm{Gal}(\overline{K}/K)$. Let $K_0$ be the maximal
absolutely unramified subfield of $K$. Let $G_K^{\mathrm{ab}}$
denote the maximal abelian quotient of $G_K$.

Let $\chi_\cyc$ be the cyclotomic character of $G_K$, $H_K$ the
kernel of $\chi_\cyc$ and $\Gamma_K$ the quotient $G_K/H_K$. Then
$\chi_\cyc$ induces an isomorphism from $\Gamma_K$ onto an open
subgroup of $\BZ_p^\times$.

Let $E$ be a finite extension of $K$ such that all embeddings of $K$
into an algebraic closure of $E$ are contained in $E$,
$\mathrm{Emb}(K,E)$ the set of embeddings of $K$ into $E$. We
consider $E$ as a coefficient field and let $G_K$ acts trivially on
$E$.

Let $\mathrm{rec}_K$ be the reciprocity map of local class field
theory such that $\mathrm{rec}_K(\pi_K)$ is a lifting of the inverse
of $q$th power Frobenius of $k$, where $\pi_K$ is a uniformizing
element of $K$ and $k$ is the residue field of $K$ with cardinal
number $q$. Note that the image of $\mathrm{rec}_K$ coincides with
the image of the Weil group $W_K\subset G_K$ by the quotient map
$G_K\rightarrow G_K^{\mathrm{ab}}$. Let $\mathrm{rec}_K^{-1}: W_K
\rightarrow K^\times$ be the converse map of $\mathrm{rec}_K$.

\section{$(\varphi,\Gamma_K)$-modules and $B$-pairs}
\label{sec:B-pairs}

\subsection{Fontaine's rings}

We recall the construction of Fontaine's period rings. Please
consult \cite{Fontaine, Ber02} for more details.

Let $\BC_p$ be a completed algebraic closure of $\BQ_p$ with
valuation subring $\mathfrak{o}_{\BC_p}$ and $p$-adic valuation
$v_p$ normalized such that $v_p(p)=1$.

Let $\widetilde{\RE}$ be $\{ (x^{(i)})_{i\geq 0} \ | \ x^{(i)}\in
\BC_p, \ (x^{(i+1)})^p=x^{(i)} \ \forall \  i\in\BN \},$ and let
$\widetilde{\RE}^+$ be the subset of $\widetilde{\RE}$ such that
$x^{(0)}\in \mathfrak{o}_{\BC_p}$. If $x,y\in \widetilde{\RE}$, we
define $x+y$ and $xy$ by
$$ (x+y)^{(i)}=\lim_{j\rightarrow \infty} (x^{(i+j)}+y^{(i+j)})^{p^j} , \hskip 10pt (xy)^{(i)}=x^{(i)}y^{(i)} .
$$ Then $\widetilde{\RE}$ is a field of characteristic $p$.  Define a function
$v_\RE:\widetilde{\RE}\rightarrow \: \BR\cup\{+\infty\}$ by putting
$v_\RE((x^{(n)}))=v_p(x^{(0)})$. This is a valuation for which
$\widetilde{\RE}$ is complete and $\widetilde{\RE}^+$ is the ring of
integers in $\widetilde{\RE}$.  If we let
$\varepsilon=(\varepsilon^{(n)})$ be an element of
$\widetilde{\RE}^+$ with $\epsilon^{(0)}=1$ and $\epsilon^{(1)}\neq
1$, then $\widetilde{\RE}$ is a completed algebraic closure of
$\BF_p((\varepsilon-1))$. Put $\omega= [\varepsilon]-1$. Let
$\tilde{p}$ be an element of $\widetilde{\RE}$ such that
$\tilde{p}^{(0)}=p$.

Let $\widetilde{\RA}^+$ be the ring $\RW(\widetilde{\RE}^+)$ of Witt
vectors with coefficients in $\widetilde{\RE}^+$, $\widetilde{\RA}$
the ring of Witt vectors $\RW(\widetilde{\RE})$, and
$\widetilde{\RB}^+=\widetilde{\RA}[1/p]$. The map $$\theta:
\widetilde{\RB}^+\rightarrow \BC_p, \hskip 10pt \sum_{n\gg -\infty}
p^k[x_k] \mapsto \sum_{n\gg -\infty} p^k x_k^{(0)} $$ is surjective.
Let $\RB^+_\dR$ be the $\ker(\theta)$-adic completion of
$\widetilde{\RB}^+$. Then $t_\cyc=\log[\varepsilon]$ is an element
of $\RB^+_\dR$, and put $\RB_\dR=\RB^+_\dR[1/t_\cyc]$. There is a
filtration $\Fil^\bullet$ on $\RB_\dR$ such that $\Fil^i
\RB_\dR=\bigoplus_{j\geq i} \RB^+_\dR t_\cyc^j$.

Let $\RB^+_\max$ be the subring of $\widetilde{\RB}^+$ consisting of
elements of the form $\sum_{n\geq 0} b_n ([\tilde{p}]/p)^n$, where
$b_n\in \widetilde{\RB}^+$ and $b_n\rightarrow 0$ when $n\rightarrow
+\infty$. Put $\RB_\max=\RB_\max^+[1/t_\cyc]$; $\RB_\max$ is
equipped with a $\varphi$-action. Put
$\RB_\log=\RB_\max[\log[\tilde{p}]]$;
 $\RB_\log$ is equipped with a
$\varphi$-action and a monodromy $N$; $\RB_\log^{N=0}=\RB_\max$;
$\RB_\log$ is a subring of $\RB_\dR$. Put
$\RB_e=\RB_\max^{\varphi=1}$. We have the following fundamental
exact sequence
\[ \xymatrix{  0 \ar[r] & \BQ_p \ar[r] & \RB_e \ar[r] & \RB_\dR/\RB^+_\dR \ar[r] & 0 .  }\]

If $r$ and $s$ are two elements in $\BN[1/p]\cup \{+\infty\}$, we
put $\widetilde{\RA}^{[r,s]} =\widetilde{\RA}^+\{
\frac{p}{[\bar{\omega}^r]}, \frac{[\bar{\omega}^s]}{p}  \} $ and
$\widetilde{\RB}^{[r,s]}=\widetilde{\RA}^{[r,s]}[1/p]$ with the
convention that $p/[\bar{\omega}^{+\infty}]=1/[\bar{\omega}]$ and
$[\bar{\omega}^{+\infty}]/p=0$. We equip these rings with the
$p$-adic topology. There are natural continuous $G_K$-actions on
$\widetilde{A}_{[r,s]}$ and $\widetilde{B}_{[r,s]}$. Frobenius
induces isomorphisms $\varphi:
\widetilde{A}_{[r,s]}\xrightarrow{\sim} \widetilde{A}_{[pr,ps]}$ and
$\varphi: \widetilde{B}_{[r,s]}\xrightarrow{\sim}
\widetilde{B}_{[pr,ps]}$. If $r\leq r_0\leq s_0\leq s$, then we have
the $G_K$-equivariant injective natural map $\widetilde{A}_{[r,
s]}\hookrightarrow \widetilde{A}_{[r_0,s_0]}$. For $r>0$ we put
$\widetilde{\BB}^{\dagger, r}_\rig= \bigcap_{s\in [r,+\infty)}
\widetilde{B}_{[r,s]}$ (equipped with certain Frechet topology) and
$\widetilde{\BB}^\dagger_\rig= \cup_{r>0}\widetilde{\BB}^{\dagger,
r}_\rig$ (equipped with the inductive limit topology). Frobenius
induces isomorphisms $\varphi: \widetilde{\BB}^{\dagger,
r}_\rig\xrightarrow{\sim} \widetilde{\BB}^{\dagger, pr}_\rig$ and
$\varphi: \widetilde{\BB}^{\dagger}_\rig\xrightarrow{\sim}
\widetilde{\BB}^{\dagger}_\rig$.

Put $$ A_{K'_0}=\{\sum_{k\geq -\infty}^{+\infty}a_k\omega^k \ | \
a_k\in \mathfrak{o}_{K'_0} , \ a_k\rightarrow 0 \ \text{ when
}k\rightarrow-\infty)\} $$ and $B_{K'_0}=A_{K'_0}[1/p]$. Here $K'_0$
is the maximal absolutely unramified subfield of
$K_\infty=K(\mu_{p^\infty})$. Then $A_{K'_0}$ is a complete discrete
valuation ring with $p$ as a prime element, and $B_{K'_0}$ is the
fractional field of $A_{K'_0}$. The $G_K$-action and $\varphi$
preserve $A_{K'_0}$: $\varphi(\omega)=(1+\omega)^p-1$ and
$g(\omega)=(1+\omega)^{\chi_\cyc(g)}-1$. Let $\BA$ be the $p$-adic
completion of the maximal unramified extension of $A_{K'_0}$ in
$\widetilde{\BA}$, $\BB$ its fractional field. Then $\varphi$ and
the $G_K$-action preserve $\BA$ and $\BB$.

We put $\BB_K=\BB^{H_K}$ and $\BB^{\dagger,r}_K=\BB_K\cap
\widetilde{\BB}^{\dagger, r}$. Let $\BB^{\dagger,r}_{\rig,K}$ be the
Frechet completion of $\BB^{\dagger,r}_K$ for the topology induced
from that on $\widetilde{\BB}^{\dagger,r}_\rig$, and put
$\BB^\dagger_{\rig,K}=\cup_{r>0} \BB^{\dagger,r}_{\rig,K}$ equipped
with the inductive limit topology. Frobunius induces injections
$\BB^{\dagger,r}_{\rig,K}\hookrightarrow \BB^{\dagger,pr}_{\rig,K}$
and $\BB^{\dagger}_{\rig,K}\hookrightarrow\BB^{\dagger}_{\rig,K}$;
there are continuous $\Gamma_K$-actions on
$\BB^{\dagger,r}_{\rig,K}$ and $\BB^{\dagger}_{\rig,K}$.

\vskip 5pt

We end this subsection by the definition of
$E$-$(\varphi,\Gamma_K)$-modules \cite{Naka}.

\begin{defn} An $E$-$(\varphi,\Gamma_K)$-module is a finite
$\BB^{\dagger}_{\rig,K}\otimes_{\BQ_p}E$-module $M$ equipped with a
Frobenius semilinear action $\varphi_M$ and a comtinuous semilinear
$\Gamma_K$-action such that $M$ is free as a
$\BB^{\dagger}_{\rig,K}$-module, that
$\mathrm{id}_{\BB^\dagger_{\rig,K}}\otimes \varphi_M :
\BB^\dagger_{\rig,K}\bigotimes_{\varphi, \BB^\dagger_{\rig,K}} M
\rightarrow M $ is an isomorphism, and that $\varphi_M$ and the
$\Gamma_K$-action commute with each other.
\end{defn}

By \cite[Lemma 1.30]{Naka} if $M$ is an
$E$-$(\varphi,\Gamma_K)$-module,
then $M$ is free over $\BB^\dagger_{\rig,K}\otimes_{\BQ_p}E$.

\subsection{$B$-pairs}

We recall the theory of $E$-$B$-pairs \cite{Ber08, Naka}.

Put $\BB_{e,E}=\BB_e\otimes_{\BQ_p}E$,
$\BB^+_{\dR,E}=\BB^+_\dR\otimes_{\BQ_p}E$ and $\BB_{\dR,
E}=\BB_\dR\otimes_{\BQ_p}E$. We extend the $G_K$-actions
$E$-linearly to these rings.

\begin{defn} An {\it $E$-$B$-pair} of $G_K$ is a couple $W=(W_e, W_\dR^+)$
such that

$\bullet$ $W_e$ is a finite $\BB_{e,E}$-module with a continuous
semilinear action $G_K$-action which is free as a $\BB_e$-module.

$\bullet$ $W^+_\dR\subset W_\dR=\BB_\dR\otimes_{\BB_e} W_e$ is a
$G_K$-stable $\BB^+_{\dR,E}$-lattice.
\end{defn}

By \cite[Remark 1.3]{Naka} $W_e$ is free over $\BB_{e,E}$ and
$W^+_\dR$ is free over $\BB^+_{\dR,E}$.

If $V$ is an $E$-representation of $G_K$, then
$W(V)=(\BB_{e,E}\otimes_E V, \BB^+_{\dR,E}\otimes_E V)$ is an
$E$-$B$-pair, called the {\it $E$-$B$-pair attached to $V$}.

If $S$ is a Banach $E$-algebra, we can define $S$-$B$-pairs
similarly; to each $S$-representation $V$ of $G_K$ is associated an
$S$-$B$-pair $W(V)=(\BB_{e,E}\otimes_E V, \BB^+_{\dR,E}\otimes_E
V)$.

If $W_1=(W_{1,e}, W_{1,\dR}^+)$ and $W_2=(W_{2,e}, W^+_{2,\dR})$ are
two $E$-$B$-pairs, we define $W_1\bigotimes W_2$ to be
$$(W_{1,e}\bigotimes\limits_{\BB_{e,E}}W_{2,e},
W^+_{1,\dR}\bigotimes\limits_{\BB^+_{\dR,E}}W^+_{2,\dR}).$$ Here,
$W_{1,e}\bigotimes\limits_{\BB_{e,E}}W_{2,e}$ is equipped with the
diagonal $G_K$-action, and
$W^+_{1,\dR}\otimes_{\BB^+_{\dR,E}}W^+_{2,\dR}$ is naturally
considered as a $G_K$-stable $\BB^+_{\dR,E}$-lattice of
$$\BB_\dR\otimes_{\BB_e}(W_{1,e}\bigotimes_{\BB_{e,E}}W_{2,e})=
W_{1,\dR}\bigotimes_{\BB_{\dR,E}}W_{2,\dR},$$ where
$W_{1,\dR}=\BB_\dR\otimes_{\BB_e} W_{1,e}$ and
$W_{2,\dR}=\BB_\dR\otimes_{\BB_e} W_{2,e}$.

If $W=(W_e, W^+_\dR)$ is an $E$-$B$-pair with
$W_\dR=\BB_\dR\otimes_{\BB_e} W_e$, we define the dual of $W$ to be
$W^*=(W^*_e, W^{*,+}_\dR)$, where $W^*_e$ is
$\mathrm{Hom}_{\BB_e}(W,\BB_e)$ equipped with the natural
$G_K$-action, and $W^{*,+}_\dR$ is the $G_K$-stable lattice of
$\BB_\dR\otimes_{\BB_e} W^*_e\cong
\mathrm{Hom}_{\BB_\dR}(W_\dR,\BB_\dR)$ defined by
$$ \{\ell\in \mathrm{Hom}_{\BB_\dR}(W_\dR,\BB_\dR): \ell(x)\in \BB^+_\dR \text{ for all } x\in W^+_\dR \}.
$$

The relation between $(\varphi,\Gamma_K)$-modules and $B$-pairs is
built by Berger \cite{Ber08}. We recall Berger's construction below.

Let $M$ be a $(\varphi,\Gamma_K)$-module of rank $d$ over the Robba
ring $\BB^{\dagger}_{\rig,K}$. Berger \cite{Ber08} showed that
$$W_e(M):=(\widetilde{\BB}^\dagger_{\rig}[1/t]\otimes
_{\BB^\dagger_{\rig,K}}M)^{\varphi=1}$$ is a free $\BB_e$-module of
rank $d$ and equipped with a continuous semilinear $G_K$-action.

