Proof The following proof is the r-dimensional generalization of the first part of the proof of [Reference Bley and CobbeBC17, Lemma 5.2.1].

By definition, we have $T^{\mathrm {nr}}=u\varphi (T^{\mathrm {nr}})$ , and by induction, we deduce $F_N(T^{\mathrm {nr}})=\varphi ^{d_N}(T^{\mathrm {nr}})=u^{-d_N}T^{\mathrm {nr}}$ , and hence $u^{d_N}F_N(T^{\mathrm {nr}})=T^{\mathrm {nr}}$ .

First of all, recall that the completion $\overline {\mathbb Z}_p^{\mathrm {nr}}$ of ${\mathbb Z}_p^{\mathrm {nr}}$ is contained both in $B_{\mathrm {cris}}$ and ${B_{\mathrm {dR}}}$ . We now prove that the elements $e_i^*$ are fixed by the absolute Galois group $G_{{\mathbb Q}_{p}}$ , which will show that the $e_i^*$ are contained in both $D_{\mathrm {cris}}^N(V^*(1))$ and $D_{\mathrm {dR}}^N(V^*(1))$ . We note that the inertia group $I_{{\mathbb Q}_{p}}$ acts trivially on $V^*(1)$ , and hence, it remains to prove that $e_i^*$ is fixed by $\varphi $ . We first need to calculate $\varphi (v_i^*)$ . Here, we use the definitions and the fact that the elements $v_i$ constitute the canonical basis of ${\mathbb Q}_p^r(1)(\rho ^{\mathrm {nr}}_{{\mathbb Q}_{p}})$ :

$$ \begin{align*} \varphi(v_i^*)(v_j)=v_i^*(u^{-1}v_j)=v_i^*\left(\sum_{k=1}^r (u^{-1})_{k,j}v_k\right)=(u^{-1})_{i,j}.\end{align*} $$

Hence,

$$ \begin{align*} \varphi(v_i^*)=\sum_{j=1}^r (u^{-1})_{i,j}v_j^*,\end{align*} $$

and we conclude that

$$ \begin{align*} \begin{aligned} \varphi(e_i^*)&=\left(\sum_{n=1}^r (T^{\mathrm{nr}})^{-1}_{i,n}\otimes v_n^*\right)^{\varphi}=\sum_{n=1}^r \sum_{h=1}^r (T^{\mathrm{nr}})^{-1}_{i,h}u_{h,n}\otimes \sum_{k=1}^r\left(u^{-1}\right)_{n,k}v_{k}^*\\ &=\sum_{h=1}^r\sum_{k=1}^r (T^{\mathrm{nr}})^{-1}_{i,h} \delta_{h,k}\otimes v_k^*=\sum_{k=1}^r (T^{\mathrm{nr}})^{-1}_{i,k} \otimes v_k^*=e_i^*. \end{aligned}\end{align*} $$

Because $T^{\mathrm {nr}} \in \mathrm {Gl}_r(\overline {\mathbb Z}_p^{\mathrm {nr}}) \subseteq \mathrm {Gl}_r(B_{\mathrm {cris}})$ , the elements $e_1^*,\dots ,e_r^*$ are a $B_{\mathrm {cris}}$ -basis of $B_{\mathrm {cris}} \otimes _{{\mathbb Q}_{p}} V^*(1)$ . As $N_1$ is a subfield of $B_{\mathrm {cris}}$ , we see that $e_1^*,\dots ,e_r^*$ are linearly independent over $N_1$ . This concludes the proof that the elements $e_i^*$ constitute a basis of $D_{\mathrm {cris}}^N(V^*(1))$ , because $\dim _{N_1}D_{\mathrm {cris}}^N(V^*(1))\leq \dim _{{\mathbb Q}_{p}}(V^*(1))=r$ . In particular, this also proves that $V^*(1)$ is cristalline. Then, the elements $e_1^*,\dots ,e_r^*$ must also be a basis of the N-vector space $D_{\mathrm {dR}}^N(V^*(1))=N\otimes _{N_1}D_{\mathrm {cris}}^N(V^*(1))$ .▪

Proof For $\sigma \in I_{{\mathbb Q}_{p}}$ we compute

$$ \begin{align*} \sigma(e_i) = \sum_{n=1}^r \sigma(t^{-1} T^{\mathrm{nr}}_{n,i}) \otimes \sigma(v_n) = \sum_{n=1}^r \chi_{cyc}(\sigma^{-1})t^{-1} T^{\mathrm{nr}}_{n,i} \otimes \chi_{cyc}(\sigma)v_n = e_i. \end{align*} $$

Hence, the elements $e_i$ are fixed by the inertia group, and a similar computation as in the proof of Lemma 4.1 shows that $\varphi (e_i) = e_i$ . The proof follows as above.▪

Proof Similar as above.▪

Proof Proofs are as for [Reference Bley and CobbeBC17, Lemma 5.1.2]. For the proof of part (c), we also need that by Lemma 4.7 below the endomorphism $1 - \phi $ of $D_{\mathrm {cris}}^N(V)$ is an isomorphism.▪

Proof By Gaussian elimination, we obtain the matrix

$$ \begin{align*} \begin{pmatrix}1&0&0&\cdots&0&B_1\\ 0&1&0&\cdots&0&B_2-AB_1\\ 0&0&1&\cdots&0&B_3-AB_2+A^2B_1\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&\sum_{i=0}^{n-2}(-A)^i B_{n-1-i}\\ 0&0&0&\cdots&0&1+\sum_{i=0}^{n-1}(-A)^i B_{n-i}\end{pmatrix}. \end{align*} $$

▪

Proof We have to compute $\phi (e_i^*)$ . Using the $\varphi $ -semilinearity of $\phi $ , we compute

$$ \begin{align*}\begin{aligned}\phi(e_i^*)&=\sum_{n=1}^r\varphi((T^{\mathrm{nr}})^{-1})_{i,n}\otimes v_n^*=\sum_{n=1}^r ((T^{\mathrm{nr}})^{-1} u)_{i,n}\otimes v_n^*\\ &=\sum_{n=1}^r ((T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}}(T^{\mathrm{nr}})^{-1})_{i,n}\otimes v_n^*=\sum_{n=1}^r\sum_{\ell=1}^r ((T^{\mathrm{nr}})^{-1} u T^{\mathrm{nr}})_{i,\ell}(T^{\mathrm{nr}})^{-1}_{\ell,n}\otimes v_n^*\\ &=\sum_{\ell=1}^r ((T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}})_{i,\ell}e_{\ell}^*.\end{aligned}\end{align*} $$

We fix a normal basis element $\theta $ of $N_1/{{\mathbb Q}}_p$ . Then, $w_{i,j}:=\varphi ^{-j}(\theta ) e_i^*$ , for $i=1,\dots ,r$ and $j=0,\dots ,d_K-1$ , is a ${\mathbb Q}_p[G/I]$ -basis of $D_{\mathrm {cris}}^N(V^*(1))$ . Let $\psi _{i,j}\in D_{\mathrm {cris}}^N(V^*(1))^*$ for $i=1,\dots ,r$ and $j=0,\dots ,d_K-1$ be the dual ${\mathbb Q}_p[G/I]$ -basis.