For sufficiently large $r_0>0$ we can take a unique
$\Gamma_K$-stable finite free $\BB^{\dagger,r}_{\rig,K}$-submodule
$M^{r}\subset M$ such that
$$\BB^{\dagger}_{\rig,K}\otimes_{\BB^{\dagger,r}_{\rig,K}}M^r=M$$ and
$$\mathrm{id}_{\BB^{\dagger, pr}_{\rig,K}}\otimes \varphi_M:
\BB^{\dagger,
pr}_{\rig,K}\otimes_{\BB^{\dagger,r}_{\rig,K}}M^r\xrightarrow{\sim}M^{pr}$$
for any $r\geq r_0$. Berger \cite{Ber08} showed that the
$\BB^+_\dR$-module
$$W^+_\dR(M):=\BB^+_\dR\otimes_{i_n, \BB^{\dagger,
(p-1)p^{n-1}}_{\rig,K}} M^{(p-1)p^{n-1}}$$  is independent of any
$n$ such that $(p-1)p^{n-1}\geq r_0$, and showed that there is a
canonical $G_K$-equivariant isomorphism
$\BB_\dR\otimes_{\BB_e}W_e(M)\xrightarrow{\sim}\BB_\dR\otimes_{\BB^+_\dR}W^+_\dR(M)$.

Put $W(M)=(W_e(M), W^+_\dR(M))$. This is an $E$-$B$-pair of rank
$d=\mathrm{rank}_{\BB^{\dagger}_{\rig,K}}M$.

The following is a variant version of Berger's result \cite[Theorem
2.2.7]{Ber08}.

\begin{prop}\label{thm:berger} \cite[Theorem 1.36]{Naka} The functor $M\mapsto W(M)$ is an exact functor and this
gives an equivalence of categories between the category of
$E$-$(\varphi,\Gamma_K)$-modules and the category of $E$-$B$-pairs
of $G_K$.
\end{prop}

\begin{prop} The functor $M\mapsto W(M)$ respects the tensor
products and duals.
\end{prop}
\begin{proof} Let $M_1$ and $M_2$ be two $E$-$(\varphi,\Gamma_K)$-modules. By taking $\varphi$-invariants, the
isomorphism
$$ (\widetilde{\BB}^\dagger_{\rig}[1/t]\otimes_{\widetilde{\BB}^{\dagger,r}_{\rig,K}}M_1
) \otimes_{\widetilde{\BB}^\dagger_{\rig}\otimes_{\BQ_p}E[1/t]}
(\widetilde{\BB}^\dagger_{\rig}[1/t]\otimes_{\widetilde{\BB}^{\dagger,r}_{\rig,K}}M_2
) \xrightarrow{\sim}
\widetilde{\BB}^\dagger_{\rig}[1/t]\otimes_{\widetilde{\BB}^\dagger_{\rig,K}}(M_1\otimes
M_2) $$ induces a $G_K$-equivariant injective map
$$ W_e(M_1)\otimes_{\BB_{e,E}} W_e(M_2) \rightarrow W_e(M_1\otimes M_2) .
$$ Here, $M_1\otimes
M_2$ denotes the $E$-$(\varphi,\Gamma_K)$-module $
M_1\otimes_{\BB^\dagger_{\rig,K}\otimes_{\BQ_p}E} M_2$. Comparing
dimensions and using \cite[Lemma 1.10]{Naka} we see that this map is
in fact an isomorphism. From the above Berger's construction we see
that the natural map $$
W^+_\dR(M_1)\otimes_{\BB^+_\dR\otimes_{\BQ_p}E}
W^+_\dR(M_2)\rightarrow W^+_\dR(M_1 \otimes M_2)$$ is an
isomorphism. This proves that the functor $M\mapsto W(M)$ respects
tensor products. The proof of that it respects duals is similar.
\end{proof}

\subsection{Semistable $E$-$B$-pairs}

\begin{defn} An {\it $E$-$(\varphi,N)$-module} over $K$ is a
$K_0\otimes_{\BQ_p}E$-module $D$ with a $\varphi\otimes
1$-semilinear isomorphism $\varphi_D: D\rightarrow D$, and a
$K_0\otimes_{\BQ_p}E$-linear map $N_D:D\rightarrow D$ such that
$N_D\varphi_D=p \varphi_D N_D$. A {\it filtered
$E$-$(\varphi,N)$-module} over $K$ is an $E$-$(\varphi,N)$-module
with an exhaustive $\BZ$-indexed descending filtration
$\Fil^\bullet$ on $K\otimes_{K_0}D$.
\end{defn}

We have an isomorphism of rings
\begin{equation} \label{eq:vector-element}
K\otimes_{\BQ_p}E\xrightarrow{\sim} \bigoplus\limits_{\tau\in
\mathrm{Emb}(K,E)}E_\tau, \ \  a\otimes b \mapsto ( \tau(a)b
)_{\tau}, \end{equation} where $E_\tau$ is a copy of $E$ for each
$\tau\in \mathrm{Emb}(K,E)$. Let $e_\tau$ be the unity of $E_\tau$.
Then $1=\sum_{\tau}e_\tau$. Put $D_\tau= e_\tau (K\otimes_{K_0}D)$.
Then
$K\otimes_{K_0}D=\bigoplus\limits_{\tau\in\mathrm{Emb}(K,E)}D_\tau$.
Let $\Fil_\tau$ denote the induced filtration on $D_\tau$.

\begin{defn} Let $W=(W_e, W^+_\dR)$ be an $E$-$B$-pair. We define
$\BD_\cris(W)=(\BB_{\mathrm{max}}\otimes_{\BB_e}W_e)^{G_K}$,
$\BD_\st(W)=(\BB_\log\otimes_{\BB_e}W_e)^{G_K}$ and
$\BD_\dR(W)=(\BB_\dR\otimes_{\BB_e}W_e)^{G_K}$.  Then we have
$\dim_{K_0}(\BD_?(W))\leq \mathrm{rank}_{\BB_e}W_e$ for $?=\cris,
\st$, and $\dim_{K}(\BD_\dR(W))\leq \mathrm{rank}_{\BB_e}W_e$. We
say that $W$ is {\it crystalline} (resp. {\it semistable}) if
$\dim_{K_0}(\BD_?(W)):= \mathrm{rank}_{\BB_e}W_e$ for $?=\cris$
(resp. $\st$).
\end{defn}

If $W$ is a semistable $E$-$B$-pair, we attach to $W$ a filtered
$E$-$(\varphi,N)$-module as follows. The underlying
$E$-$(\varphi,N)$-module is $\BD_\st(W)$; the filtration on
$\BD_\dR(W)=K\otimes_{K_0}\BD_\st(W)$ is given by $\Fil^i
\BD_\dR(W)= t^i W^+_\dR\cap \BD_\dR(W)$.

\begin{prop}\label{prop:berger}
\begin{enumerate}
\item\label{it:berger-a} The functor $W\mapsto \BD_\st(W)$ realizes an equivalence of
categories between the category of semistable $E$-$B$-pairs of $G_K$
and the category of filtered $E$-$(\varphi,N)$-modules over $K$.
\item\label{it:berger-b-2} If $W_1$ and $W_2$ are semistable, then
so is $W_1\otimes W_2$.
\item\label{it:berger-d} The functor $W\mapsto \BD_\st(W)$ respects the tensor products
and duals.
\item\label{it:berger-b} If
\[\xymatrix{ 0 \ar[r] & W_1 \ar[r] & W \ar[r] & W_2 \ar[r] & 0 }\]
is a short exact sequence of $E$-$B$-pairs, and $W$ is semistable,
then $W_1$ and $W_2$ are semistable.
\item\label{it:berger-c} The functor $W\mapsto \BD_\st(W)$ is exact.
\end{enumerate}
\end{prop}
\begin{proof} Assertion (\ref{it:berger-a}) follows from \cite[Proposition 2.3.4]{Ber08}. See also \cite[Theorem 1.18
(2)]{Naka}.

Let $W_1$ and $ W_2$ be two $E$-$B$-pairs. The isomorphism
$$ (\BB_{\log} \otimes_{\BB_e} W_1)
\otimes_{\BB_\log\otimes_{\BQ_p}E} (\BB_{\log} \otimes_{\BB_e} W_2)
\xrightarrow{\sim} \BB_{\log} \otimes_{\BB_e} (W_1 \otimes W_2)
$$ induces an injective map
\begin{equation}\label{eq:inj-iso}
\BD_\st(W_1)\otimes_{K_0\otimes_{\BQ_p}E} \BD_\st(W_2) \rightarrow
\BD_\st(W_1\otimes W_2).\end{equation} When $W_1$ and $W_2$ are
semistable, the dimension of the source over $K_0$ is
$\frac{\mathrm{rank}_{\BB_e}W_1
\mathrm{rank}_{\BB_e}W_2}{[E:\BQ_p]}$. The dimension of the target
over $K_0$ is always  equal to or less than  $\mathrm{rank}_{\BB_e}
(W_1\otimes W_2)=\frac{\mathrm{rank}_{\BB_e}W_1
\mathrm{rank}_{\BB_e}W_2}{[E:\BQ_p]}$. Hence, (\ref{eq:inj-iso}) is
an isomorphism, and so $W_1\otimes W_2$ is semistable. This proves
(\ref{it:berger-b-2}). Similarly, the isomorphism
\begin{equation} \label{eq:iso-deRham}
(\BB_{\dR} \otimes_{\BB_e} W_1) \otimes_{\BB_\dR\otimes_{\BQ_p}E}
(\BB_{\dR} \otimes_{\BB_e} W_2) \xrightarrow{\sim} \BB_{\dR}
\otimes_{\BB_e} (W_1 \otimes W_2)
\end{equation} induces an isomorphism
$$\BD_\dR(W_1)\otimes_{K\otimes_{\BQ_p}E} \BD_\dR(W_2) \rightarrow
\BD_\dR(W_1\otimes W_2).$$ Via the isomorphism (\ref{eq:iso-deRham})
the filtration on $(\BB_{\dR} \otimes_{\BB_e} W_1)
\otimes_{\BB_\dR\otimes_{\BQ_p}E} (\BB_{\dR} \otimes_{\BB_e} W_2)$
coincides with that on $\BB_{\dR} \otimes_{\BB_e} (W_1 \otimes W_2)
$. Therefore, the filtration on
$\BD_\dR(W_1)\otimes_{K\otimes_{\BQ_p}E} \BD_\dR(W_2) $ and that on
$\BD_\dR(W_1\otimes W_2)$ coincide. Indeed, they are the
restrictions of the filtrations on $(\BB_{\dR} \otimes_{\BB_e} W_1)
\otimes_{\BB_\dR\otimes_{\BQ_p}E} (\BB_{\dR} \otimes_{\BB_e} W_2)$
and $\BB_{\dR} \otimes_{\BB_e} (W_1 \otimes W_2)$ respectively.
Similarly we can show that $W\mapsto \BD_\st(W)$ respects duals.
This proves (\ref{it:berger-d}).

For (\ref{it:berger-b}) we have the following exact sequence
\begin{equation}\label{eq:exact-3-W} \xymatrix{ 0 \ar[r] & \BD_\st(W_1) \ar[r] & \BD_\st(W)
\ar[r] & \BD_\st(W_2)  .} \end{equation} So (\ref{it:berger-b})
follows from a dimension argument. Furthermore, when $W$ is
semistable,  $\BD_\st(W)\rightarrow \BD_\st(W_2)$ is surjective. For
any $i\in\BZ$ we write $d_i(W)$ for $\dim_{K}\Fil^i \BD_\st(W)$. As
the maps in the exact sequence (\ref{eq:exact-3-W}) respect
filtrations, we have $d_i(W)\leq d_i(W_1)+ d_i(W_2)$. Similarly, we
have $d_{1-i}(W^*)\leq d_{1-i}(W^*_1)+ d_{1-i}(W^*_2)$. As $W\mapsto
\BD_\st(W)$ respects duals, we have $d_i(W)=\dim_{K}(\BD_\dR(W))-
d_{1-i}(W^*)$. Then
\begin{eqnarray*} d_i(W) & = & \dim_{K}(\BD_\dR(W))- d_{1-i}(W^*) \\ & \geq
& (\dim_{K}(\BD_\dR(W_1))- d_{1-i}(W_1^*) ) +
\dim_{K}(\BD_\dR(W_2))- d_{1-i}(W_2^*) \\ &=&
d_i(W_1)+d_i(W_2).\end{eqnarray*} Thus we must have
$d_i(W)=d_i(W_1)+d_i(W_2)$ for all $i\in\BZ$. In other words, the
maps in (\ref{eq:exact-3-W}) are strict for the filtrations, which
shows (\ref{it:berger-c}).
\end{proof}

By \cite[Proposition 2.3.4]{Ber08} the quasi-inverse of the functor
$\BD_\st$
is given by
\begin{equation} \label{eq:def-DB}
\BD_B(D)=((\BB_{\log}\otimes_{K_0 }D)^{\varphi=1, N=0},
\Fil^0(\BB_{\dR }\otimes_{K_0 }D)).\end{equation}

For a filtered $E$-$(\varphi,N)$-module $D$ we put $$ \BX_\log(D)
=(\BB_\log\otimes_{K_0}D)^{\varphi=1, N=0} \  \ \text{and} \ \
\BX_\dR(D)= \BB_\dR \otimes_{K_0}D  / \Fil^0( \BB_\dR \otimes_{K_0}D
).
$$ If $\BD_B(D)=(W_e, W^+_\dR)$, then
$\BX_\log(D)=W_e$ and
$\BX_\dR(D)=(\BB_\dR\otimes_{\BB_e}W_e)/W^+_\dR$.

\section{$S$-$B$-pairs of rank $1$ and triangulations}
\label{sec:family-B-pairs}

\subsection{$S$-$B$-pairs of rank $1$}

Let $S$ be a Banach $E$-algebra.

For any $a\in S^\times$ we define a filtered $S$-$\varphi$-module
$D_a$ as follows. As a $K_0\otimes_{\BQ_p}S$-module,
$$D_a=K_0\otimes_{\BQ_p}S=\oplus_{\tau: K_0\hookrightarrow E}S
e_\tau;$$ the $\varphi\otimes 1$-semilinear action $\varphi$ on
$D_a$ satisfies
$$\varphi(e_{\mathrm{id}}) = e_{\varphi^{-1}}, \ \varphi(e_{\varphi^{-1}}) = e_{\varphi^{-2}}, \ \cdots, \ \varphi(e_{\varphi^{1-f}}) = a e_{\mathrm{id}};$$ the
descending filtration on $D_{a,K}=K\otimes_{\BQ_p}S$ is given by
$\Fil^0 D_{a,K}=D_{a,K}$ and $\Fil^1 D_{a,K}=0$.

\begin{lem} \label{lem:kisin} If $a\in S$ satisfies that $a-1$ is topologically nilpotent,
then there exists a unit $u_0\in \BB_\max\widehat{\otimes}_{K_0}S$
such that $\varphi^{[K_0:\BQ_p]}(u_0)= au_0$. Consequently
$$ \{x\in \BB_\max\widehat{\otimes}_{K_0}S: \varphi^{[K_0:\BQ_p]}(x)= ax\} = (\BB_{e,K_0}\widehat{\otimes}_{K_0}S) u_0 . $$
\end{lem}
\begin{proof} Let $\BQ_p^{\mathrm{ur}}$ be the completed unramified extension
of $\BQ_p$. Then there exists an inclusion
$\BQ_p^{\mathrm{ur}}\hookrightarrow \BB_\max$ that is compatible
with $\varphi$.

As $\varphi^{[K_0:\BQ_p]}-1$ is surjective on $\BQ_p^{\mathrm{ur}}$,
there exists a sequence $c_0=1, c_1, \cdots$ of elements in
$\BQ_p^{\mathrm{ur}}$ such that
$$(\varphi^{[K_0:\BQ_p]}-1)c_i=c_{i-1}$$ for $i\geq 1$. The image of
$c_i$ by the map
$$\BQ_p^{\mathrm{ur}}\hookrightarrow\BB_\max\rightarrow
\BB_\max\widehat{\otimes}_{K_0}S$$ is again denoted by $c_i$. Put
$$u_0=\sum_{i=0}^{\infty}c_i(a-1)^i.$$ Then $u_0$ is a unit and we have
$\varphi^{[K_0:\BQ_p]}u_0=au_0$.
\end{proof}

\begin{prop}\label{prop:S-B-pair} If $a\in S$ satisfies that $a-1$ is topologically
nilpotent, then $\BD_B(D_a)$ is an $S$-$B$-pair of rank $1$. Here
$\BD_B$ is the functor defined by $($\ref{eq:def-DB}$)$.
\end{prop}
\begin{proof} For each $z\in \BB_\max\widehat{\otimes}_{\BQ_p}D_a$ we write $z=\sum c_\tau
e_\tau$ with $c_\tau\in \BB_\max\widehat{\otimes}_{K_0,\tau}S$. Then
$\varphi(z)=z$ if and only if
$\varphi(c_{\varphi^i})=c_{\varphi^{i-1}}$ ($i=1,\cdots,
[K_0:\BQ_p]$) and
$\varphi^{[K_0:\BQ_p]}(c_{\mathrm{id}})=ac_{\mathrm{id}}$. Our
assertion follows from Lemma \ref{lem:kisin}.
\end{proof}

For any $a\in S^\times$, let $\delta_a: K^\times \rightarrow
S^\times$ denote the character such that $\delta_a(\pi_K)=a$ and
$\delta_a|_{\mathfrak{o}_K^\times}=1$.

\begin{rem} In the case of $S=E$, for any $u\in E^\times$,  $\BD_B(D_u)$ coincides with the $E$-$B$-pair
$W(\delta_u)$ defined in \cite{Naka} (see \cite[\S 1.4]{Naka}). From
now on the base change of $W(\delta_u)$ from $E$ to $S$ is again
denoted by $W(\delta_u)$.
\end{rem}

Let $\delta: K^\times \rightarrow S^\times$ be a continuous
character such that $\delta(\pi_K)$ is of the form
$\delta(\pi_K)=au$, where $u\in E^\times$ and $a\in S$ satisfies
that $a-1$ is topologically nilpotent. We call such a character {\it
a good character}. Let $W_a$ be the resulting $S$-$B$-pair in
Proposition \ref{prop:S-B-pair}. Let $\delta'$ be the unitary
continuous character $K^\times\rightarrow E^\times$ such that
$\delta'|_{\mathfrak{o}^\times_K}=\delta|_{\mathfrak{o}^\times_K}$
and $\delta'(\pi_K)=1$. By local class field theory, this induces a
continuous character $\widetilde{\delta}': G_K\rightarrow S^\times$
such that $\widetilde{\delta}'\circ \mathrm{rec}_K=\delta'$. Then we
put $$ W(\delta)=W(S(\widetilde{\delta}'))\otimes W(\delta_u)
\otimes W_a, $$ where $W(S(\widetilde{\delta}'))$ is the
$S$-$B$-pair attached to the Galois representation
$S(\widetilde{\delta}')$.

If $\delta$ is a continuous character $\delta: K^\times\rightarrow
S^\times$, we write $\log(\delta)$ for the logarithmic of
$\delta|_{\mathfrak{o}_K^\times}$, which is a $\BZ_p$-linear
homomorphism $\log(\delta): K \rightarrow S$.

For any $\tau\in\mathrm{Emb}(K,E)$ we use the same notation $\tau$
to denote the composition of $\tau: K\hookrightarrow E$ and $E
\hookrightarrow S$. Then $\{\tau: K\hookrightarrow S\}$ is a basis
of $\Hom_{\BZ_p}(E, S)$ over $S$. Write
$\log(\delta)=\sum_{\tau}k_\tau \tau$, $k_\tau\in S$. We call
$(k_\tau)_{\tau}$ the {\it weight vector} of $\delta$ and denote it
by $\vec{w}(\delta)$. We use $w_\tau(\delta)$ to denote $k_\tau$.

\begin{rem} Let $S$ be an affinoid algebra over $E$. For any continuous character $\delta: K^\times
\rightarrow S^\times$ and any point $z_0$ of $\Max(S)$, there exists
an affinoid neighborhood $U=\Max(S')$ of $z_0$ in $\Max(S)$ such
that the restriction of $\delta$ to $U$ is good.
\end{rem}

\begin{lem}\label{lem:varph-gamma-fil} Let $\delta$ be a character of $K^\times$ with values in
$S=E[Z]/(Z^2)$, $\bar{\delta}$ the character of $K^\times$ with
values in $E$ obtained from $\delta$ modulo $(Z)$. Write
$\delta=\bar{\delta}_S(1+Z\epsilon)$, where $\bar{\delta}_S$ is the
character $K^\times \xrightarrow{\bar{\delta}} E^\times
\hookrightarrow S^\times$. Let $\epsilon'$ be the additive character
of $G_K$ such that $\epsilon' \circ\mathrm{rec}_K(p)=0$ and
$\epsilon'\circ\mathrm{rec}_K|_{\fo_K^\times}=\epsilon|_{\fo_K^\times}$.

Assume that $W(\bar{\delta})$ is crystalline and
$\varphi^{[K_0:\BQ_p]}$ acts on $\BD_\cris(W(\bar{\delta}))$ by
$\alpha$. Then there is a nonzero element $$ x\in
(\BB_{\max,E}\otimes_{\BB_{e,E}}W(\delta)_e)^{\varphi^{[K_0:\BQ_p]}=\alpha
(1+Zv_p(\pi_K)\epsilon(p)), G_K= (1+Z\epsilon')} $$ whose reduction
modulo $Z$ is a basis of $\BD_\st(W(\bar{\delta}))$ over
$K\otimes_{\BQ_p}E$.
\end{lem}
\begin{proof} This follows from the fact that
$W(\delta)= W(\bar{\delta}_S)\otimes
W_{\delta_{1+Zv_p(\pi_K)\epsilon(p)}}\otimes W(1+Z\epsilon') .$
\end{proof}

\subsection{Triangulations and refinements}

Now let $S$ be an affinoid algebra over $E$. For any open affinoid
subset $U$ of $S$ and any $S$-$B$-pair $W$ let $W_U$ denote the
restriction to $U$ of $W$.

\begin{defn} Let $W$ be an $S$-$B$-pair of rank $n$, $z_0$ a point of $\Max(S)$. If there
is
\begin{quote}
$\bullet$ an affinoid neighborhood $U=\Max(S_U)$ of $z_0$,

$\bullet$ a strictly increasing filtration $$ \{0\}= \Fil_0 W_U
\subset \Fil_1 W_U \subset\cdots \subset \Fil_n W_U = W_U $$ of
saturated free sub-$S_U$-$B$-pairs, and

$\bullet$ $n$ good continuous characters $\delta_i:
\BQ_p^\times\rightarrow S_U^\times$

\end{quote}
\noindent such that for any $i=1,\cdots, n$,
$$\Fil_i W_U/\Fil_{i-1} W_U \simeq W(\delta_i),$$
we say that $W$ is {\it locally triangulable at $z_0$};  we call
$\Fil_\bullet$ a {\it local triangulation} of $W$ at $z_0$, and call
$(\delta_1,\cdots, \delta_n)$ the {\it local triangulation
parameters} attached to $\Fil_\bullet$.
\end{defn}

Please consult \cite{Cz2010,Be2011} for more knowledge on
triangulations.

To discuss the relation between triangulations and refinements, we
restrict ourselves to the case of $S=E$.

Let $D$ be a filtered $E$-$(\varphi,N)$-module of rank $n$. The
operator $\varphi^{[K_0:\BQ_p]}$ on $D$ is
$K_0\otimes_{\BQ_p}E$-linear. We assume that the eigenvalues of
$\varphi^{[K_0:\BQ_p]}:D\rightarrow D$ are all in
$K_0\otimes_{\BQ_p}E$, i.e. there exists a basis of $D$ over
$K_0\otimes_{\BQ_p}E$ such the matrix of $\varphi^{[K_0:\BQ_p]}$
with respect to this basis is upper-triangular.

Following Mazur \cite{Mazur} we define a {\it refinement} of $D$ to
be a filtration on $D$
$$ 0 = \CF_0 D \subset \CF_1 D \subset \cdots \subset \CF_n D= D $$ by
$E$-subspaces stable by $\varphi_D$ and $N_D$, such that each factor
$\mathrm{gr}^\CF_i D= \CF_i D/\CF_{i-1} D$ ($i=1,\cdots, n$) is of
rank $1$ over $K_0\otimes_{\BQ_p}E$. Any refinement fixes an
ordering $\alpha_1,\cdots, \alpha_n$ of eigenvalues of
$\varphi^{[K_0:\BQ_p]}$ and an ordering $\vec{k}_1,\cdots,
\vec{k}_n$ of Hodge-Tate weights of $K\otimes_{K_0}D$ taken with
multiplicities such that the eigenvalue of $\varphi^{[K_0:\BQ_p]}$
on $\mathrm{gr}^\CF_i D$ is $\alpha_i$ and the Hodge-Tate weight of
$\mathrm{gr}^\CF_i\CD$ is $\vec{k}_i$.

We have the following analogue of \cite[Proposition 1.3.2]{Ben}.

\begin{prop} Let $W$ be a semistable $E$-$B$-pair,
$D=\BD_\st(W)$.
\begin{enumerate}
\item\label{it:trian-refine-a}
The equivalence of categories between the category of semistable
$E$-$B$-pairs and the category of filtered $E$-$(\varphi,N)$-modules
induces a bijection between the set of triangulations on $W$ and the
set of refinements on $D$.
\item\label{it:trian-refine-b}
If $(\Fil_i W)$ is a triangulation of $W$ with triangulation
parameters $(\delta_1,\cdots, \delta_n)$ that correspond to a
refinement $\CF_\bullet D$ of $D$ with the ordering of Hodge-Tate
weights being $\vec{k}_1, \cdots, \vec{k}_n$, then
$\delta_i=\tilde{\delta}_i\prod\limits_{\tau\in\mathrm{Emb}(K,E)}\tau(x)^{k_{i,\tau}}$,
where $\tilde{\delta}_i$ is a smooth character.
\end{enumerate}
\end{prop}
\begin{proof} Assertion (\ref{it:trian-refine-a}) follows from the
fact that $\BD_\st$ is an exact. Assertion (\ref{it:trian-refine-b})
follows from \cite[Lemma 4.1]{Naka}.
\end{proof}

\section{Cohomology Theory} \label{sec:coh}
\subsection{Cohomology of $(\varphi,\Gamma_K)$-modules and cohomology of
$B$-pairs} \label{sec:coh-B-pair}

Let $M$ be a $(\varphi,\Gamma_K)$-module. Assume that $\Gamma_K$ has
a topological generator $\gamma$. Define the cohomology
$H^\bullet_\PGam(M)$ by the complex $C^\bullet(M)$ defined by
\[ C^0(M)=M \xrightarrow{(\gamma-1, \varphi-1)} C^1(M)=M\oplus M \rightarrow C^2(M)=M,
\] where the map $C^1(M) \rightarrow C^2(M)$ is given by $(x,y)\mapsto
(\varphi-1)x-(\gamma-1)y$. Denote the kernel of $C^1(M)\rightarrow
C^2(M)$ by $Z^1(M)$.

There is a one-to-one correspondence between $H^1(M)$ and the set of
extensions of $M_0$ by $M$ in the category of
$(\varphi,\Gamma_K)$-modules, where $M_0=\BB^{\dagger}_{\rig,K}e_0$
is the trivial $(\varphi,\Gamma_K)$-module with
$\varphi(e_0)=\gamma(e_0)=e_0$. Let $\tilde{M}$ be an extension of
$M_0$ by $M$, and let $\tilde{e}$ be any lifting of $e_0$ in
$\tilde{M}$. Then the element in $H^1(M)$ corresponding to the
extension $\tilde{M}$ is the class of $((\gamma-1)\tilde{e},
(\varphi-1)\tilde{e})\in Z^1(M)$.

In \cite{Naka} Nakamura introduced a cohomology for $B$-pairs and
use it to compute the cohomology of $(\varphi,\Gamma_K)$-modules.

If $W=(W_e, W^+_\dR)$ is an $E$-$B$-pair, let $C^\bullet(W)$ be the
complex of $G_K$-modules defined by
$$ C^0(W):=W_e \rightarrow C^1(W):= W_\dR/W^+_\dR.
$$ Here, $W_e\rightarrow W_\dR/W^+_\dR$ is the natural map.

\begin{defn} Let $W=(W_e, W^+_\dR)$ be an $E$-$B$-pair. We
define the {\it Galois cohomology of $W$} by $H^i_B(W):= H^i(G_K,
C^\bullet(W))$.
\end{defn}

By definition there is a long exact sequence \begin{equation}
\label{eq:long-exact-sq}
\cdots \rightarrow H^i_B(W) \rightarrow
H^i(G_K, W_e) \rightarrow H^i(G_K, W_\dR/W^+_\dR) \rightarrow \cdots
.
\end{equation}

For a $G_K$-module $M$ put $C^0(M)=M$ and let $C^i(M)$ be the space
of continuous functions from $(G_K)^{\times i}$ to $M$. Let
$\delta_0: C^0(M)\rightarrow C^1(M)$ be the map $x\mapsto (g \mapsto
g(x)-x)$ and let $\delta_1: C^1(M)\rightarrow C^2(M)$ be the map $f
\mapsto ((g_1,g_2)\mapsto f(g_1g_2)-f(g_1)-g_1f(g_2))$.