For $0<i,h\leq r$ , $0\leq j,k<d_K-1$ , and $0\leq n\leq d-1$ , we have

$$ \begin{align*} \begin{aligned} \phi^{*}(\psi_{i,j}) (F^n w_{h,k}) &= \psi_{i,j} (\phi(F^n \varphi^{-k}(\theta) e_h^{*})) \\&= \psi_{i,j}\!\left(\!F^n \varphi^{-k+1}(\theta) \sum_{\ell=1}^r ((T^{\text{nr}})^{-1} uT^{\text{nr}})_{h,\ell}e_{\ell}^{*}\right)\\ &=\sum_{\ell=1}^r ((T^{\text{nr}})^{-1} u T^{\text{nr}})_{h,\ell}\psi_{i,j}(F^n \varphi^{-k+1}(\theta) e_{\ell}^{*})\\&=\sum_{\ell=1}^r ((T^{\text{nr}})^{-1} uT^{\text{nr}})_{h,\ell}\psi_{i,j}(F^n w_{\ell,k-1}). \end{aligned} \end{align*} $$

If $n=0$ , $k=j+1$ , and any $0<h\leq r$ , this is equal to $((T^{\mathrm {nr}})^{-1} uT^{\mathrm {nr}})_{h,i}$ ; it is $0$ otherwise. Hence,

$$ \begin{align*}\phi^*(\psi_{i,j})=\sum_{h=1}^r ((T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}})_{h,i}\psi_{h,j+1}.\end{align*} $$

Analogously, for $0<i\leq r$ and $j=d_K-1$ , we have

$$ \begin{align*} \phi^*(\psi_{i,d_K-1})=\sum_{h=1}^r ((T^{\mathrm{nr}})^{-1} u T^{\mathrm{nr}})_{h,i}F^{-1}\psi_{h,0}. \end{align*} $$

Note here that $\phi ^*$ is indeed defined over ${\mathbb Q}_p[G/I]$ , because $(T^{\mathrm {nr}})^{-1}uT^{\mathrm {nr}}\in M_r({\mathbb Q}_p)$ .

With respect to the ${{\mathbb Q}_{p}}[G/I]$ -basis $\psi _{i,j}$ the matrix associated to $1-\phi ^*$ is given by

$$ \begin{align*} \begin{pmatrix}1&0&0&\cdots&0&-F^{-1}(T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}}\\ -(T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}}&1&0&\cdots&0&0\\ 0&-(T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}}&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\\ 0&0&0&\cdots&-(T^{\mathrm{nr}})^{-1} uT^{\mathrm{nr}}&1\end{pmatrix}. \end{align*} $$

In this matrix, each entry is an $r\times r$ matrix with coefficients in ${\mathbb Q}_p[G/I]$ (recall that $F=F_K$ generates $G/I$ ). In the following, we write $\det $ for the determinant over the commutative ring ${\mathbb Q}_p[G/I]$ . By Lemma 4.5, the determinant of the above matrix is $\det (1-F^{-1}(T^{\mathrm {nr}})^{-1} u^{d_K}T^{\mathrm {nr}})=\det (1-F^{-1}u^{d_K})$ . Because this determinant computes the reduced norm of $1-\phi ^*$ , the equality in the lemma follows.

To conclude that $1-\phi ^*$ is an isomorphism, it is now enough to notice that

$$ \begin{align*} (1-F^{-1}u^{d_K})(1+F^{-1}u^{d_K}+\dots+(F^{-1}u^{d_K})^{d_{N/K-1}})=1-U_N, \end{align*} $$

which is invertible, because, by Hypothesis (F), we always have $\det (U_N-1)\neq 0$ .▪

Proof The proof is analogous to the proof of the previous lemma.

To show that $\det (1-F(pu)^{-d_K})$ is invertible in ${{\mathbb Q}_{p}}[G/I]$ , it is enough to notice that $\sum _{i=0}^{\infty } (F^{-1}(pu)^{d_K})^i$ converges in the matrix ring $M_r({{\mathbb Z}_p}[G/I])$ and is the inverse of

$$ \begin{align*} 1-F^{-1}(pu)^{d_K}=-F^{-1}(pu)^{d_K}(1-F(pu)^{-d_K}).\\[-35pt] \end{align*} $$

▪

Proof The proof is easily achieved by combining (4.1) and Lemmas 4.6 and 4.7.▪

Proof Let $u_{\mathrm {cris},N/K}\in Z({{\mathbb Q}_{p}}[G])^{\times }$ be such that $\hat \partial _{{{\mathbb Z}_p[G]}, {\mathbb Q_p[G]}}^1(u_{\mathrm {cris},N/K})=U_{\mathrm {cris},N/K}$ . We will use an analogous notation for all the other Galois extensions involved in the proof.

Then, for any irreducible character $\chi $ of G, we can take

$$ \begin{align*} (u_{\mathrm{cris},N/K})_{\chi}= \begin{cases} \frac{\det(1-\chi(F)(pu)^{-d_K})}{\det(1-u^{d_K}\chi(F)^{-1})}&\text{if }\chi|_{I_{N/K}}=1,\\ 1&\text{if }\chi|_{I_{N/K}} \neq 1. \end{cases} \end{align*} $$

For two (virtual) characters $\chi _1$ and $\chi _2$ of a finite group J, we write $\langle \chi _1, \chi _2\rangle _J$ for the standard scalar product.

1. (1) By [Reference Bley and WilsonBW09, Section 6.1], we have

   $$ \begin{align*} \rho^G_H(u_{\mathrm{cris},N/K})= \left( \prod_{\chi\in\mathrm{Irr}(G)}(u_{\mathrm{cris},N/K})_{\chi}^{\langle\chi,\mathrm{Ind}^G_H\psi\rangle_G} \right)_{\psi\in \mathrm{Irr}({H})}. \end{align*} $$
   Because $\langle \chi ,\mathrm {Ind}^G_H\psi \rangle _G = \langle \chi |_H,\psi \rangle _H$ by Frobenius reciprocity, we obtain
   $$ \begin{align*} (u_{\mathrm{cris},N/K})_{\chi}^{\langle \chi,\mathrm{Ind}_H^G\psi\rangle_G} =\begin{cases}1&\text{if }\chi|_{I_{N/K}}\neq 1,\\ u_{\mathrm{cris},N/K}& \text{if }\chi|_{I_{N/K}}=1 \text{ and } \chi|_H = \psi,\\ 1&\text{if }\chi|_{I_{N/K}}=1 \text{ and } \chi|_H \neq \psi. \end{cases} \end{align*} $$
   If $\psi |_{I_{N/L}} \ne 1$ , then $\chi |_{I_{N/K}} \ne 1$ whenever $\langle \chi |_H, \psi \rangle _H \ne 0$ . Thus, $\rho ^G_H(u_{\mathrm {cris},N/K})_{\psi } = 1$ for those characters $\psi $ .