Nakamura \cite{Naka} showed that $H^1_B(W)$ is isomorphic to
$\ker(\tilde{\delta}_1)/\mathrm{im}(\tilde{\delta}_0)$, where
$\tilde{\delta_0}$ and $\tilde{\delta}_1$ are defined by
\begin{eqnarray*} \tilde{\delta}_0: C^0(W_e)\oplus C^0(W^+_\dR)
& \rightarrow & C^1(W_e)\oplus C^1(W^+_\dR)\oplus C^0(W_\dR) :\\
(x,y) & \mapsto & (\delta_0(x), \delta_0(y), x-y), \\
\tilde{\delta}_1: C^1(W_e)\oplus C^1(W^+_\dR)\oplus C^0(W_\dR)
& \rightarrow &  C^2(W_e)\oplus C^2(W^+_\dR)\oplus C^1(W_\dR):\\
(f_1,f_2, x) & \mapsto & (\delta_1(f_1), \delta_1(f_2),
f_1-f_2-\delta_0(x)).
\end{eqnarray*}
The map $H^1_B(W)\rightarrow H^1(G_K, W_e)$ is induced by the
forgetful map $$ C^1(W_e)\oplus C^1(W^+_\dR)\oplus
C^0(W_\dR)\rightarrow C^1(W_e). $$

There is a one-to-one correspondence between $H^1(G_K,W)$ and the
set of extensions of $W_0$ by $W$ in the category of $E$-$B$-pairs.
Here, $W_0=(\BB_e\otimes_{\BQ_p}E, \BB^+_{\dR}\otimes_{\BQ_p}E)$ is
the trivial $E$-$B$-pair. Let $\tilde{W}=(\tilde{W}_e,
\tilde{W}^+_\dR)$ be an extension of $W_0$ by $W$. Let
$(\tilde{w}_e, \tilde{w}^+_\dR)$ be a lifting in $\tilde{W}$ of
$(1,1)\in W_0$. Then the element in $H^1_B(W)$ corresponding to the
extension $\tilde{W}$ is just the class of $( (\sigma\mapsto
(\sigma-1)\tilde{w}_e), (\sigma\mapsto (\sigma-1)\tilde{w}^+_\dR),
\tilde{w}_e-\tilde{w}^+_\dR)\in \ker(\tilde{\delta}_1)$.

By Proposition \ref{thm:berger} there is a one-to-one correspondence
between $\Ext(M_0,M)$ and $\Ext(W_0, W(M))$. It induces a natrual
isomorphism
$$ i_M: H^1_{\PGam}(M) \rightarrow H^1_B( W(M)). $$

\subsection{$1$-cocycles from infinitesimal
deformations} \label{ss:cocycle-inform}

Let $S$ be the $E$-algebra $E[Z]/(Z^2)$, $\tilde{M}$ an
$S$-$(\varphi,\Gamma_K)$-module. Let $\{e_1, \cdots, e_n\} $ be an
$S$-basis of $\tilde{M}$, $\{e^*_1, \cdots, e^*_n\}$ the dual basis
of $\tilde{M}^*$. Put $M=\tilde{M}\otimes_S E$ and
$M^*=\tilde{M}^*\otimes_SE$. Let $e_{i,z}$ denote $e_i \ \mathrm{
mod } \ Z$, and $e^*_{j,z} $ denote $e^*_j \ \mathrm{ mod } \ Z$.
Then $\{e_{1,z},\cdots, e_{n,z}\}$ is an $E$-basis of $M$, and
$\{e^*_{1,z},\cdots, e^*_{n,z}\}$ is an $E$-basis of $M^*$.

The matrices of $\varphi$ and $\gamma$ with respect to
$\{e_1,\cdots, e_n\}$ are denote by $\tilde{A}_\varphi$ and
$\tilde{A}_\gamma$ respectively, so that $ \varphi(e_j) = \sum_{i}
(\tilde{A}_{\varphi})_{ij} e_i$ and $\gamma (e_j)=\sum_i
(\tilde{A}_\gamma)_{ij} e_i$. Write $\tilde{A}_\varphi = (I_n+
ZU_\varphi)A_\varphi$ and $\tilde{A}_\gamma = (I_n+
ZU_\gamma)A_\gamma$. Put
$$  c_{\PGam}(\tilde{M})=( \sum\limits_{i,j} (U_{\varphi})_{ i,j} e^*_{j,z}\otimes e_{i,z}  , \sum\limits_{i,j}
(U_{\gamma})_{ i,j} e^*_{j,z}\otimes e_{i,z} ).$$

Write $\BD_B(\tilde{M})=(\tilde{W}_e, \tilde{W}^+_\dR)$,
$\BD_B(M)=W$ and $\BD_B(M^*)=W^*$.

Let $f_1, \cdots, f_n$ be a basis of $\tilde{W}_e$ over $\BB_{e,E}$,
and let
 $g_1,\cdots, g_n$ be a basis of
$\tilde{W}^+_\dR$ over $\BB^+_{\dR,E}$. We write the matrix of
$\sigma\in G_K$ with respect to the basis $\{f_1, \cdots, f_n\}$ by
$(I_n + Z U_{e,\sigma})A_{e,\sigma}$, and the matrix of $\sigma$
with respect to the basis $\{g_1, \cdots, g_n\}$ by $(I_n + Z
U^+_{\dR,\sigma})A^+_{\dR,\sigma}$. Here, $$ U_{e, \sigma} \in
\mathrm{M}_{n }(\BB_{e,E}), \ U^+_{\dR,\sigma}\in \mathrm{
M}_{n}(\BB^+_{\dR,E}), \ A_{e, \sigma} \in \GL_{n}(\BB_{e,E}), \
\text{and} \  A^+_{\dR,\sigma}\in \GL_{n}(\BB^+_{\dR,E}). $$ Write
$(f_1, \cdots, f_n) = (g_1, \cdots, g_n) (I_n +Z U_\dR)A_\dR$ and
put {\small
$$c_{B}(\tilde{M}) =\Big( (\sigma\mapsto \sum_{i,j}(U_{e, \sigma})_{ij}f_{j,z}^*\otimes f_{i,z}),
(\sigma\mapsto \sum_{i,j}(U^+_{\dR, \sigma})_{ij}g_{j,z}^*\otimes
g_{i,z}),  \sum_{i,j}(U_{\dR})_{ij}g_{j,z}^*\otimes g_{i,z} \Big).
$$ }

\begin{prop}
\begin{enumerate}
\item\label{it:PGam-B-a} $c_\PGam(\tilde{M})$ is in $Z^1(M^*\otimes
M)$.
\item\label{it:PGam-B-b} $c_B(\tilde{M})$ is in $\ker(\tilde{\delta}_{1, W^*\otimes W})$.
\item\label{it:PGam-B-c} We have $i_M([c_\PGam (\tilde{M})])=[c_B(\tilde{M})]$.
\end{enumerate}
\end{prop}
\begin{proof} It is easy to verify (\ref{it:PGam-B-a}) and (\ref{it:PGam-B-b}).

Put $M^*_S=M^*\otimes_ES$. We consider $M^*_S\otimes_S \tilde{M}$ as
an extension of $M^*\otimes_E M$ by itself, and form the following
commutative diagram
$$ \xymatrix{ && & M_0\ar[d] \\ 0 \ar[r] & M^*\otimes_E M \ar[r] & M^*_S\otimes_S \tilde{M} \ar[r] & M^*\otimes_E M \ar[r] & 0,}
$$ where the vertical map $M_0\rightarrow M^*\otimes_E M$ is given
by $1\mapsto \sum_{i=1}^n e^*_{i,z}\otimes e_{i,z}$, which does not
depend on the choice of the basis $\{e_1,\cdots, e_n\}$. Pulling
back $M^*_{S}\otimes_S \tilde{M}$ via $M_0\rightarrow M^*\otimes_E
M$ we obtain an extension of $M_0$ by $M^*\otimes_E M$. Let
$\mathcal{M}$ denote the resulting extension. Then $\mathcal{M}$ is
a sub-$E$-$B$-pair of $M^*_S\otimes_S \tilde{M}$. Put
$\BD_B(\mathcal{M})=(\mathcal{W}_e, \mathcal{W}^+_\dR)$.

A lifting of $1$ in $\mathcal{W}_e$ is $\sum_j f^*_{j,z}\otimes
f_j$, and a lifting of $1$ in $\mathcal{W}^+_\dR$ is $\sum_j
g^*_{j,z}\otimes g_j$. We have \begin{eqnarray*} (\sigma-1)\sum_j
f^*_{j,z}\otimes f_j &=& \sigma (f^*_{1,z},\cdots, f^*_{n,z})
\otimes \sigma \left(
\begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_n \end{array}\right) - (f^*_{1,z},\cdots, f^*_{n,z})
\otimes \left(
\begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_n \end{array}\right)
\\
&=& (f^*_{1,z},\cdots, f^*_{n,z}) (A_{e,\sigma}^t)^{-1} \otimes
A_{e,\sigma}^t(1+zU_{e,\sigma}^t) \\
&=& (f^*_{1,z},\cdots, f^*_{n,z}) \otimes U_{e,\sigma}^t z \left(
\begin{array}{c} f_1 \\ f_2 \\ \vdots \\ f_n \end{array}\right).
\end{eqnarray*} Similarly,
\begin{eqnarray*} (\sigma-1)\sum_j
g^*_{j,z}\otimes g_j = (g^*_{1,z},\cdots, g^*_{n,z}) \otimes
(U^+_{\dR,\sigma})^t z \left(
\begin{array}{c} g_1 \\ g_2 \\ \vdots \\ g_n \end{array}\right),
\end{eqnarray*}
and
$$ \sum_j
f^*_{j,z}\otimes f_j - \sum_j g^*_{j,z}\otimes g_j =
(g^*_{1,z},\cdots, g^*_{n,z}) \otimes U_{\dR}^t z \left(
\begin{array}{c} g_1 \\ g_2 \\ \vdots \\ g_n \end{array}\right). $$
Hence the element in $H^1_B(\BD_B(M^*\otimes_EM))$ attached to the
extension $\BD_B(\mathcal{M})$ is $[c_B(\tilde{M})]$.

A similar computation shows that the element in
$H^1_\PGam(M^*\otimes_E M)$ attached to the extension $\mathcal{M}$
is $[c_\PGam(\tilde{M})]$. Now (\ref{it:PGam-B-c}) follows.
\end{proof}

\section{The reciprocity law and an application} \label{sec:aux-for}

\subsection{Reciprocity law}

In \cite[Section 2]{Zhang} using local class field theory Zhang
precisely described the perfect pairing
$$ H^1(G_K, E) \times H^1(G_K,E(1)) \rightarrow H^2(G_K, E(1)). $$
We recall it below.

The Kummer theory gives us a canonical isomorphism so called the
Kummer map
\begin{eqnarray*}
\lim_{\overleftarrow{\;\;
n\;\:}}(K^\times/(K^\times)^{p^n})\otimes_{\BZ_p} E & \rightarrow &
H^1(G_K, E(1)) \\ \sum_i \alpha_i\otimes a_i & \mapsto & \sum_i a_i
[(\alpha_i)].
\end{eqnarray*} Here $(\alpha)$ is the $1$-cocycle such that
$$ \frac{g (\sqrt[p^n]{\alpha})}{\alpha} = \varepsilon_n^{(\alpha_g)}
$$ for $\alpha\in K^\times$ and $g\in G_K$, where
$(\sqrt[p^{n+1}]{\alpha})^p=\sqrt[p^n]{\alpha}$. Combining the
Kummer map and the exponent map $$\exp: p\fo_K\rightarrow K^\times$$
and extending it by linearity we obtain an embedding from
$K\otimes_{\BQ_p}E$ to $H^1(G_K, E(1))$, again denoted by $\exp$.
Then we have $$ H^1(G_K, E(1))=\exp(K\otimes_{\BQ_p}E)\oplus E \cdot
[(p)].$$

Let $\Hom(G_K,E)$ be the group of additive characters of $G_K$ with
values in $E$. As the action of $G_K$ on $E$ is trivial,
$H^1(G_K,E)$ is naturally isomorphic to $\Hom(G_K, E)$. Let
$\psi_0:G_K\rightarrow E$ be the additive character that vanishes on
the inertial subgroup of $G_K$ and maps the geometrical Frobenius to
$[K_0:\BQ_p]$. For any $\tau\in \mathrm{Emb}(K,E)$ let $\psi_\tau$
be the composition $\tau\circ \log \circ \mathrm{rec}_K^{-1}$
\footnote{ Since the character $\psi_\tau$ of the Weil group $W_K$
sends any lifting of the $q$th power Frobenius to $0$, it can be
extended to a character of $G_K$ which is again denoted by
$\psi_\tau$}, where $\log$ is normalized such that $\log (p)=0$.
Then $\{\psi_0, \psi_\tau : \tau\in \mathrm{Emb}(K,E) \}$ is an
$E$-basis of $H^1(G_K,E)$.

\begin{lem} \label{lem:zhang}
\cite[Proposition 2.1]{Zhang} The cup product of $a_0 \psi_0 +
\sum_{\tau\in \mathrm{Emb}(K,E)} a_\tau \psi_\tau $ $(a_0, a_\tau\in
E)$ and $b_0[(p)]+  \exp(b)$ $(b_0\in E, b\in K\otimes_{\BQ_p} E)$
is
$$\Big( a_0 b_0 - \mathrm{tr}_{K/\BQ_p} ((a_\tau)_\tau \cdot b) \Big) (\psi_0\cup [(p)]). $$
Here,  $(a_\tau)_{\tau}$ is considered as an element in
$K\otimes_{\BQ_p}E$ via the isomorphism $(\ref{eq:vector-element})$.
\end{lem}

\begin{lem}\label{lem:de-Rham-cocycle}
For $\lambda_0, \lambda_\tau\in E$ $(\tau\in \mathrm{Emb}(K,E) )$,
the extension of $E$ $($as a trivial $G_K$-module$)$ by $E$
corresponding to the cocycle $\lambda_0\psi_0+\sum_{\tau\in
\mathrm{Emb}(K,E)}\lambda_\tau\psi_\tau$ is de Rham if and only if
$\lambda_\tau=0$ for each $\tau$.
\end{lem}
\begin{proof} By \cite[Lemma 4.3]{Naka}, the subspace of extensions
of $E$ by $E$ that are de Rham is $1$-dimensional, and so consists
of those corresponding to the cocycles $\lambda_0\psi_0$
($\lambda_0\in E$).
\end{proof}

\subsection{An auxiliary formula}

Let $\vec{\CL}=(\CL_\sigma)_{\sigma: K\hookrightarrow E}$ be a
vector. We consider $\vec{\CL}$ as an element of $K\otimes_{\BQ_p}E$
via the isomorphism (\ref{eq:vector-element}).