   On the other hand, if $\psi |_{I_{N/L}} = 1$ , then $\psi $ is a character of the cyclic group $\overline {H} := H/I_{N/L} = \langle F_L \rangle $ . Each character $\chi \in \mathrm {Irr}(G)$ with $(u_{\mathrm {cris},N/K})_{\chi } \ne 1$ is actually a character of $\overline {G} := G / I_{N/K} = \langle F_K \rangle $ . Note that we can naturally identify $\overline {H}$ with a subgroup of $\overline {G}$ and recall that $|\overline {G} / \overline {H}| = d_{L/K}$ .

   We therefore obtain

   $$ \begin{align*} \prod_{\chi\in\mathrm{Irr}(G)}(u_{\mathrm{cris},N/K})_{\chi}^{\langle\chi,\mathrm{Ind}^G_H\psi\rangle_G} = \prod_{\chi \in \mathrm{Irr}(\overline{G}), \chi|_{\overline{H}} = \psi} \frac{\det(1 - \chi(F_K)(pu)^{-d_K})}{\det(1 - u^{d_K}\chi(F_K)^{-1})}. \end{align*} $$
   We consider the numerator and the denominator separately and use, in each case, the polynomial identity
   $$ \begin{align*} \prod_{\chi \in \mathrm{Irr}(\overline{G}), \chi|_{\overline{H}} = \psi} (X - \chi(F_K)) = X^{d_{L/K}} - \psi(F_L). \end{align*} $$

   For the numerator, we compute

   $$ \begin{align*} & \prod_{\chi \in \mathrm{Irr}(\overline{G}), \chi|_{\overline{H}} = \psi} \det(1 - \chi(F_K)(pu)^{-d_K}) \\ &= \det \left( \prod_{\chi \in \mathrm{Irr}(\overline{G}), \chi|_{\overline{H}} = \psi} (pu)^{-d_K} \left( (pu)^{d_K} - \chi(F_K) \right) \right) \\ &= \det \left( (pu)^{-d_K d_{L/K}} \left( (pu)^{d_K d_{L/K}} - \psi(F_L) \right) \right) \\ &= \det \left( 1 - (pu)^{-d_L} \psi(F_L)\right), \end{align*} $$
   and a similar computation for the denominator shows claim (a).

2. (2) For any character $\psi $ of $G/H$ , we write $\mathrm {infl}(\psi )$ for the inflated character of G. By [Reference Bley and WilsonBW09, Section 6.3], $q^G_{G/H}(u_{\mathrm {cris},N/K})_{\psi }=(u_{\mathrm {cris},N/K})_{\mathrm {infl}(\psi )}$ for any $\psi \in \mathrm {Irr}(G/H)$ . This is equal to $(u_{\mathrm {cris},L/K})_{\psi }$ , because $I_{L/K} = I_{N/K}H/H$ and $\mathrm {infl}(\psi )(F_K)=\psi (F_K)$ .▪

Proof This proof is an expanded version of the arguments of [Reference Izychev and VenjakobIV16, p. 509].

Recall the isomorphism $\theta :\mathcal F\to \mathbb {G}_m^r$ from (2.1), which satisfies $\theta (x)\equiv \varepsilon ^{-1}x\pmod {\deg \geq 2}$ . Let $\pi _{{B_{\mathrm {dR}}}}:{B_{\mathrm {dR}}}^+\to \mathbb C_p$ be the natural projection to the residue field. Similarly to [Reference Bloch, Kato, Cartier, Illusie, Katz, Laumon, Manin and RibetBK90, p. 360], we can construct a commutative diagram of exact sequences:

Note that (differently from [Reference Bloch, Kato, Cartier, Illusie, Katz, Laumon, Manin and RibetBK90]) some of the objects are twisted by $\rho ^{\mathrm {nr}}$ in order to make all maps $G_N$ -invariant; $\varepsilon $ always denotes multiplication by the element $\varepsilon \in \overline {{\mathbb Q_p^{\mathrm {nr}}}}$ which occurs in (2.1). We observe that by Lemma 3.1 we have the equality $\log _p\circ \varepsilon \circ \theta =\log _{\mathcal F}$ .

Taking $G_N$ -fixed elements and cohomology, we obtain

where the map s is such that $s(v_i)=e_i$ for all i. Note that this diagram is the higher dimensional version of [Reference Izychev and VenjakobIV16, equation (3.4)]. It makes the identification s of $N^r$ and the tangent space $t_V(N)$ explicit.

We rewrite $\lambda _2$ in terms of the maps in the last diagram and get

By diagram (6.3), we see that $(\partial _{Ku}^{-1}\circ H^0(\xi )\circ f_{{\mathcal F},{\mathcal N}})(X(\mathcal {L})) = X(\mathcal {L})$ , so that it remains to show

$$ \begin{align*} [X(\mathcal{L}), s \circ \log_{\mathcal F}, \bigoplus_{i=1}^r \mathcal{L} e_i] = 0, \end{align*} $$

which is immediate from $\log _{\mathcal F}(X(\mathcal {L})) = \mathcal {L}^{(r)}$ and $s(v_i)=e_i$ for $i=1, \ldots ,r$ .▪

Proof It is easy to see that in the r-dimensional setting, we also have a diagram as in [Reference Izychev and VenjakobIV16, equation (6.1)]. If we denote the vertical maps in this diagram by $f_1$ and $f_2$ , then

$$ \begin{align*} \begin{aligned} &\left[\bigoplus_{i=1}^r\mathcal L e_i,\mathrm{comp}_V,\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T\right]+ [\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T,f_2,\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1)]\\ &\qquad\qquad=\left[\bigoplus_{i=1}^r\mathcal L e_i,f_1,\bigoplus_{i=1}^r\mathcal L \tilde e_i\right]+ \left[\bigoplus_{i=1}^r\mathcal L \tilde e_i,\mathrm{comp}_{V(-1)},\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1)\right]. \end{aligned} \end{align*} $$

Because $f_1$ sends each basis element $e_i$ to $\tilde e_i$ , the first summand on the right-hand side is trivial. Both $\mathrm {Ind}_{N/{{\mathbb Q}_{p}}}T$ and $\mathrm {Ind}_{N/{{\mathbb Q}_{p}}}T(-1)$ are isomorphic to ${{\mathbb Z}_p}[G]^{rm}$ as ${{\mathbb Z}_p}[G]$ -modules. Via these isomorphisms, the map $f_2$ corresponds to multiplication by t, and so we obtain

$$ \begin{align*} [\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T,f_2,\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1)]= \left[{{\mathbb Z}_p}[G]^{rm},t,{{\mathbb Z}_p}[G]^{rm}\right]=rm\hat\partial_{{{\mathbb Z}_p[G]}, {{B_{\mathrm{dR}}}[G]}}^1(t). \end{align*} $$

▪

Proof We let $T_{\mathrm {triv}} = {\mathbb Z}_p^{(r)}$ and $V_{\mathrm {triv}}=\mathbb Q_p^{(r)}$ denote the trivial representations. Let $z_1, \ldots , z_r$ denote the canonical ${{\mathbb Z}_p}$ -basis of $T_{\mathrm {triv}}$ .

In the following, we choose to use for each $G_N$ -representation W

$$ \begin{align*} \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}(W) = \{ x\colon G_{{\mathbb Q}_{p}} \longrightarrow W \mid x(\tau\sigma) = \tau x(\sigma) \text{ for all } \tau \in G_N, \sigma \in G_{{\mathbb Q}_{p}} \} \end{align*} $$

as the definition for the induction. Note that if $L/{{\mathbb Q}_{p}}$ is any field extension (e.g., $L = {B_{\mathrm {dR}}}$ ) which carries an action of $G_{{\mathbb Q}_{p}}$ and W is an L-space, then $ \mathrm {Ind}_{N/{{\mathbb Q}_{p}}}(W)$ is also an L-space with $(\alpha x)(\sigma ) = \sigma (\alpha ) x(\sigma )$ for all $\alpha \in L$ and $\sigma \in G_{{\mathbb Q}_{p}}$ . We also note that

$$ \begin{align*} \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}(W) \longrightarrow L[G_{{\mathbb Q}_{p}}] \otimes_{L[G_N]} W, \quad x \mapsto \sum_{\sigma \in G_{{\mathbb Q}_{p}} / G_N} \sigma \otimes x(\sigma^{-1}), \end{align*} $$

is a well-defined isomorphism of $L[G]$ -modules. For the comparison isomorphism $\mathrm {comp}_W$ , we then obtain the following simple description:

$$ \begin{align*} \mathrm{comp}_W \colon L \otimes_{{\mathbb Q}_{p}} \left( L \otimes_{{\mathbb Q}_{p}} W \right)^{G_N} &\longrightarrow \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}(L \otimes_{{\mathbb Q}_{p}} W), \\ l \otimes z &\mapsto l y_z , \end{align*} $$

where $l \in L$ , $z \in \left ( L \otimes _{{\mathbb Q}_{p}} W \right )^{G_N}$ and $y_z(\sigma ) := z$ for all $\sigma \in G_{{\mathbb Q}_{p}}$ (and hence, $(ly_z)(\sigma ) = \sigma (l)z$ ).

We define a G-equivariant isomorphism

$$ \begin{align*} \tilde h \colon \left( {B_{\mathrm{dR}}} \otimes_{{\mathbb Q}_{p}} V_{\mathrm{triv}} \right)^{G_N} \longrightarrow \left( {B_{\mathrm{dR}}} \otimes_{{\mathbb Q}_{p}} V(-1) \right)^{G_N}, \quad 1 \otimes z_i \mapsto \tilde e_i, \end{align*} $$

and

$$ \begin{align*} h \colon \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}\left( {B_{\mathrm{dR}}} \otimes_{{\mathbb Q}_{p}} V_{\mathrm{triv}} \right) \longrightarrow \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}\left( {B_{\mathrm{dR}}} \otimes_{{\mathbb Q}_{p}} V(-1) \right), \quad x \mapsto \tilde h \circ x. \end{align*} $$

Then, similar as in the proof of [Reference Izychev and VenjakobIV16, Lemma 6.1], we obtain a commutative diagram:

As a consequence, we derive

$$ \begin{align*}\begin{aligned} & \left[\bigoplus_{i=1}^r\mathcal L \tilde e_i,\mathrm{comp}_{V(-1)},\mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1)\right] \\ &\quad = \left[\bigoplus_{i=1}^r\mathcal L z_i,{B_{\mathrm{dR}}} \otimes \tilde h, \bigoplus_{i=1}^r\mathcal L \tilde e_i \right] + \left[\bigoplus_{i=1}^r\mathcal L z_i, \mathrm{comp}_{V_{\mathrm{triv}}}, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} \right] \\ &\qquad\qquad + \left[ \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} , h, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1) \right] \\ &\quad = \left[\bigoplus_{i=1}^r\mathcal L z_i, \mathrm{comp}_{V_{\mathrm{triv}}}, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} \right] + \left[ \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} , h, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1) \right] \\ &\quad = r \left[ \mathcal{L}, \mathrm{id}, {\mathcal{O}_K[G]} \beta \right] + \left[\bigoplus_{i=1}^r({\mathcal{O}_K[G]} \beta z_i), \mathrm{comp}_{V_{\mathrm{triv}}}, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} \right] \\ &\qquad\qquad + \left[ \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} , h, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1) \right] \end{aligned}\end{align*} $$

The computations in [Reference Izychev and VenjakobIV16, pp. 512–513] show that

$$ \begin{align*} \left[\bigoplus_{i=1}^r\mathcal O_K[G]\beta z_i, \mathrm{comp}_{V_{\mathrm{triv}}}, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} \right] = r \hat\partial_{{{\mathbb Z}_p[G]}, {{B_{\mathrm{dR}}}[G]}}^1(\rho_{\beta}). \end{align*} $$

It finally remains to prove that

$$ \begin{align*} \left[ \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T_{\mathrm{triv}} , h, \mathrm{Ind}_{N/{{\mathbb Q}_{p}}}T(-1) \right] = m U_{tw}(\rho^{\mathrm{nr}}_{{\mathbb Q}_{p}}). \end{align*} $$

To that end, we write

$$ \begin{align*} G_{{\mathbb Q}_{p}} = \bigcup_{\bar\rho\in G} \bigcup_{\sigma_i\in G_K\backslash G_{{\mathbb Q}_{p}}} G_N\rho\sigma_i \end{align*} $$

and define elements $x_{ij} \in \mathrm {Ind}_{N/{{\mathbb Q}_{p}}}(T_{\mathrm {triv}})$ and $y_{ij} \in \mathrm {Ind}_{N/{{\mathbb Q}_{p}}}(T(-1))$ , for $i = 1, \ldots ,m$ and $j = 1, \ldots , r$ , by

$$ \begin{align*} x_{ij}(\rho\sigma_k) = \begin{cases} z_j, & \text{ if } i=k \text{ and } \bar\rho = 1, \\ 0, & \text{ otherwise}, \end{cases} \end{align*} $$

and

$$ \begin{align*} y_{ij}(\rho\sigma_k) = \begin{cases} \tilde v_j, & \text{ if } i=k \text{ and } \bar\rho = 1, \\ 0, & \text{ otherwise}. \end{cases} \end{align*} $$

Without loss of generality, we assume $\rho =1$ for $\bar \rho =1$ . Then, the $x_{ij}$ , respectively, the $y_{ij}$ , constitute a ${{\mathbb Z}_p[G]}$ -basis of $\mathrm {Ind}_{N/{{\mathbb Q}_{p}}}(T_{\mathrm {triv}})$ , respectively, $\mathrm {Ind}_{N/{{\mathbb Q}_{p}}}(T(-1))$ .