Let $D$ be a filtered $E$-$(\varphi,N)$-module: the underlying
$E$-$(\varphi,N)$-module $D$ is a $(K_0\otimes_{\BQ_p}E)$-module
with a basis $\{f_1, f_2, f_3\}$ such that
$$\varphi^{[K_0:\BQ_p]} f_1 = p^{-[K_0:\BQ_p]} f_1, \
\varphi^{[K_0:\BQ_p]} f_2 =  f_2, \ \varphi^{[K_0:\BQ_p]} f_3 = f_3,
$$ and $$ N(f_1)=0, \ N(f_2)= - f_1, \ N(f_3)=f_1 ;  $$ the
filtration on $$ K\otimes_{K_0}D=(K\otimes_{\BQ_p}E) f_1 \oplus
(K\otimes_{\BQ_p}E) f_2 \oplus (K\otimes_{\BQ_p}E) f_3 $$ satisfies
$$ \Fil^iD = \left\{\begin{array}{cl} (K\otimes_{\BQ_p}E)(f_2- \vec{\CL}f_1) \oplus (K\otimes_{\BQ_p}E)(f_3 +
\vec{\CL}f_1) & \text{ if } i=0, \\ 0 & \text{ if } i>0 .\end{array}
\right.
$$

Let $\pi_i$ be the projection map
$$ \BX_\log(D)
\rightarrow \BB_{\log,E}, \ \ \ \sum_{j=1}^3 a_j f_j \mapsto a_i. $$

\begin{lem}\label{lem:aux} Let $c: G_K\rightarrow \BX_\log(D)$ be a
$1$-cocycle whose class in $H^1(G_K,\BX_\log(D))$ belongs to
$\ker(H^1(G_K,\BX_\log(D))\rightarrow H^1(G_K,\BX_\dR(D)))$. Then
there exist $$\gamma_{2,0}, \gamma_{2,\tau}, \gamma_{3,0},
\gamma_{3,\tau}\in E$$ $(\tau\in \mathrm{Emb}(K,E))$ such that
$$ \pi_2(c)=\gamma_{2,0}\psi_0+\sum_{\tau\in
\mathrm{Emb}(K,E)}\gamma_{2,\tau}\psi_\tau $$ and
$$ \pi_3(c)=\gamma_{3,0}\psi_0+\sum_{\tau\in
\mathrm{Emb}(K,E)}\gamma_{3,\tau}\psi_\tau. $$ Furthermore,
$$ \gamma_{2,0}-\gamma_{3,0}=\sum_{\tau\in
\mathrm{Emb}(K,E)}\CL_\tau(\gamma_{2,\tau}-\gamma_{3,\tau}). $$
\end{lem}

In our proof of Lemma \ref{lem:aux} we need the following

\begin{lem} \label{lem:same-kernel}
Let $D$ be an $E$-$(\varphi,N)$-module. If $\Fil_1$ and $\Fil_2$ are
two filtrations on $K\otimes_{K_0}D$ such that
$\Fil_1^0(K\otimes_{K_0}D)=\Fil^0_2(K\otimes_{K_0}D)$, then the
kernel of $$H^1(G_K, \BX_\log(D))\rightarrow H^1(G_K, \BX_\dR(D,
\Fil_1))$$ coincides with the kernel of $$H^1(G_K,
\BX_\log(D))\rightarrow H^1(G_K, \BX_\dR(D, \Fil_2)).$$
\end{lem}
\begin{proof} The proof is similar to that of \cite[Proposition 2.5]{Xie2015}
\end{proof}

\noindent{\it Proof of Lemma \ref{lem:aux}.} The argument is similar
to the proof of \cite[Lemma 5.1]{Xie2015}. We only give a sketch.

Write
$c_\sigma=\lambda_{1,\sigma}f_1+\lambda_{2,\sigma}f_2+\lambda_{3,\sigma}f_3$.
As $c$ takes values in $\BX_\log(D)$, we have $\lambda_{2,\sigma},
\lambda_{3,\sigma}\in E$. This ensures the existence of
$\gamma_{2,0}, \gamma_{2,\tau}, \gamma_{3,0},\gamma_{3,\tau}$.

Let $Fil$ be the filtration on $D$ such that $Fil^{-1}D=D$ and
$Fil^iD=\Fil^iD$ if $i\geq 0$. Then $(D, Fil)$ is admissible. Let
$V$ be the semistable $E$-representation of $G_K$ attached to
$D_V=(D, Fil)$. By Lemma \ref{lem:same-kernel}, $[c]$ is in the
kernel of $ H^1(G_K,\BX_\log(D_V))\rightarrow H^1(G_K,\BX_\dR(D_V))$
and so there exists a $1$-cocycle $c^{(1)}: G_K\rightarrow V$ such
that the image of $[c^{(1)}]$ by $H^1(G_K,V)\rightarrow
H^1(G_K,\BX_\log(D_V))$ is $[c]$.

We form the following commutative diagram
\begin{equation}\label{eq:comm-diag-galois-repn}
\xymatrix{ && V' \ar[d]  \\ 0 \ar[r] & V_0 \ar[r] \ar@{=}[d] & V
\ar[r]\ar[d]^{\pi_{V,V_1}} & T \ar[r]\ar[d] & 0
\\  0 \ar[r] & V_0 \ar[r] & V_1 \ar[r] & T_1 \ar[r] &
0 }\end{equation} with the horizontal lines being exact, where $V_0$
(resp. $V'$) is the subrepresentation of $V$ corresponding to the
filtered $E$-$(\varphi,N)$-submodule of $D_V$ generated by $f_1$
(resp. by $f_2 + f_3$) which is admissible. From
(\ref{eq:comm-diag-galois-repn}) we obtain the following commutative
diagram
$$ \xymatrix{ H^1(G_K, V) \ar[r]\ar[d]^{\pi_{V, V_1}} & H^1(G_K, T) \ar[r]\ar[d] & H^2(G_K, V_0)\ar@{=}[d] \\
H^1(G_K, V_1)\ar[r] & H^1(G_K, T_1) \ar[r] & H^2(G_K, V_0)  , } $$
where the horizontal lines are exact.

Write $c^{(2)}$ for the $1$-cocycle $G_K\xrightarrow{c^{(1)}}
V\rightarrow T\rightarrow T_1.$ By a simple computation we obtain
$$[c^{(2)}]=  [\Big( (\gamma_{2,0}
-\gamma_{3,0})\psi_0 +
\sum_{\tau\in\mathrm{Emb}(K,E)}(\gamma_{2,\tau}-\gamma_{3,\tau})\psi_\tau
\Big)\bar{f}_2],
$$ where $\bar{f}_2$ is the image of $f_2\in V$ in $T_1$.
Note that $T_1$ is isomorphic to $E$, and $V_0$ is isomorphic to
$E(1)$. Being the image of $[\pi_{V,V_1}(c^{(1)})]$ in $H^1(T_1)$,
$[c^{(2)}]$ lies in the kernel of $H^1(G_K,T_1)\rightarrow
H^2(G_K,V_0)$. By \cite[Lemma 5.5]{Zhang}, as an extension of $E$ by
$E(1)$, $V_1$ corresponds to the element $[(p)] + \exp(\vec{\CL})$.
Now Lemma \ref{lem:zhang} yields our second assertion. \qed

\section{$L$-invariants} \label{sec:L-inv}

Let $D$ be a filtered $E$-$(\varphi,N)$-module of rank $n$. Fix a
refinement $\CF$ of $D$. Then $\CF$ fixes an ordering $\alpha_1,
\cdots, \alpha_n$ of the eigenvalues of $\varphi^{[K_0:\BQ_p]}$ and
an ordering $\vec{k}_1, \cdots, \vec{k}_n$ of the Hodge-Tate
weights.

\subsection{The operator $N_{\CF}$}

The operator $\varphi$ induces a $K_0\otimes_{\BQ_p}E$-semilinear
operator $\varphi_\CF$ on $\gr^{\CF}_\bullet
D=\bigoplus\limits_{i=1}^n \CF_{i}D/\CF_{i-1}D$.

We define a $K_0\otimes_{\BQ_p}E$-linear operator $N_\CF$ on
$\gr^\CF_\bullet D$. The definition is similar to the one defined in
\cite{Xie2015}, so we omit some details.

For any $i\in \{1, \cdots, n\}$, if $N(\CF_i D)=N (\CF_{i-1}D)$, we
demand that $N_\CF$ maps $\gr^\CF_i D$ to zero.

Now we assume that $N(\CF_i D) \supsetneq N (\CF_{i-1} D)$. Let $j$
be the minimal integer such that $$ N (\CF_i D) \subseteq
N(\CF_{i-1}D) + \CF_j D.
$$

\begin{prop}
$N (\CF_{i-1}D) \cap \CF_j D = N (\CF_{i-1}D)\cap \CF_{j-1}D$.
\end{prop}
\begin{proof} Note that $\CF_jD$, $\CF_{j-1}D$, $N (\CF_{i-1}D) + \CF_j D$ and
$N(\CF_{i-1}D)+\CF_{j-1}D$ are stable by $\varphi$. Thus $(N
(\CF_{i-1}D) + \CF_j D)/(N (\CF_{i-1}D) + \CF_{j-1} D)$ is a
$\varphi$-module, and so must be free over $K_0\otimes_{\BQ_p}E$.
Hence the map
\begin{equation} \label{eq:N-CF-iso}
\CF_jD/\CF_{j-1}D \rightarrow  (N (\CF_{i-1}D) + \CF_j D)/(N
(\CF_{i-1}D) + \CF_{j-1} D) \end{equation} is an isomorphism. It
follows that $N (\CF_{i-1}D) \cap \CF_j D = N (\CF_{i-1}D)\cap
\CF_{j-1}D$.
\end{proof}

The operator $N$ induces a $K_0\otimes_{\BQ_p}E$-linear map
$$ \CF_iD/\CF_{i-1}D \rightarrow (N
(\CF_{i-1}D) + \CF_j D)/(N (\CF_{i-1}D) + \CF_{j-1} D). $$ We define
the map $N_\CF: \gr_i^\CF D \rightarrow \gr^\CF_{j}D$ to be the
composition of this map and the inverse of (\ref{eq:N-CF-iso}).

Finally we extend $N_\CF$ to the whole $\gr^\CF_\bullet D$ by
$K_0\otimes_{\BQ_p}E$-linearity. Note that
$N_\CF\varphi_\CF=p\varphi_\CF N_\CF$. By definition, for any $i$ we
have either $N(\gr^\CF_i D) = 0$ or $N(\gr^\CF_i D)= \gr^\CF_j D$
for some $j$.

\begin{defn} For $j\in \{1,\cdots, n-1\}$ we say that $j$ is {\it marked} (or a {\it marked index}) for $\CF$
if there is some $i\in \{2, \cdots, n\}$ such that $N_\CF(\gr^\CF_i
D)= \gr^\CF_j D$.
\end{defn}

Note that $i$ and $j$ in the above definition are determined by each
other. We write $i=t_\CF(j)$ and $j=s_\CF(i)$.

\begin{prop}  The following two assertions are
equivalent:
\begin{enumerate}
\item\label{it:critical-a} $s$ is marked and $t=t_\CF(s)$.
\item\label{it:critical-b} $N\CF_{t-1}D \cap \CF_s D = N\CF_{t-1}D \cap \CF_{s-1} D$ and
$N\CF_{t}D \cap \CF_s D \supsetneq N\CF_{t}D \cap \CF_{s-1} D$.
\end{enumerate}
\end{prop}
\begin{proof} We have already seen that, if (\ref{it:critical-a}) holds, then (\ref{it:critical-b}) holds.
Conversely, we assume that (\ref{it:critical-b}) holds. Then
$N\CF_{t}D \cap \CF_s D \supsetneq N \CF_{t-1}D\cap \CF_sD$. Thus
$N\CF_{t}D \supsetneq N \CF_{t-1}D$.

We show that $N\CF_tD \varsubsetneq N \CF_{t-1}D +\CF_{s-1}D$. If it
is not true, then there exists $y\in \CF_tD \backslash \CF_{t-1}D$
which is a lifting of a basis of $\gr^\CF_tD$ over
$K_0\otimes_{\BQ_p}E$ such that $N(y)\in \CF_{s-1}D$. For any $z\in
\CF_tD$, write $z=w+\lambda y$ with $w\in \CF_{t-1}D$ and
$\lambda\in K_0\otimes_{\BQ_p}E$. If $N(z)$ is in $\CF_s D$, then
$N(w)$ is also in $\CF_s D$. But $N\CF_{t-1}D\cap \CF_s D
=N\CF_{t-1}D\cap \CF_{s-1}D$. Thus $N(w)$ is in $\CF_{s-1}D$, which
implies that $N(z)=N(w)+\lambda N(y)$ is also in $\CF_{s-1}D$. So,
$N\CF_tD\cap \CF_s D=N\CF_tD\cap \CF_{s-1}D$, a contradiction.

From $N\CF_{t}D \cap \CF_s D \supsetneq N \CF_{t-1}D\cap \CF_sD$ we
see that there is $x\in \CF_tD \backslash \CF_{t-1}D$ such that
$N(x)\in \CF_sD$. We must have  $N \CF_t D \subseteq N \CF_{t-1}D +
\CF_sD$. Otherwise, let $j$ be the smallest integer such that $N
\CF_t D \subseteq N \CF_{t-1}D + \CF_jD$ and assume that $j>s$. Then
$N_\CF(x+\CF_{t-1}D)=0$, which contradicts the fact that $N_\CF:
\gr^\CF_t D \rightarrow \gr^\CF_j D$ is an isomorphism.
\end{proof}

\subsection{Strongly marked indices and
$\CL$-invariants} \label{ss:strong-crit-L-inv}

Assume that $s$ is marked for $\CF$ and $t=t_\CF(s)$. We consider
the decompositions
$$ \CF_t D / \CF_{s-1} D = (K_0\otimes_{\BQ_p} E)\cdot \bar{e}_s \oplus L \oplus (K_0\otimes_{\BQ_p} E) \bar{e}_t
$$
that satisfy the following conditions:

$\bullet$ $\overline{\CF}_1 (\CF_t D / \CF_{s-1}
D)=(K_0\otimes_{\BQ_p}E)\bar{e}_s$ and
$\overline{\CF}_{t-s}(\CF_tD/\CF_{s-1} D) = (K_0\otimes_{\BQ_p}E)
\bar{e}_s\oplus L$, where $\overline{\CF}$ is the refinement on
$\CF_tD/\CF_{s-1} D$ induced by $\CF$.

$\bullet$ Both $L$ and $(K_0\otimes_{\BQ_p}E) \bar{e}_s \oplus
(K_0\otimes_{\BQ_p}E) \bar{e}_t$ are stable by $\varphi$ and $N$;
$\varphi^{[K_0:\BQ_p]}(\bar{e}_t)=\alpha_t \bar{e}_t$ and
$N(\bar{e}_t)=\bar{e}_s$.

\noindent Such a decomposition is called an {\it $s$-decomposition.}

\begin{rem} $s$-decompositions may be not exist. However, if $\varphi$ is
semisimple, then $s$-decompositions always exist (see
\cite{Xie2015}).
\end{rem}

Let $\mathrm{dec}$ denote an $s$-decomposition $ \CF_t D / \CF_{s-1}
D = E \bar{e}_s \oplus L \oplus E \bar{e}_t. $

There is a natural isomorphism $E \bar{e}_s \oplus E
\bar{e}_t\rightarrow (\CF_tD/\CF_{s-1}D)/L$ of
$(\varphi,N)$-modules. Usually the filtration on the filtered
$E$-$(\varphi,N)$-submodule $E \bar{e}_s \oplus E \bar{e}_t$ and
that on $(\CF_tD/\CF_{s-1}D)/L$ are different.

When these two filtrations satisfy certain compatible condition, we
say the decomposition $\mathrm{dec}$ is perfect. Precisely, we say
that $\mathrm{dec}$ is {\it perfect} if for any $\tau:
K\hookrightarrow E$ we have $k_{s,\tau}< k_{t, \tau}$, and if there
exist $k'_{s,\tau}, k'_{t,\tau}$ and $\CL_{\mathrm{dec},\tau}\in E$
satisfying $k_{s,\tau}\leq k'_{s,\tau}< k'_{t,\tau}\leq k_{t,\tau}$
such that the following conditions hold.