For fixed i and j and for $1 \le k \le m$ , we compute

$$ \begin{align*} \left( h(x_{ij}) \right) (\rho\sigma_k) = \begin{cases} \tilde e_j, & \text{ if } i=k \text{ and } \bar\rho = 1, \\ 0, & \text{ otherwise}. \end{cases} \end{align*} $$

For $1 \le s \le m$ and $1 \le t \le r$ , let $\xi _{st}$ be indeterminates (with values in ${B_{\mathrm {dR}}}$ ). Then,

$$ \begin{align*} \left( \sum_{s,t} \xi_{st} y_{st} \right) (\rho\sigma_k) = \begin{cases} \sum_t (\rho\sigma_k)(\xi_{kt}) \tilde v_t, & \text{ if } \bar\rho = 1, \\ 0, & \text{ otherwise}. \end{cases} \end{align*} $$

Because $\tilde e_j = \sum _t T^{\mathrm {nr}}_{tj} \otimes \tilde v_t$ , we obtain

$$ \begin{align*} \sum_t (\rho\sigma_k)(\xi_{kt}) \tilde v_t = \begin{cases} \sum_t T^{\mathrm{nr}}_{tj} \otimes \tilde v_t, & \text{ if } i=k, \\ 0, & \text{ if } i \ne k, \end{cases} \end{align*} $$

and hence, for all $1\leq t\leq r$ ,

$$ \begin{align*} \xi_{it} = \sigma_i^{-1}(T^{\mathrm{nr}}_{tj}), \quad \xi_{kt} = 0 \text{ for } k \ne i. \end{align*} $$

We conclude that with respect to the chosen basis, the map h is represented by a block matrix of the form

$$ \begin{align*} C = \left( \begin{array}{ccc} \sigma_1^{-1}(T^{\mathrm{nr}}) & & \\ & \ddots & \\ & & \sigma_m^{-1}(T^{\mathrm{nr}}) \end{array} \right). \end{align*} $$

We recall that $\varphi (T^{\mathrm {nr}}) = u^{-1}T^{\mathrm {nr}}$ and fix $\alpha _i \in \hat {\mathbb z}$ such that $\sigma _i |_{\mathbb Q_p^{\mathrm {nr}}} = \varphi ^{\alpha _i}$ . Note that for all $n\in {\mathbb z}$ , $\varphi ^{-n}(T^{\mathrm {nr}})\cdot (T^{\mathrm {nr}})^{-1}=u^n$ , which coincides with $\rho ^{\mathrm {nr}}(\varphi ^n)$ . By a continuity argument, we have $\varphi ^{-\alpha _i}(T^{\mathrm {nr}})\cdot (T^{\mathrm {nr}})^{-1}=u^{\alpha _i}$ , where $u^{\alpha _i}$ is well defined by [Reference CobbeCob18, Lemma 1.5]. Then,

$$ \begin{align*} \det(C) = \prod_{i=1}^m \det\left( u^{\alpha_i}T^{\mathrm{nr}} \right) = \det(u)^{\alpha} \det(T^{\mathrm{nr}})^m \end{align*} $$

with $\alpha = \sum _i \alpha _i$ . Because $\det (u)^{\alpha } \in {\mathbb Z}_p^{\times } \subseteq {{\mathbb Z}_p[G]}^{\times }$ , the result follows from the definition of $U_{tw}(\rho ^{\mathrm {nr}}_{{\mathbb Q}_{p}})$ .▪

Proof The key point in the proof is that by Proposition 2.2 the cohomology modules of $M^{\bullet }(\mathcal {L})$ are perfect, so that we can compute the refined Euler characteristic of $M^{\bullet }(\mathcal {L})$ in terms of cohomology without explicitly using the complex. In a little more detail, we note that the mapping cone of

$$ \begin{align*} \mathcal F(\mathfrak p_N^{(r)}) / X(\mathcal L)[1] \to M^{\bullet}(\mathcal{L}), \end{align*} $$

where the map in degree $1$ is induced by $f_{{\mathcal F},{\mathcal N}}$ , is isomorphic to $H^2(N, T)[2]$ . We also recall from Section 2 that we identify $H^2(N, T)$ with $\mathcal {Z} / (F_N -1)\mathcal {Z}$ . Hence, we conclude from [Reference Breuning and BurnsBB05, Theorem 5.7] that

$$ \begin{align*} \begin{aligned} &\chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}(M^{\bullet}(\mathcal{L}), 0)\\ &\qquad=\chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}\left( \mathcal F(\mathfrak p_N^{(r)})/X(\mathcal L)[1],0 \right) + \chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}(\mathcal{Z} / (F_N -1)\mathcal{Z}, 0). \end{aligned} \end{align*} $$

To compute the first summand, we observe that by Proposition 3.4 we have $X(\mathcal {L}) = \mathcal F\left ( (\mathfrak p_N^{\nu })^{(r)}\right )$ . Because, for each integer $i \ge 0$ , the identity map induces isomorphisms

(6.12) $$ \begin{align} \mathcal F\left( (\mathfrak p_N^i)^{(r)}\right) / \mathcal F\left( (\mathfrak p_N^{i+1})^{(r)}\right) \cong (\mathfrak p_N^i)^{(r)} / (\mathfrak p_N^{i+1})^{(r)}, \end{align} $$

a standard argument shows that

$$ \begin{align*} \chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}\left( \mathcal F(\mathfrak p_N^{(r)})/X(\mathcal L)[1],0 \right) &= \chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}\left( \mathfrak p_N^{(r)} / (\mathfrak p_N^{\nu})^{(r)} [1],0 \right) \\& = r [\mathfrak p_N^{\nu}, \mathrm{id}, \mathfrak p_N]. \end{align*} $$

For the computation of the second term, we consider the short exact sequence of G-modules

$$ \begin{align*} 0 \to {\mathbb Z}_p^r[G/I] \xrightarrow{F^{-1}u^{d_K}-1} {\mathbb Z}_p^r[G/I] \xrightarrow{\pi} \mathcal{Z} / (F_N -1)\mathcal{Z} \to 0, \end{align*} $$

where $\pi (z(\bar g))=g\cdot (z+(F_N-1)\mathcal {Z})$ for all $z\in {\mathbb Z}_p^r$ and $g\in G$ and where G acts on $\mathcal {Z} / (F_N -1)\mathcal {Z}$ through any lift of its elements to $G_K$ (which is well defined, because elements of $G_N$ act trivially).

Let $x=\sum _{i=0}^{d-1}\alpha _i F^{-i}\in {\mathbb Z}_p^r[G/I]$ be an element in the kernel of the map on the left. Then, $u^{d_K}\alpha _{i-1}-\alpha _i=0$ for all i. Hence, $(u^{dd_K}-1)\alpha _i=0$ , and by the assumption $\mathcal Z^{G_N}=1$ , it follows that $\alpha _i=0$ for all i. Hence, the map on the left is injective.

Next, we see that $\pi ((F^{-1}u^{d_K}-1)e_i)=(\rho ^{\mathrm {nr}}(F^{-1})u^{d_K}-1)e_i=0$ .

Conversely, let $x=\sum _{i=0}^{d-1}\alpha _i F^{-i}\in {\mathbb Z}_p^r[G/I]$ be such that $\pi (x)=0$ . Modulo the image of $F^{-1}u^{d_K}-1$ , x has a representative $y\in {\mathbb Z}_p^r$ . We must show that $y\in \mathrm {im}(F^{-1}u^{d_K}-1)$ . Because $\pi (y)=0$ , there exists $z\in {\mathbb Z}_p^r$ such that $y=(u^{dd_K}-1)z=((F^{-1}u^{d_K})^d-1)z$ , which is in the image of $F^{-1}u^{d_K}-1$ . Hence, we have exactness in the middle term.

To prove the exactness of the sequence, it remains to check the surjectivity of the map on the right, which is obvious.

Because we are considering the case of a tame extension, ${\mathbb Z}_p^r[G/I]$ is a projective ${\mathbb Z}_p[G]$ -module and we have:

$$ \begin{align*} \chi_{{\mathbb Z}_p[G], {B_{\mathrm{dR}}}[G]}(H^2(N,T)[2],0)=-[{\mathbb Z}_p^r[G/I],F^{-1}u^{d_K}-1,{\mathbb Z}_p^r[G/I]]. \end{align*} $$

The results follows.▪

Proof For $n \ge 2$ , the formal logarithm induces an isomorphism $\mathcal F((\mathfrak p_N^n)^{(r)}) \cong (\mathfrak p_N^n)^{(r)}$ of ${{\mathbb Z}_p[G]}$ -modules by Proposition 3.4. Hence, the first assertion follows from [Reference KöckKöc04, Theorem 1.1 and Proposition 1.3].

We henceforth assume $n=1$ . By Lemma 2.3, we know that $\mathcal F(\mathfrak p_N^{(r)})$ is cohomologically trivial, if and only if Hypothesis (I) holds. Hence, it suffices to prove that $\mathcal F(\mathfrak p_N^{(r)})$ is torsion-free. By [Reference CobbeCob18, Lemma 2.1], the module $\mathcal F(\mathfrak p_N^{(r)})$ is isomorphic to $\left ( \prod _r \widehat {N_0^{\times }} (\rho ^{\mathrm {nr}})\right )^{G_N}$ which is torsion-free. Indeed, any tuple $(\zeta _1, \ldots , \zeta _r)$ of pth roots of unity $\zeta _1, \ldots , \zeta _r \in \widehat {N_0^{\times }} \cong {{\mathbb Z}_p} \times U_{N_0}^{(1)}$ must be contained in N (because $N_0/N$ is unramified). Hence, $(\zeta _1, \ldots , \zeta _r)$ is fixed by $G_N$ , if and only if it is fixed by $F_N$ , if and only if $\zeta _1 = \cdots = \zeta _r = 1$ (using Hypothesis (I)).▪

Proof Let $H=\mathrm {Gal}(K'/\mathbb Q_p)$ and let $f \colon H\to K'$ be defined by $f(\sigma )=(A\theta _2)^{\sigma ^{-1}}$ . Then, applying [Reference WashingtonWas97, Lemma 5.26(a)], we obtain

$$ \begin{align*} \det((A\theta_2)^{\tau\sigma^{-1}})_{\sigma,\tau\in H}=\prod_{\chi\in \hat H}\sum_{\sigma\in H}(A\theta_2)^{\sigma^{-1}}\chi(\sigma). \end{align*} $$

Because $A\theta _2$ is an integral normal basis generator and $K'/\mathbb Q_p$ is unramified, the left-hand side $\det (\tau \sigma ^{-1}(A\theta _2))_{\sigma ,\tau \in H}$ is a unit (whose square is the discriminant of $K'/\mathbb Q_p$ ). Therefore, for each character $\chi \in \hat H$ , the factor

$$ \begin{align*} \sum_{\sigma\in H}(A\theta_2)^{\sigma^{-1}}\chi(\sigma)=\sum_{i=0}^{dm-1}(A\theta_2)^{\varphi^{i}}\chi(\varphi^{-i}) \end{align*} $$

is a unit, and hence, $\sum _{i=0}^{dm-1}(A\theta _2)^{\varphi ^{i}}\varphi ^{-i}$ is a unit in the maximal order ${\mathcal M}$ of $K'[H]$ . Because $\sum _{i=0}^{dm-1}(A\theta _2)^{\varphi ^{i}}\varphi ^{-i}\in \mathcal O_{K'}[H]$ , we deduce from the well-known fact

$$ \begin{align*} {\mathcal M}^{\times} \cap \mathcal O_{K'}[H] = \mathcal O_{K'}[H]^{\times} \end{align*} $$

that $\sum _{i=0}^{dm-1}(A\theta _2)^{\varphi ^{i}}\varphi ^{-i}\in \mathcal O_{K'}[H]^{\times }$ . We now apply the character $\rho ^{\mathrm {nr}}_{{\mathbb Q}_{p}}$ and easily derive (a).

For the proof of (b), we can assume Hypothesis (T) and we compute

$$ \begin{align*} \varphi(E) = \sum_{i=0}^{dm-1} (A\theta_2)^{\varphi^{i+1}}u^{-(i+1)} u \equiv Eu \equiv uE \pmod{p\mathcal{O}_{K'}}, \end{align*} $$

where the congruences hold, because we have $u^{md} \equiv 1\pmod {p}$ by hypothesis (T).

If Hypothesis (T) holds, then part (c) is an immediate consequence of [Reference CobbeCob18, Lemma 1.9]. Under Hypothesis (I), it follows from the definitions.▪

Proof We write $\nu =\sum _{h,l,k}\nu _{h,l,k}a^hb^l\alpha _k$ for some $\nu _{h,l,k}\in {\mathbb Z}_p$ . Then,

$$ \begin{align*}f_{3,W}\left(\sum_{h,l,k}\nu_{h,l,k}a^hb^le_i\alpha_k w_j\right)=\sum_{h,l,k}\nu_{h,l,k}a^hb^l \cdot E \alpha_k (a-1)^j \theta e_i\in\mathfrak p_N^{j+1}e_i\end{align*} $$

by [Reference Bley and CobbeBC16, Lemma 3.2.5]. Using the isomorphism in (6.12), this is congruent to

$$ \begin{align*}\sum_{h,l,k}\nu_{h,l,k}a^hb^l E \alpha_k (a-1)^j \theta e_i \pmod{\mathfrak p_N^{j+2}}.\\[-48pt]\end{align*} $$

▪

Proof For $j=p$ , $W_{\geq p}=\{0\}$ and $\mathcal F((\mathfrak p_N^{p+1})^{(r)})/\mathcal F((\mathfrak p_N^{p+1})^{(r)})=\{0\}$ , so the result is trivial.

We assume the result for $j+1$ and proceed by descending induction. Let $x\in \mathcal F((\mathfrak p_N^{j+1})^{(r)})$ . As in the proof of [Reference Bley and CobbeBC17, Lemma 7.2.4], there exist $\nu _{h,\ell ,i,k}\in {\mathbb Z}_p$ such that

$$ \begin{align*} \begin{aligned} x&\equiv E\sum_{h,\ell,i,k} \nu_{h,\ell,i,k}a^hb^{\ell}\alpha_k(a-1)^j\theta e_i\pmod{\mathfrak p_N^{j+2}}\\ &\equiv \sum_{h,\ell,i,k}\tilde u^{m\ell} E^{\varphi^{-m\ell}} \nu_{h,\ell,i,k}a^hb^{\ell}\alpha_k(a-1)^j\theta e_i\pmod{\mathfrak p_N^{j+2}}\\ &\equiv \sum_{h,\ell,i,k}\tilde u^{m\ell} \nu_{h,\ell,i,k}a^hb^{\ell} E \alpha_k(a-1)^j\theta e_i\pmod{\mathfrak p_N^{j+2}}\\ &\equiv \tilde f_{3,W}\left(\sum_{h,\ell,i,k}\tilde u^{ml} \nu_{h,\ell,i,k}a^hb^{\ell} e_i\alpha_k w_j\right)\pmod{\mathfrak p_N^{j+2}}. \end{aligned} \end{align*} $$