$\bullet$ The filtration on the filtered $E$-$(\varphi,N)$-submodule
$E \bar{e}_s \oplus E \bar{e}_t$ satisfies
$$\Fil^{i}_\tau (E \bar{e}_s \oplus E \bar{e}_t) = \left\{ \begin{array}{ll} E \bar{e}_{s,\tau}\oplus E \bar{e}_{t,\tau} & \text{ if }i \leq k_{s,\tau}, \\
E(\bar{e}_{t,\tau} + \CL_{\mathrm{dec,\tau}} \bar{e}_{s,\tau}) & \text{ if } k_{s,\tau} < i\leq k'_{t,\tau} ,\\
0 & \text{ if } i > k'_{t,\tau},
\end{array}\right.$$

$\bullet$ The filtration on the quotient of $\CF_tD/\CF_{s-1}D$ by
$L$ satisfies
$$\Fil^{i}_\tau \CF_tD/\CF_{s-1}D = \left\{ \begin{array}{ll} E \bar{e}_{s,\tau}\oplus E \bar{e}_{t,\tau} & \text{ if }i \leq k'_{s,\tau}, \\
E(\bar{e}_t + \CL_{\mathrm{dec},\tau} \bar{e}_s) & \text{ if } k'_{s,\tau} < i\leq k_{t,\tau} ,\\
0 & \text{ if } i > k_{t,\tau},
\end{array}\right.$$ where the images of $\bar{e}_{s}$ and
$\bar{e}_t$ in $\CF_tD/\CF_{s-1}D$ are again denoted by $\bar{e}_s$
and $\bar{e}_t$.

\begin{defn}\label{defn:Fontaine-Mazur}
If there exists a perfect $s$-decomposition, we say that $s$ is {\it
strongly marked} (or a {\it strongly marked index}). In this case we
attached to each pair $(s,t)$  with $t=t_\CF(s)$ an invariant
$\vec{\CL}_{\CF,s,t}=(\CL_{\mathrm{dec},\tau})_{\tau}$, where
$\mathrm{dec}$ is a perfect $s$-decomposition. Proposition
\ref{prop:fon-mazur-inv} below tells us that $\vec{\CL}_{\CF,s,t}$
is independent of the choice of perfect $s$-decompositions. We call
$\vec{\CL}_{\CF,s,t}$ the {\it Fontaine-Mazur $\CL$-invariant}
associated to $(\CF,s,t)$, and denote $\CL_{\mathrm{dec},\tau}$ by
$\CL_{\CF, s,t, \tau}$.
\end{defn}

In the case of $t=s+1$, $s$ is strongly marked if and only if
$k_{s,\tau}<k_{t,\tau}$ for all $\tau$.

\begin{prop} \label{prop:fon-mazur-inv}
If $\mathrm{dec}_1$ and $\mathrm{dec}_2$ are two perfect
$s$-decompositions, then $\CL_{\mathrm{dec}_1,
\tau}=\CL_{\mathrm{dec}_2,\tau}$ for any $\tau$.
\end{prop}
\begin{proof} The argument is similar to the proof of \cite[Proposition
4.9]{Xie2015}.
\end{proof}

Let $D^*$ be the filtered $E$-$(\varphi, N)$-module that is the dual
of $D$. Let $\check{\CF}$ be the refinement on $D^*$ such that
$$ \check{\CF}_i D^* : = (\CF_{n-i}D)^\bot=\left\{y\in D^*: \langle y, x
\rangle =0 \text{ for all }x\in \CF_{n-i}D \right\}. $$ We call
$\check{\CF}$ the {\it dual refinement} of $\CF$.

If $L\subset M$ are submodules of $D$, then $M^\bot\subset L^\bot$.
The pairing $\langle \cdot, \cdot \rangle:  L^\bot\times M$ induces
a non-degenerate pairing on $L^\bot/M^\bot \times M/L  $, so that we
can identify $L^\bot/M^\bot$ with the dual of $M/L$ naturally. In
particular, $\gr^{\check{\CF}}_i D^*$ is naturally isomorphic to the
dual of $\gr^{\CF}_{n+1-i}D$. Thus $\gr^{\check{\CF}}_\bullet D^*$
is naturally isomorphic to the dual of $\gr^\CF_\bullet D$.

\begin{prop}\label{prop:duality-critical}
\begin{enumerate}
\item \label{it:duality-critical-a}
$N_{\check{\CF}}$ is dual to $-N_\CF$.
\item \label{it:duality-critical-b}
$s$ is marked for $\CF$ if and only if $n+1-t_\CF(s)$ is marked for
$\check{\CF}$.
\item \label{it:duality-critical-c}
$s$ is strongly marked for $\CF$ if and only if $n+1-t_\CF(s)$ is
strongly marked for $\check{\CF}$.
\end{enumerate}
\end{prop}
\begin{proof} The proof of (\ref{it:duality-critical-a}) is similar to that of \cite[Proposition 4.14]{Xie2015}. The proof of
(\ref{it:duality-critical-b}) is similar to that of
\cite[Proposition 4.13]{Xie2015}. The proof of
(\ref{it:duality-critical-c}) is similar to that of
\cite[Proposition 4.15 (a)]{Xie2015}.
\end{proof}

\section{Projection vanishing property} \label{sec:proj-van-prop}

Put $S=E[Z]/(Z^2)$. Let $z$ be the closed point defined by the
maximal ideal $(Z)$ of $S$.

Let $W=(W_e, W^+_\dR)$ be an $S$-$B$-pair. Let $\{w_1, \cdots,
w_n\}$ be a $\BB_{e,S}$-basis of $W_e$. Suppose that $W$ admits a
triangulation $\Fil_\bullet$. Let $(\delta_{1}, \cdots, \delta_{n})$
be the corresponding triangulation parameters. Then for each
$i=1,\cdots, n$ there exists a continuous additive character
$\epsilon_i$ of $K^\times$ with values in $E$ such that $\delta_i =
\delta_{i,z} (1+Z \epsilon_i)$.

Suppose that $W_z$, the evaluation of $W$ at $z$, is semistable, and
let $D_z$ be the filtered $E$-$(\varphi,N)$-module attached to
$W_z$. Let $\CF$ be the refinement of $D_z$ corresponding to the
induced triangulation of $W_z$, and let $\{e_{1,z}, e_{2,z}, \cdots,
e_{n,z}\}$ be a $(K_0\otimes_{\BQ_p}E)$-basis of $D_z$ that is
compatible with $\CF$ i.e. $\CF_i D= (K_0\otimes_{\BQ_p}E)e_{1,z}
\oplus \cdots \oplus (K_0\otimes_{\BQ_p}E) e_{i,z}$. Let
$\alpha_{i,z}\in E$ be such that
$\varphi^{[K_0:\BQ_p]}(e_{i,z})=\alpha_{i,z}e_{i,z} \mod \CF_{i-1}$.

Let $x_{ij}\in \BB_{\log,E}$ ($i,j=1,\cdots, n$) be such that
\begin{equation}\label{eq:relation-e-and-v}
e_{i,z} = x_{1i} w_{1,z} +\cdots + x_{ni} w_{n,z} .   \end{equation}
Then $X=(x_{ij})$ is in $\GL_n(\BB_{\log,E})$. Write the matrix of
$\sigma\in G_K$ with respect to the basis $\{w_{1}, \cdots, w_{n}\}$
by $(I_n + Z U_{e,\sigma})A_{e,\sigma}$. As $e_{1, z},\cdots,
e_{n,z}$ are fixed by $G_{K}$, we have $X^{-1}A_{e,\sigma}
\sigma(X)=I_n$ for all $\sigma\in G_{K}$.

For $i=1,\cdots, n$ put $e_i = x_{1i} w_1 +\cdots + x_{ni}  w_n$.
Then $\{e_1,\cdots, e_n\}$ is a basis of $\BB_{\log,S}\otimes_S W_e$
over $\BB_{\log,S}$.

\begin{lem} \label{lem:trivial-useful} If $T$ is the matrix of $\varphi_{D_z}$ for the basis $\{e_{1,z}, \cdots,
e_{n,z}\}$, then $T$ is also the matrix of
$\varphi_{\BB_{\log,S}\otimes_S W_e}$ for the basis $\{e_{1},
\cdots, e_{n}\}$.
\end{lem}
\begin{proof} The assertion follows from the definition of $\{e_1, \cdots, e_n\}$ and the fact that $w_{1, z}, \cdots, w_{n,z}, w_1, \cdots,
w_n$ are fixed by $\varphi$.
\end{proof}

In Section \ref{sec:coh-B-pair} we attach to $W$ an element $c_B(W)$
in $H^1_B(W_z^*\otimes W_z)$. Consider the composition
$$ H^1_B(W_z^*\otimes W_z) \rightarrow H^1(G_K, W_{e,z}^*\otimes_{\BB_{e,E}} W_{e,z}) \rightarrow H^1(G_K,
\BB_{\log,E}\otimes_E (D_z^*\otimes D_z)) . $$ As the matrix of
$\sigma\in G_K$ for the basis $\{e_1,\cdots, e_n\}$ is $I_n + Z
X^{-1}U_{e,\sigma} X$, from the discussion in Section \ref{sec:coh}
we see that the image of $c_B$ in $H^1(G_K, \BB_{\log,E}\otimes_E
(D_z^*\otimes D_z))$ is the class of the $1$-cocycle
$$(U_{e,\sigma})_{ij} w^*_{j,z}\otimes w_{i,z}  =(X^{-1}U_{e,\sigma}X)_{ij} e^*_{j,z}\otimes e_{i,z} . $$
Let $\pi_{h \ell}$ be the projection
\begin{equation}\label{eq:proj}
\BB_{\log,E}\otimes_E (D_z^*\otimes D_z)\rightarrow \BB_{\log,E} , \
\ \ \sum_{j,i} b_{ji} e^*_{j,z}\otimes e_{i,z} \mapsto b_{h \ell} .
\end{equation}

For $h=1, \cdots, n$, let $\epsilon'_h$ be the additive character of
$G_K$ such that $\epsilon'_h \circ\mathrm{rec}_K(p)=0$ and
$\epsilon'_h\circ\mathrm{rec}_K|_{\fo_K^\times}=\epsilon_h|_{\fo_K^\times}$.

\begin{thm} \label{prop:middle-step}
\begin{enumerate}
\item\label{it:middle-step-a} For any pair of integers $(h,\ell)$ such that $h<\ell$ we have
$\pi_{h \ell}([c])=0$.
\item\label{it:middle-step-b} For any $h=1,\cdots, n$,
$\pi_{h,h}([c])$ coincides with the image of $[\epsilon'_h]$ in
$H^1(G_K, \BB_{\log,E})$.
\end{enumerate}
\end{thm} We call (\ref{it:middle-step-a}) the {\it projection
vanishing property}.
\begin{proof} The filtered $E$-$(\varphi,N)$-module attached to $W_z/
\Fil_{h-1}W_z$ is $D_z/ \CF_{h-1}D_z$. We denote the image of
$e_{\ell,z}$ ($\ell\geq h$) in $D_z/\CF_{h-1}D_z$ again by
$e_{\ell,z}$.

Let $\delta'_h$ be the character of $G_K$ such that
$\delta'_h=1+Z\epsilon'_h$. By Lemma \ref{lem:varph-gamma-fil} there
exists an element
$$x\in (\BB_{\mathrm{max},E}\otimes_{ \BB_{e,E} } (W/
\Fil_{h-1}W)_e)^{G_K=\delta'_h,
\varphi^{[K_0:\BQ_p]}=\alpha_{i,z}(1+Zv_p(\pi_K)\epsilon_h(p))}$$
whose image in $D_z/\CF_{h-1} D_z$ is $e_{h,z}$.  Write $x= e_{h}+
Z\sum\limits_{\ell\geq h} \lambda_\ell e_{\ell}$ with
$\lambda_\ell\in \BB_{\log,E}$.

As the matrix of $\sigma\in G_K$ for the basis $\{e_{1}, \cdots,
e_n\}$ is $I_n+ZX^{-1}U_{e,\sigma}X$, we have \begin{eqnarray*} &&
[1+Z\epsilon'_h(\sigma)]x =[1+Z\epsilon'_h(\sigma)]( e_{h}+ Z\sum_{\ell\geq h} \lambda_\ell e_{\ell}) \\
& = &\sigma(x)=   e_{h} + Z\sum_{\ell\geq h}
(X^{-1}U_{e,\sigma}X)_{\ell h}e_{\ell} + Z\sum_{\ell\geq
h}\sigma(\lambda_\ell)e_{\ell} .
\end{eqnarray*} For $\ell>h$,
comparing the coefficients of $e_{\ell}$ we obtain
$$(X^{-1}U_{e,\sigma}X)_{\ell h}= (1-\sigma)\lambda_\ell, $$ which shows
(\ref{it:middle-step-a}). Similarly, comparing coefficients of
$e_{h}$ we obtain
\begin{equation} \label{eq:Uhh-episilon}
(X^{-1}U_{e,\sigma}X)_{hh}-\epsilon'_h(\sigma) = (1-\sigma)\lambda_h
,\end{equation} which implies (\ref{it:middle-step-b}).
\end{proof}

\section{The proof of Theorem \ref{thm:main}} \label{sec:main}

We will need the following lemmas.

\begin{lem}\label{cor:inc-iso} The inclusion $ E \hookrightarrow \BB_{e,E} $ induces an
isomorphism
$$ H^1(G_K,E) \xrightarrow{\sim} \ker(N: H^1(G_K, \BB_{e,E})\rightarrow H^1(G_K, \BB_{\log, E})).
$$
\end{lem}
\begin{proof} The proof is identical to that of \cite[Corollary
1.4]{Xie2015}.
\end{proof}

\begin{lem} \label{lem:N-surj} The map
$N: \BB_{\log,E}^{\varphi=p}\rightarrow \BB_{\log, E}^{\varphi=1}$
is surjective.
\end{lem}
\begin{proof} The proof is identical to that of \cite[Lemma 1.2]{Xie2015}.
\end{proof}

For the proof of Theorem \ref{thm:main} we may assume that
$S=E[Z]/(Z^2)$, and $z$ is the closed point defined by the maximal
ideal $(Z)$. Let $W$ be as in Theorem \ref{thm:main}. Replacing $W$
by the $E$-$B$-pair $\CF_tW/\CF_{s-1}W$ and replacing $\CF$ by the
induced refinement on $\CF_tW/\CF_{s-1}W$, we may assume that $s=1$
and $t=n=\mathrm{rank}_{\BB_{e,E}}(W_e)$. Let $e_{1,z}, e_{2,z},
\cdots, e_{n,z}$ be a $K_0\otimes_{\BQ_p}E$-basis of $D_z$ such that
\begin{equation} \label{eq:s-decomp}
(K_0\otimes_{\BQ_p}E)e_{1,z} \bigoplus  L \bigoplus
(K_0\otimes_{\BQ_p}E) e_{n,z}
\end{equation} with $L=\oplus_{i=2}^{n-1}(K_0\otimes_{\BQ_p}E)e_{i,z}$
a perfect $1$-decomposition of $D_z$ for $\CF$ (see \S
\ref{ss:strong-crit-L-inv} for the meaning of perfect
decompositions). Let $e_{1,z}^*, e_{2,z}^*, \cdots, e^*_{n,z}$ be
the dual basis of $D^*_z$ over $K_0\otimes_{\BQ_p}E$.

Let $D_1$ be the quotient of $D_z$ by $L$, $D_2^*$ the quotient of
$D_z^*$ by $\oplus_{i=2}^{n-1}(K_0\otimes_{\BQ_p}E)e^*_{i,z}$. Put
$\mathscr{D}=D_2^*\otimes D_1$. The images of $e_{1,z}$ and
$e_{n,z}$ in $D_1$ are again denoted by $e_{1,z}$ and $e_{n,z}$, and
the images of $e^*_{1,z}$ and $e^*_{n,z}$ in $D_2^*$ are again
denoted by $e^*_{1,z}$ and $e^*_{n,z}$ respectively. So
$e^*_{1,z}\otimes e_{1,z}$, $e^*_{1,z}\otimes e_{n,z}$,
$e^*_{n,z}\otimes e_{1,z}$, $e^*_{n,z}\otimes e_{n,z}$ form a
$K_0\otimes_{\BQ_p}E$-basis of $\mathscr{D}$. Let $\mathscr{D}_0$ be
the filtered $E$-$(\varphi,N)$-submodule of $\mathscr{D}$ with a
$K_0\otimes_{\BQ_p}E$-basis $\{e^*_{1,z}\otimes e_{1,z}, \
e^*_{n,z}\otimes e_{1,z}, \ e^*_{n,z}\otimes e_{n,z}\}$. Let
$\mathscr{W}=(\mathscr{W}_e,\mathscr{W}^+_\dR)$ (resp.
$\mathscr{W}_0$) be the $E$-$B$-pair attached to $\mathscr{D}$
(resp. $\mathscr{D}_0$). Note that
$$\varphi^{[K_0:\BQ_p]}(e^*_{1,z}\otimes e_{1,z})=e^*_{1,z}\otimes e_{1,z}, \
\varphi^{[K_0:\BQ_p]}(e^*_{n,z}\otimes e_{n,z})=e^*_{n,z}\otimes
e_{n,z},
$$
$$ \varphi^{[K_0:\BQ_p]}(e^*_{n,z}\otimes e_{1,z})= p^{-[K_0:\BQ_p]}
e^*_{n,z}\otimes e_{1,z},
$$ and
$$ -N(e^*_{1,z}\otimes e_{1,z}) = N(e^*_{n,z}\otimes e_{n,z}) = e^*_{n,z}\otimes e_{1,z}, \ N(e^*_{n,z}\otimes e_{1,z}) = 0 $$

Let $\vec{\CL}_{\CF}=\vec{\CL}_{\CF,s,t}$ be the $\CL$-invariant
defined in Definition \ref{defn:Fontaine-Mazur}. As
(\ref{eq:s-decomp}) is a prefect decomposition, we have
\begin{eqnarray*}
\Fil^0 (K\otimes_{K_0}\mathscr{D})  = && E e^*_{n,z} \otimes
(e_{n,z} +\vec{\CL}_{\CF}e_{1,z}) \oplus E
(e^*_{1,z}-\vec{\CL}_{\CF}e^*_{n,z})\otimes e_{1,z} \\ && \oplus E
(e^*_{1,z}-\vec{\CL}_{\CF}e^*_{n,z})\otimes (e_{n,z}
+\vec{\CL}_{\CF}e_{1,z}). \end{eqnarray*} and \[ \Fil^0
(K\otimes_{K_0}\mathscr{D}_0) = E e^*_{n,z} \otimes (e_{n,z}
+\vec{\CL}_{\CF}e_{1,z}) \oplus E
(e^*_{1,z}-\vec{\CL}_{\CF}e^*_{n,z})\otimes e_{1,z} .
\]

Consider $W$ as an infinitesimal deformation of $W_z$. In Section
\ref{ss:cocycle-inform} we attach to this infinitesimal deformation
an element $c_B(W)$ in $H^1_B(W_z^*\otimes W_z)$. Let $[c]$ be the
image of $c_B(W)$ by the composition
$$ H^1_B(W_z^*\otimes W_z)\rightarrow H^1(G_K, W_{e,z}^*\otimes_{\BB_{e,E}}W_{e,z})\rightarrow H^1(G_K,
\BB_{\log,E}\otimes_{K_0\otimes_{\BQ_p}E} (D_z^*\otimes D_z)), $$
and choose a $1$-cocyle $c$ representing $[c]$. Write $c$ in the
form
$$ c =  \sum_{j,i} c_{j,i} e^*_{j,z}\otimes e_{i,z}
$$ with $c_{i,j}$ being a $1$-cocycle of $G_K$ with values in $\BB_{\log,E}$. By the projection vanishing property (Theorem \ref{prop:middle-step}
(\ref{it:middle-step-a})) we have $[c_{1,n}]=0$.

\begin{lem}\label{lem:constants} There exist $\xi_1, \xi_n \in \BB_{e,E}$ and $\gamma_{1,0},
\gamma_{1,\tau}, \gamma_{n,0}, \gamma_{n,\tau}$ $(\tau\in
\mathrm{Emb}(K,E))$ such that
$$ c_{1,1}(\sigma) =(\sigma-1)\xi_1 +  \gamma_{1,0} \psi_0(\sigma) +\sum_{\tau\in \mathrm{Emb}(K,E)} \gamma_{1,\tau} \psi_\tau (\sigma)  $$
and
$$ c_{n,n}(\sigma) =(\sigma-1)\xi_n +  \gamma_{n,0} \psi_0(\sigma) +\sum_{\tau\in \mathrm{Emb}(K,E)} \gamma_{n,\tau} \psi_\tau (\sigma)
$$ for any $\sigma\in G_K$. \end{lem}
\begin{proof} Let $\bar{c}_B$ be the image of $c_B$ in $H^1_B(\mathscr{W})$, and let $\bar{c}$ be the $1$-cocycle
$$ \bar{c} =  \sum_{j,i\in \{1,n\}} c_{j,i} e^*_{j,z}\otimes e_{i,z}
$$ of $G_K$ with values in $\BB_{\log,E}\otimes_{K_0\otimes_{\BQ_p}E}
\mathscr{D}$. Then the image of $\bar{c}_B$ in
$$H^1(G_K,\BB_{\log,E}\otimes_{K_0\otimes_{\BQ_p}E} \mathscr{D} )$$ is
$[\bar{c}]$.

Note that $\bar{c}$ has values in
$\mathscr{W}_e=(\BB_{\log,E}\otimes_{K_0\otimes_{\BQ_p}E}
\mathscr{D})^{\varphi=1,N=0}$. So, in particular $c_{1,1}$ and
$c_{n,n}$ have values in $\BB_{e,E}$.
As $N\bar{c}=0$, we have
\begin{eqnarray*}  N(c_{n,1})= c_{1,1} - c_{n,n}, \ \ \
 -N(c_{1,1}) = N(c_{n,n})= c_{1,n}.
\end{eqnarray*}
As $[c_{1,n}]=0$, the statement follows from Lemma
\ref{cor:inc-iso}.
\end{proof}

Write $ \delta_i=\delta_{i,z}(1+Z\epsilon_i) $. Let $\epsilon'_i$ be
the additive character of $G_K$ with values in $E$ such that
$\epsilon'_i\circ \mathrm{rec}_K(p)=0$ and $\epsilon'_i\circ
\mathrm{rec}_K|_{\fo_K^\times}=\epsilon_i|_{\fo_K^\times}$. Then
there are $\epsilon_{i,\tau}$ ($\tau\in \mathrm{Emb}(K, E)$) such
that $\epsilon'_i=\sum\limits_{\tau\in\mathrm{Emb}(K,E)}
\epsilon_{i,\tau} \psi_{\tau}$.

\begin{lem} \label{lem:constant-gamma}
For $h=1,n$ we have
$[K_0:\BQ_p]\gamma_{h,0}=-v_p(\pi_K)\epsilon_h(p)$ and
$\gamma_{h,\tau}=\epsilon_{h,\tau}$.
\end{lem}
\begin{proof}
We keep to use notations in the proof of Theorem
\ref{prop:middle-step}. By (\ref{eq:Uhh-episilon}) and Lemma
\ref{lem:constants} we have
\begin{eqnarray*} (\sigma-1)(\lambda_h ) & = &-(X^{-1} U_\sigma
X)_{hh}+\sum_{\tau\in \mathrm{Emb}(K,E)}
\epsilon_{h,\tau}\psi_\tau(\sigma) \\
&=&
-(\sigma-1)\xi_h-\gamma_{h,0}\psi_0(\sigma)+\sum_{\tau\in\mathrm{Emb}(K,E)}(\epsilon_{h,\tau}-\gamma_{h,\tau})\psi_\tau(\sigma).
\end{eqnarray*} Note that there exists $\omega\in
\mathrm{W}(\overline{\BF}_p)$ such that $\varphi(\omega)-\omega=1$,
where $\mathrm{W}(\overline{\BF}_p)$ is the ring of Witt vectors
with coefficients in the algebraic closure of $\BF_p$. Then
$(\sigma-1)\omega= \psi_0(\sigma) $. Hence
$$ \sum_{\tau\in\mathrm{Emb}(K,E)}(\epsilon_{h,\tau}-\gamma_{h,\tau})\psi_\tau(\sigma)
= (\sigma-1)(\lambda_h +\xi_h +\gamma_{h,0} \omega).
$$ In other words, the cocycle
$\sum\limits_{\tau\in\mathrm{Emb}(K,E)}(\epsilon_{h,\tau}-\gamma_{h,\tau})\psi_\tau(\sigma)$
is de Rham. By Lemma \ref{lem:de-Rham-cocycle} we have
$\gamma_{h,\tau} = \epsilon_{h,\tau}$ and $\lambda_h+\xi_h
+\gamma_{h,0} \omega \in E$. Then \begin{equation}
\label{eq:lambda-a} (\varphi^{[K_0:\BQ_p]}-1)\lambda_h =
-(\varphi-1)\xi_h - \gamma_{h,0} (\varphi^{[K_0:\BQ_p]}-1)\omega =
-[K_0:\BQ_p]\gamma_{h,0} .
\end{equation}

By our choice of the basis $\{e_{1,z}, \cdots, e_{n,z}\}$,  $Y_1=
\oplus_{i=2}^n Z e_{i,z} $ is stable by $\varphi$. Put $Y_n=0$. Let
$x$ be as in the proof of Theorem \ref{prop:middle-step}. By Lemma
\ref{lem:trivial-useful} we have $\varphi^{[K_0:\BQ_p]} e_{h,z}
=\alpha_{h,z}e_{h,z}$. Thus for $h=1, n$ we have
$$ \varphi^{[K_0:\BQ_p]}(x) =  ( 1+Z \varphi^{[K_0:\BQ_p]}(\lambda_h) )\alpha_{h,z} e_h  \hskip 10pt (\text{mod } Y_h).
$$ On the other hand,
\begin{eqnarray*} \varphi^{[K_0:\BQ_p]}(x) & = & (1+Z v_p(\pi_K)\epsilon_h(p)) \alpha_{h,z} x \\
& = & (1+Z  v_p(\pi_K)\epsilon_h(p))   \alpha_{h,z} (1+ Z
\lambda_{h}) e_h \hskip 10pt (\text{mod } Y_h).
\end{eqnarray*} Hence we obtain
\begin{equation} \label{eq:lambda-h}
(\varphi^{[K_0:\BQ_p]}-1)\lambda_h =
v_p(\pi_K)\epsilon_h(p).\end{equation} By (\ref{eq:lambda-a}) and
(\ref{eq:lambda-h}) we have
$$ [K_0:\BQ_p]\gamma_{h,0} = -(\varphi^{[K_0:\BQ_p]}-1) \lambda_h = -v_p(\pi_K)\epsilon_h(p), $$ as
wanted.
\end{proof}

By Lemma \ref{lem:N-surj} there exists some $y\in
\BB_{\log,E}^{\varphi=p}$ such that $N(y)=\xi_1-\xi_n$. Let
$\bar{c}'$ be the $1$-cocycle of $G_K$ with values in
$\BB_{\log,E}\otimes_{K_0\otimes_{\BQ_p}E} \mathscr{D}_0$
 such that $$ \bar{c}'= c'_{1,1}
e^*_{1,z}\otimes e_{1,z} + c'_{n,n} e^*_{n,z}\otimes e_{n,z} +
c'_{n,1}e_{n,z}^*\otimes e_{1,z} $$ with
$$ c'_{1,1} = \gamma_{1,0} \psi_0  +\sum_{\tau\in \mathrm{Emb}(K,E)} \gamma_{1,\tau} \psi_\tau
, \ \ c'_{n,n} = \gamma_{n,0} \psi_0  +\sum_{\tau\in
\mathrm{Emb}(K,E)} \gamma_{n,\tau} \psi_\tau  $$ and
$$ c'_{n,1}(\sigma) = c_{n,1}(\sigma) - (\sigma-1)y , \ \ \sigma\in G_K.
$$ It is easy to check that $\varphi(\bar{c}')=\bar{c}'$ and
$N(\bar{c}')=0$. Hence $\bar{c}'$ is a $1$-cocycle of $G_K$ with
values in $\BX_\log(\mathscr{D}_0)$.

\begin{prop} The image of $[\bar{c}']$ in $H^1(G_K,
\BX_\log(\mathscr{D}_0))$ belongs to the kernel of $$ H^1(G_K,
\BX_\log(\mathscr{D}_0))\rightarrow H^1(G_K,
\BX_\dR(\mathscr{D}_0)).$$
\end{prop}
\begin{proof}
Consider the following commutative diagram
\[ \xymatrix{ H^1(G_K, \BX_\log(\mathscr{D}_0)) \ar[r] \ar[d] & H^1(G_K,
\BX_\dR(\mathscr{D}_0)) \ar[d] \\
H^1(G_K, \BX_\log(\mathscr{D})) \ar[r]  & H^1(G_K,
\BX_\dR(\mathscr{D})). } \] The right vertical arrow in the above
diagram is injective (see \cite[Corollary 2.4]{Xie2015}). So we only
need to show that the image of $[\bar{c}']$ in $H^1(G_K,
\BX_\dR(\mathscr{D}))$ is zero. Note that
$$[\bar{c}']=[\bar{c}]-[c_{1,n} e^*_{1,z}\otimes e_{n,z}]= -[c_{1,n}
e^*_{1,z}\otimes e_{n,z}]$$ in $H^1(G_K, \BX_{\dR}(\mathscr{D}))$.
As the image of $[c_{1,n}]$ in $H^1(G_K, \BB_{\log,E})$ is zero, so
is its image in $H^1(G_K, \BB_{\dR, E}/ \Fil^{f}\BB_{\dR,E})$, where
$f$ is the smallest integer such that $e^*_{1,z}\otimes e_{n,z}\in
\Fil^{-f}\mathscr{D}_K$. Hence, the image of $[\bar{c}']$ in
$H^1(G_K, \BX_{\dR}(\mathscr{D}))$ is zero.
\end{proof}

Now, applying Lemma \ref{lem:aux} to $\mathscr{D}_0$ with $f_1=
e^*_{n,z}\otimes e_{1,z}$, $f_2=e^*_{1,z}\otimes e_{1,z}$ and
$f_3=e^*_{n,z}\otimes e_{n,z}$, we get
$$ \gamma_{n,0}-\gamma_{1,0} = \sum_{\tau\in\mathrm{Emb}(K,E)} \CL_{\tau} (\gamma_{n,\tau}-\gamma_{1,\tau}).  $$
Hence, by Lemma \ref{lem:constant-gamma} we have
$$ \frac{v_p(\pi_K)}{[K_0:\BQ_p]} ( \epsilon_n(p) -\epsilon_1(p) ) + \sum_{\tau\in\mathrm{Emb}(K,E)} \CL_{\tau} (\epsilon_{n,\tau}-\epsilon_{1,\tau}) =0. $$
As $\frac{\mathrm{d} \delta_h (p) }{\delta_h(p)} = \epsilon_h(p)
\mathrm{d}Z$ and
$\mathrm{d}\vec{w}(\epsilon_h)=(\epsilon_{h,\tau}\mathrm{d}Z)_\tau$,
we obtain
$$ \frac{1}{[K:\BQ_p]}\left(\frac{\mathrm{d} \delta_n(p)}{\delta_n(p)} - \frac{\mathrm{d}
\delta_1(p)}{\delta_1(p)}\right)
 + \vec{\CL}_{\CF} \cdot (\mathrm{d}\vec{w}(\delta_n)-\mathrm{d}\vec{w}(\delta_1)) =
0,
$$ as desired. This finishes the proof of Theorem \ref{thm:main}.