This means that the class $\pi (x)$ of x in $\mathcal F(\mathfrak p_N^{(r)})/\mathcal F((\mathfrak p_N^{p+1})^{(r)})$ is the sum of an element in the image of $W_{\geq j}$ and an element in $\mathcal F((\mathfrak p_N^{j+2})^{(r)})/\mathcal F((\mathfrak p_N^{p+1})^{(r)})$ , which is by assumption in the image of $W_{\geq j+1}\subseteq W_{\geq j}$ .▪

Proof As in the proof of [Reference Bley and CobbeBC17, Lemma 7.2.5], we see that the formal subtraction of X and Y takes the form

$$ \begin{align*}X-_{\mathcal F} Y=X-Y+(X^tA_hY)_h-(Y^tA_hY)_h+(\deg \ge 3),\end{align*} $$

with $A_h\in M_r({\mathbb Z}_p)$ for $h=1,\dots ,r$ . In analogy to the proof of [Reference Bley and CobbeBC17, Lemma 7.2.5], we set

$$ \begin{align*}x := E\alpha_k (a-1)^ja\theta e_i, \quad y := E\alpha_k(a-1)^j\theta e_i, \quad z := x-y=E\alpha_k(a-1)^{j+1}\theta e_i,\end{align*} $$

and we obtain that $x -_{\mathcal F} y -_{\mathcal F} z \equiv 0\pmod {\mathcal F((\mathfrak p_N^{j+3})^{(r)})}$ . Therefore,

$$ \begin{align*}\tilde f_{3,W}(\alpha_k(a-1) e_i w_j-\alpha_k e_i w_{j+1})\equiv 0\pmod{\mathcal F((\mathfrak p_{N}^{j+3})^{(r)})},\end{align*} $$

and we conclude the proof of the lemma using Lemma 6.11.▪

Proof We denote by $v_j$ the jth component of a vector v. Using [Reference Bley and CobbeBC16, Lemma 3.2.2] and Lemma 6.9(c), we calculate

$$ \begin{align*} \begin{aligned}(\mathcal{N}_{N_0/K_0}(\varepsilon^{-1}E\alpha_k\theta e_i))_j&=\mathcal{N}_{N_0/K_0}(\theta_1){(\varepsilon^{-1}E\alpha_k\theta_2 e_i)_j}^p\\ &\equiv -\alpha_k^p\theta_2^p\alpha_1^{1-p}p{(\varepsilon^{-1}E e_i)_j}^p\pmod{\mathfrak p_{N_0}^{p+1}}\\ &\equiv -\alpha_k^p\theta_2^p\alpha_1^{1-p}p(\varphi(\varepsilon^{-1}E) e_i)_j\pmod{\mathfrak p_{N_0}^{p+1}}\\ &\equiv -\alpha_k^p\theta_2^p\alpha_1^{1-p}p \left(u^{-1}\tilde u\varepsilon^{-1}E e_i\right)_j\pmod{\mathfrak p_{N_0}^{p+1}}. \end{aligned} \end{align*} $$

We also compute

$$ \begin{align*} \mathcal{T}_{N_0/K_0}(\varepsilon^{-1}E\alpha_k\theta e_i)=\mathcal{T}_{N_0/K_0}(\theta_1)\varepsilon^{-1}E\alpha_k\theta_2 e_i\equiv p\alpha_k\theta_2 \varepsilon^{-1}E e_i\pmod{\mathfrak p_{N_0}^{p+1}}. \end{align*} $$

With this in mind, we can do analogous calculations to those in [Reference Bley and CobbeBC16, Lemma 4.2.6] and [Reference Bley and CobbeBC17, Lemma 7.2.6] and obtain

$$ \begin{align*} f_{3,W}(\mathcal T_a e_i \alpha_k w_0)\equiv \varepsilon\left(\alpha_k\theta_2-\alpha_{1}\left(\frac{\alpha_k}{\alpha_1}\right)^p\theta_2^{b^{-\tilde m}}u^{-1}\tilde u\right)p\varepsilon^{-1}E e_i\pmod{\mathfrak p_{N_0}^{p+1}}. \end{align*} $$

Furthermore,

$$ \begin{align*} \begin{aligned} &f_{3,W}(\varepsilon u^{-1}\tilde u^{1-m\tilde m}\varepsilon^{-1}b^{-\tilde m}e_i\alpha_{k+1}w_{p-1})\\ &\qquad\equiv \varepsilon u^{-1}\tilde u^{1-m\tilde m}\varepsilon^{-1}E^{\varphi^{m\tilde m}}\alpha_{k+1}p\theta_2^{b^{-\tilde m}} e_i\pmod{\mathfrak p_{N_0}^{p+1}}\\ &\qquad\equiv \varepsilon u^{-1}\tilde u^{1-m\tilde m}u^{m\tilde m}(\varepsilon^{-1}E)^{\varphi^{m\tilde m}}\alpha_{k+1}p\theta_2^{b^{-\tilde m}} e_i\pmod{\mathfrak p_{N_0}^{p+1}}\\ &\qquad\equiv \varepsilon u^{-1}\tilde u \varepsilon^{-1}E\alpha_{k+1}p\theta_2^{b^{-\tilde m}} e_i\pmod{\mathfrak p_{N_0}^{p+1}}. \end{aligned} \end{align*} $$

It is straightforward to adapt the remaining calculations from [Reference Bley and CobbeBC17, Lemma 7.2.6] and conclude that $r_{i,k}\in \ker f_{3,W}$ for $1\leq i\leq r$ and $1< k<m$ . The proof that $r_{i,m}\in \ker f_{3,W}$ is analogous.▪

Proof We adapt the proof of [Reference Bley and CobbeBC17, Lemmas 7.3.1]. We write the coefficients of the $e_i\alpha _kw_{p-1}$ -components, $i=1,\dots ,r$ , $k=1,\dots ,m$ , of the elements $r_{i,j}$ , $j=1,\dots ,m$ , into the columns of an $mr \times mr$ matrix, which we call ${\mathcal M}$ , and whose entries are $r\times r$ blocks,

$$ \begin{align*}{\mathcal M}=\begin{pmatrix} \varepsilon u^{-1}\varepsilon^{-1} b^{-\tilde m}\!-\!1\!\!\!\!\!\!&0&\cdots&0&0&\varepsilon u^{-1}\varepsilon^{-1} b^{-\tilde m}\\ 0&-1&\cdots&0&0&-\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}\\ 0&\!\!\!\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}\!\!\!&\cdots&0&0&-\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\!\!\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}\!\!\!\!\!\!\!&-1&-\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}\\ 0&0&\cdots&0&\varepsilon u^{-1}\varepsilon^{-1}b^{-\tilde m}&-1\!-\!\varepsilon u^{-1}\varepsilon^{-1} b^{-\tilde m} \end{pmatrix}\!. \end{align*} $$

By Lemma 4.5 and analogous computations as in [Reference Bley and CobbeBC17], we obtain

$$ \begin{align*} \det{\mathcal M}=(-1)^{r(m-1)}\det(u^{-m}b^{-1}-1). \end{align*} $$