redpajama_set_name: RedPajamaArXiv


text:
Motorist wins case after maths whizzes break speed camera code
August 11, 2005 — 10.00am
A team of Chinese maths enthusiasts have thrown NSW's speed cameras system into disarray by cracking the technology used to store data about errant motorists.
The NRMA has called for a full audit of the way the state's 110 enforcement cameras are used after a motorist escaped a conviction by claiming that data was vulnerable to hackers.
A Sydney magistrate, Laurence Lawson, threw out the case because the Roads and Traffic Authority failed to find an expert to testify that its speed camera images were secure.
The motorist's defence lawyer, Denis Mirabilis, argued successfully that an algorithm known as MD5, which is used to store the time, date, place, numberplate and speed of cars caught on camera, was a discredited piece of technology.
Mr Mirabilis yesterday said he had received more than 100 inquiries from motorists anxious to use the same defence. "People have shown it [the algorithm] has been hacked and it's open to viruses."
Designed in the early 1990s by an American academic, MD5 safeguards against tampering by turning information into a 128-bit sequence of digits. However, researchers from China's Shandong University have proved it is possible to store conflicting pieces of information as the same MD5 sequence.
Nick Ellsmore, an encryption expert at the consultancy SIFT, said this theoretically meant the RTA could change the speed at which a car was recorded and retain the same code.
"Since the research came out, we've been recommending that clients move away from MD5 and we've certainly recommended that people don't use it for new applications," he said.
The NRMA said it was crucial the public had confidence in convictions. Its policy specialist, Lisa McGill, said: "We want a full audit and a review of the system to ensure that it is working appropriately."
The RTA's spokesman, Paul Willoughby, rejected the decision as a one-off: "No one, in relation to court cases, can be a hundred per cent sure they're going to win a hundred per cent of the time."
NSW's weekly take from the cameras is more than $1 million.
Meanwhile, the RTA denied reports that cameras catching toll evaders in the Harbour Tunnel are routinely turned off.

redpajama_set_name: RedPajamaCommonCrawl


text:
Thought, Self-Creation, and the Nature of Karma
September 19, 2016 December 13, 2016 Dr. Linda Gadbois
Karma is the Habituated State of Mind that forms all of Our Perceptions
Many people believe that karma is something that's incurred only through actions of some kind, or that it's okay to think about something as long as you don't actually "do" anything. Pure thought is subtle (invisible) and seemingly inactive on the material plane, while feelings, sensation, and emotion are active forces that serve to stimulate us physically. We imagine that we can think without actually creating any consequences to our thoughts as long as we don't act on our thoughts in the material sense. Thought is always individual in nature, each person having their own unique thoughts in response to any form of stimulation, whereas emotions are more universal in nature in the sense that we all experience the same basic type of emotions. Yet, in reality, thought, as a subtle force, underlies and forms the basis for all of our emotions and actions. Thought is "action" on the spiritual plane, and forms the pattern as an idea that becomes enacted on the material plane. To think is to literally create in the most basic sense of the idea. It's not enough to control our behavior and what outward activities we engage in, we have to also control our thoughts which forms the basis for our activities. Thoughts form the images and patterns of material form and processes, and become the template for producing an analogous outer reality through corresponding behaviors..
Thought that we dwell in and repeat consistently becomes habitual (normal), and forms the basis for our karma. As we think, we become our thoughts in form, and we view our life "through" our thoughts and mental impressions as our "perceptual lens" that reformulates neutral elements in the outer world to match the thoughts (thematic patterns) of our inner world. As we think we perceive, and through our perceptions we reshape (recompose) the outer reality through our interpretation of it. Thought is a natural force as a vibration that has a structuring mechanism inherent in it, that reformulates light (material substance) through sympathies and antipathies, reshaping it (changing how it appears) to be of the same nature as the thoughts. Our thoughts form an idea as a mental impression or internal reality out of our paradigm (vibratory structure as a model) that becomes the perceptual lens that we "look through" to see the same thing outside of us that's inside of us. This mental impression as certain type of thoughts form internal representations of the outer world that are superimposed over it as our personal perception of it, and become a form of "organizational template" that's used to form a subjective experience in place of objective facts. In this way, we ultimately make everything "about" us by building it into our story about things.
Our internal reality of thought, organizes and forms how we experience the outer material plane, and serves to shape us because we are always "one with our thoughts" (the one having our thoughts), and "identifying" with our own self-created experiences, As we think and create, we're always spawning, integrating and assimilating our own thoughts to form a greater overall matrix and becoming the summation of our thoughts as an evolutionary process of becoming, which forms our individuality "as" our thoughts. Thoughts are being propagated and transmitted through the astral plane or mind-field of the Earth, as universal thoughts that are adapted and modified into the individual thoughts of humanity or mass consciousness, and are "picked up on" by other minds of the same nature that can accommodate and utilize the same "type of thoughts" into their mental model in a congruent manner, in a subliminal fashion as thoughts that just "come to us out of the blue". Whatever is of the same vibratory frequency as we are, or within near proximity, we resonate with and naturally absorb and perceive them as being our own, because they're the same type of thoughts we normally think.
Thought also exists in universal form, as archetypes that are metaphorical in nature, and act as a prototype or model for using to create our own personal version of the same idea through an imaginary process. As we draw in and are able to conceive of an idea that's of a universal nature, it's steadily modified through a series of adaptation to various factors – the paradigm and mental structure of the individual, their level of development and maturity, how the idea is applied to specific conditions, circumstances, and situations pertinent to the individual, how they're conformed to per-existing beliefs and assumptions, etc. – to produce a unique version of the same idea as a personalized version. Our karma is not only produced by our actions, but by way of our own thinking, which elicits complementary emotions and is what ultimately forms the basis for all of our actions. Our karma is the natural vibratory structure of our mind and how it naturally expresses to create our reality
Thought coming into the material mind in a pure, unadulterated form, has no emotional content or feeling (sensation) active in it, though it does contain feeling and emotion in its latent state, and acts to produce feeling as complementary sensations and emotions as its equivalent or that are of the same nature. The emotion elicited by thought, amplifies that intensifies the thought, producing the actions of the thoughts as our behaviors. All behavior is emotionally driven, and all emotions are produced by our thoughts. What we feel and think, we ultimately do, and what we do is the outer expression and activity of what we think. The mind is the soul, which in its spiritual aspect is the summation of our personal thoughts which form the basis for our perceptions and actions. These thoughts as mental impressions imprinted on our mind, that also produce the activities, experiences, and memories of the thoughts, is our karma as the patterns that determine the nature and course of our life events and what it is that we remain attached and connected to. As we think we become our thoughts and perceive the outer world through our thoughts about it, and we systematically produce the reality of our thoughts by changing how things "appear to us" through the story we tell ourselves about them.
We can work to develop ourselves as a soul-mind by learning how to willfully control our thoughts, and intentionally employ thoughts that will reshape our character, replace other thoughts of an undesirable and destructive nature, and align ourselves with new ideas that will serve to transform us making us able to connect with and produce a new kind of experience of reality. But we have to be able to do this while we're still experiencing old karma, disciplining ourselves to correct our own tendencies, until the old karma runs its course, creates the experience it was designed to create, and energetically burns out. Then, the karma created by the new way of thinking can begin to take hold and transform our life experiences by way of them. So we're always required to "pay forward" in terms of creating new forms of karma, knowing that they will only take effect in terms of our outer circumstances and situations once the effects of the old have passed. In some cases, as in relationships, changing our karma may only come through leaving the relationship, because we entered those relationships through "shared karma" as behavioral dynamics that the relationship will continue to hold us to, not allowing a new way of being. This is especially true when only one person in the relationship is changing and growing into new ways of being, while the other is still perceiving and acting out of the old karma..
The temptation to relapse involved in changing comes from the fact that just as our thinking and perceiving produced the outer events of our life, once the effect is produced outside of us, it then acts as the "cause" that stimulate in us more of the same type of thoughts, emotions, and behaviors in response to it. So a self-perpetuating cycle formed as the cause that produces an effect that becomes the cause for producing another effect that becomes the cause . . . . and so on, has to be neutralized. We have to break the cycle by recognizing how we're being effected by our own life circumstances and situations, and resist the tendency to think the way they naturally cause us to think, while consciously choosing instead to employ other thoughts and not react to the emotional charge they habitually produce. This forms the basis for what we think of as having to "believe" in something even though it's not happening and therefore self-evident, while knowing that it will come, because it operates by law, and is therefore always true as a kind of guarantee. The only real question becomes, how long it will take for our old way of thinking and emoting to dissolve and exhaust itself, no longer held together through a cooperative effort, and start manifesting as the changing conditions of our life circumstances.
Many of us have been taught to use our creative power by trying to change or manipulate other people and the outer conditions of our life, which is working on the "effect" and not with what's causing the effect. This is operating "against law", which is why it never works. In order to change the outer world, we have to change the inner world of the mind and individual character that's producing it as a natural expression. As long as we remain the same in terms of our thinking, perceiving, and the emotional states we live out of, we continue producing and experiencing the same type of ideas as thematic life patterns over and over. Our thoughts form the patterns of our life as a theme that weaves together all the areas of our life in a congruent and consistent manner. Because of this, no matter how drastic the changing conditions and circumstances of our life, we simply recreate the same type of ideas using new elements. This is because the mind "organizes" any of the outer elements into the same pattern as a reality that produces the same type of experiences in a faithful and automatic fashion.
The Golden Rule – what we sow, we also reap, what we put out comes back to us ten-fold (self-perpetuating as a cause and effect relationship). As we act "on the world" and others through our thoughts, we also act "on ourselves" by living out of the reality produced by our thoughts and gaining a sense of ourselves through our own self-created experiences. Our projections (reality created through our perceptual lens) are reflected back to us, and what we send out as our imagined thoughts (inner reality), produce a corresponding effect, that then becomes the cause of our thoughts, and we live out of the reality produced by our way of thinking as a rhythmic cycle. This self-perpetuating cycle as a cause and effect relationships forms our karma, determines our life experiences, and how they serve to continue shaping us as a product of our own creation. This cyclical effect can go on indefinitely, and is only changed through self-awareness, realization, and the disciplined use of will to change the very nature of our thoughts and our character, while still being stimulated by the return of previous thoughts tempting us to relapse into our old way of thinking. This takes commitment and devotion to a process with no immediate return, while disciplining ourselves to develop mental fortitude and moral stamina to persevere against the odds of our own making.
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text:
using System;
using System.Threading;
using L4p.Common.Extensions;
using L4p.Common.Loggers;

namespace L4p.Common.ForeverThreads
{
    public interface IForeverThread
    {
        void Start(string name = null);
        void Stop();
        void PostStopRequest();
        bool StopRequestIsPosted();
        bool IsStopped();
        bool ItsMyThread();
    }

    public class ForeverThread : IForeverThread
    {
        #region members

        private static int _threadCounter = 1;

        private readonly ILogFile _log;
        private readonly ForeverThreadConfig _config;
        private long _stopRequestCount;
        private readonly ManualResetEvent _isStartedEvent;
        private readonly ManualResetEvent _isStoppedEvent;
        private readonly Thread _thr;

        #endregion

        #region construction

        public static IForeverThread New(Action callback, ILogFile log = null, ForeverThreadConfig config = null)
        {
            log = log ?? LogFile.Console;
            config = config ?? new ForeverThreadConfig();

            return
                new ForeverThread(callback, log, config);
        }

        private ForeverThread(Action callback, ILogFile log, ForeverThreadConfig config)
        {
            _log = log;

            int count = Interlocked.Increment(ref _threadCounter);

            if (config == null)
            {
                config = new ForeverThreadConfig();
            }

            _config = new ForeverThreadConfig
                {
                    Name = config.Name ?? "ForeverThread.{0}".Fmt(count),
                    StartTimeout = config.StartTimeout,
                    StopTimeout = config.StopTimeout
                };

            _stopRequestCount = 0;
            _isStartedEvent = new ManualResetEvent(false);
            _isStoppedEvent = new ManualResetEvent(false);
            _thr = new Thread(() => thread_loop(callback));
        }

        #endregion

        #region private

        private void thread_loop(Action callback)
        {
            _isStartedEvent.Set();

            try
            {
                callback();
            }
            catch (Exception ex)
            {
                _log.Error(ex);
            }

            _isStoppedEvent.Set();
        }

        #endregion

        #region interface

        void IForeverThread.Start(string name)
        {
            if (name.IsNotEmpty())
                _config.Name = name;

            _thr.Start();
            bool isStarted = _isStartedEvent.WaitOne(_config.StartTimeout);

            if (isStarted)
                return;

            throw new ForeverThreadException(
                "Failed to start thread '{0}' in {1} milliseconds", _config.Name, _config.StartTimeout);
        }

        void IForeverThread.Stop()
        {
            Interlocked.Increment(ref _stopRequestCount);
            bool isStopped = _isStoppedEvent.WaitOne(_config.StopTimeout);

            if (isStopped)
                return;

            _log.Warn("Failed to stop thread '{0}' in {1} milliseconds", _config.Name, _config.StopTimeout);
        }

        void IForeverThread.PostStopRequest()
        {
            Interlocked.Increment(ref _stopRequestCount);
        }

        bool IForeverThread.StopRequestIsPosted()
        {
            bool posted = Interlocked.Read(ref _stopRequestCount) > 0;
            return posted;
        }

        bool IForeverThread.IsStopped()
        {
            bool isStopped = _isStoppedEvent.WaitOne(0);
            return isStopped;
        }

        bool IForeverThread.ItsMyThread()
        {
            return
                _thr == Thread.CurrentThread;
        }

        #endregion
    }
}

redpajama_set_name: RedPajamaGithub


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The 2018 National NM&SC Meeting, centered on the theme of Improving Human Performance and Effectiveness, focused on M&S technologies and practices having significant impact on human performance in today's environment - one characterized by 24/7 presence of M&S technologies in everyday life. Keynotes and panels addressed healthcare provider education, training, and certification, and probed the growing predominance of human-compuer coexistence in the age of Big Data, Analytics, & 24/7 interactions of individuals in daily life..
The event was held September 25-26, 2018 at the University of Nebraska Medical Center, Omaha, NE.
The complete agenda ifrom the event is available here.
Listed below are regional simulation centers by state. The Alabama Modeling & Simulation Council's (AMSC) primary goal is to … Read more about regional centers.
NM&SC seeks to provide a forum to bring together disparate organizations in order to enable the M&S community to speak with one … Read more about our affiliate organizations.
The NM&SC Research Agenda addresses both short and long term issues. Currently, the short term plan includes focuses on training the … Read more about our initiatives.