The rest of the proof works exactly as in [Reference Bley and CobbeBC17]; note that Hypothesis (I) plays the role of the assumption $\omega =0$ in the one-dimensional setting.▪

Proof We choose the elements $r_{i,k}$ and $s_{i,j,k}$ of Lemma 6.14 as a ${{\mathbb Z}_p[G]}$ -basis of $\ker (f_{3,W})$ and fix the canonical ${{\mathbb Z}_p[G]}$ -basis of W. Then,

$$ \begin{align*} \chi_{{{\mathbb Z}_p[G]}, {B_{\mathrm{dR}}}[G]}(M^{\bullet}(\mathcal{L}), 0) = [\ker(f_{3,W}), \mathrm{id}, W] \end{align*} $$

is represented by the determinant of

$$ \begin{align*} {\mathfrak M}=\begin{pmatrix} \mathcal T_aI&(a-1)I&0&\cdots&0&0\\ 0&-I&(a-1)I&\cdots&0&0\\ 0&*&-I&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&*&*&\cdots&-I&(a-1)I\\ {\mathcal M}&*&*&\cdots&*&-I \end{pmatrix}, \end{align*} $$

where in the above $pmr\times pmr$ all the entries are $mr\times mr$ blocks. Recalling that p is odd, we get:

$$ \begin{align*} \begin{aligned} \det(\chi\phi(\mathfrak M))&=\begin{cases}p^{mr}(-1)^{mr(p-1)}&\text{if } \chi=\chi_0\\ (-1)^{m^2r^2(p-1)}\det(\chi\phi({\mathcal M}))(\chi(a)-1)^{mr(p-1)}&\text{if } \chi\neq\chi_0\end{cases}\\ &=\begin{cases}p^{mr}&\text{if } \chi=\chi_0\\ (-1)^{r(m-1)}\det(u^{-m}\phi(b)^{-1}-1)(\chi(a)-1)^{mr(p-1)}&\text{if } \chi\neq\chi_0.\end{cases} \end{aligned} \end{align*} $$

▪

Proof We recall from [Reference CobbeCob18, Section 2] that the action of $\mathrm {Gal}(K_0/K) \times G$ on $\mathcal {I}_{N/K}(\rho ^{\mathrm {nr}})$ is characterized by

$$ \begin{align*} (F \times 1) [x_1, \ldots, x_d] &= [U_Nx_d^{F_N}, x_2, \ldots, x_{d-1}], \\ (F^{-n} \times \sigma) [x_1, \ldots, x_d] &= [\rho^{\mathrm{nr}}(\tilde\sigma) x_1^{\tilde\sigma}, \ldots, \rho^{\mathrm{nr}}(\tilde\sigma) x_d^{\tilde\sigma}], \end{align*} $$

where $x_i \in \prod _r \widehat {N_0^{\times }}$ , the elements $F^{-n}$ and $\sigma \in G$ have the same restriction to $N \cap K_0$ , and $\tilde \sigma \in \mathrm {Gal}(N_0/K)$ is uniquely defined by $\tilde \sigma |_{K_0} = F^{-n}$ and $\tilde \sigma |_{N} = \sigma $ . Furthermore, we remark that the action of G on $\mathcal {Z} / (U_N - 1)\mathcal {Z}$ is induced by

$$ \begin{align*} a \cdot e_i = e_i, \quad b \cdot e_i = \rho^{\mathrm{nr}}(F^{-1}) e_i = U_K^{-1}e_i. \end{align*} $$

The bottom sequence is exact by [Reference CobbeCob18, Theorem 3.3 and Lemma 2.1]. The proofs of the exactness of the top sequence as well as the proof of commutativity follow along the lines of proof in the one-dimensional case (see [Reference Bley and CobbeBC17, Lemma 7.4.1]). For example, if we denote the coefficients of $U_K = u^m$ by $u_{ij}$ , then

$$ \begin{align*} \tilde f_2\circ\delta_2(e_iz_1)&=\tilde f_2((u^mb-1)e_i)=\tilde f_2\left(\sum_{j=1}^r bu_{j,i}e_j-e_i\right)\\ &=\sum_{j=1}^r (1\times b)u_{j,i}\cdot [(\theta_1)_j,1,\dots,1]-[(\theta_1)_i,1,\dots,1] \\ &=(1\times b)\cdot [(\theta_1^{u_{1,i}},\theta_1^{u_{2,i}},\dots,\theta_1^{u_{r,i}}),1,\dots,1]-[(\theta_1)_i,1,\dots,1]\\ &=(1\times b)[u(\theta_1)_i,1,\dots,1]-[(\theta_1)_i,1,\dots,1]\\ &=(F\times 1)(F^{-1}\times b)[u(\theta_1)_i,1,\dots,1]-[(\theta_1)_i,1,\dots,1]\\ &=(F\times 1)[u^{-1}u(\theta_1)_i,1,\dots,1]-[(\theta_1)_i,1,\dots,1]\\ &=((F-1)\times 1)\circ \tilde f_3(e_iz_1).\\ \end{align*} $$

▪

Proof The proof is just the r-dimensional analogue of that of [Reference Bley and CobbeBC17, Lemma 7.4.3].▪

Proof If we recall the definition of $M^{\bullet }(\mathcal {L})$ from (6.8), then it follows readily from Lemmas 6.11 and 6.16 that we have a quasi-isomorphism of complexes

$$ \begin{align*} F^{\bullet} \longrightarrow M^{\bullet}(\mathcal{L}).\\[-34pt] \end{align*} $$

▪

Proof Let $x_2\in \mathcal {O}_{K^{\mathrm {nr}}}$ be such that $x_2/\alpha _1\pmod {\mathfrak p_{K'}}$ is a root of $X^p-X+A\theta _2$ , and let $(1+x_2\theta _1)_i$ be the element in $\prod _r U_{N_0}^{(1)}$ whose ith component is $1+x_2\theta _1$ and all the other components are $1$ . By the same proof as in [Reference Bley and CobbeBC17, Lemma 7.2.2], we obtain

$$ \begin{align*}(1+x_2\theta_1)_i^{\rho^{\mathrm{nr}}(F_N)F_N-1}\equiv (1-\alpha_1\theta_1)_i\pmod{\mathfrak p_{N_0}^2}.\end{align*} $$

Therefore, by [Reference CobbeCob18, Lemma 2.4], we may assume $\gamma _i\equiv (1+x_2\theta _1)_i\pmod {\mathfrak p_{N_0}^2}$ .

With this in mind, the proof of the lemma is the same as in [Reference Bley and CobbeBC17, Lemma 7.4.5] and [Reference Bley and CobbeBC17, Lemma 7.4.6].▪

Proof Straightforward by the same calculations as in [Reference Bley and CobbeBC16, Lemma 4.2.5].▪

Proof It is enough to follow the proof of [Reference Bley and CobbeBC17, Lemma 7.4.9]. Note that we can construct the matrix ${\mathcal M}$ as in [Reference Bley and CobbeBC17] and that it also takes exactly the same shape up to the fact that all the entries are $r\times r$ blocks.▪

Proof The proof is a straightforward adaption of the computations in the proof of [Reference Bley and CobbeBC17, Proposition 7.4.10].▪