The formula of Theorem 4.6 becomes, through (4.36) and Proposition 4.10,

$$2\lambda_{\text{fake}} = \lambda_{\text{Iw}} + (d - \lambda_{\Pi}^{-}) - (2d - \lambda_{\Pi} - 1) = \lambda_{\text{Iw}} - d + \lambda_{\Pi}^{+} + 1.$$

Since $\lambda_{\Pi}^{+} \ge 0$, the estimate (4.34) follows.

Assume the Gross and the Leopoldt conjectures for $F$ and that $F/\mathbb{Q}$ is Galois: the estimate in case (c) of Proposition 4.12 reads $\lambda(0, \mathcal{O}_{L_{\bullet}}^{\times})^{+} = d-1$, which yields $\lambda_{\Pi} = d$ through (4.36). We now proceed to prove that $\lambda_{\Pi}^{+} \ge 1$: this will imply both estimates $\lambda_{\Pi}^{-} \le d-1$ and (4.35). The proof goes along the same lines as those of Corollary 4.14. Denote by $\mathcal{G}$ the Galois group $\text{Gal}(L_{\infty}/\mathbb{Q})$ and set $\tilde{L} = L_{\infty}L_{\text{cyc}}$, where $L_{\text{cyc}}$ is the cyclotomic $\mathbb{Z}_p$-extension of $F$. Since $F/\mathbb{Q}$ is Galois, so is $L_{\text{cyc}}/\mathbb{Q}$ and the Galois group $\text{Gal}(\tilde{L}/\mathbb{Q})$ is the direct product $\mathcal{G} \times \text{Gal}(L_{\text{cyc}}/F)$. Write $p\mathcal{O}_k = \overline{q_1} \cdots \overline{q_d}$ and $\overline{q}_i\mathcal{O}_F = q_iq_{d+i}$: each prime $q_j$ is divided by the same number of primes in $L_{\infty}/F$, so let $q_j\mathcal{O}_{L_{\infty}} = \mathfrak{Q}_j^{(1)} \cdots \mathfrak{Q}_j^{(s)}$ be its factorisation in $L_{\infty}$, for $1 \le j \le 2d$. The localisations $F_{q_j}$ are all isomorphic to $\mathbb{Q}_p$, and hence admit no totally ramified $\mathbb{Z}_p^2$-extension, showing that $\mathfrak{Q} = \mathfrak{Q}_1^{(1)}$ is unramified in $\tilde{L}/L_{\infty}$ and so Lemma 4.13 guarantees that $\tilde{L} \subseteq M_{\infty}$, where $M_{\infty}$ is the maximal $p$-extension of $L_{\infty}$ which is unramified everywhere. The diagram of fields is as follows:

<figure><img src="image_26.png" /></figure>

The group $\mathcal{G}$ permutes the ideals $\mathfrak{Q}_j^{(i)}$ and acts trivially on $\text{Gal}(\tilde{L}/L_{\infty})$, so the equivalent of (4.33) reads (4.37)

$$\prod_{i,j} \text{Frob}(\mathfrak{Q}_j^{(i)}, M_{\infty}/L_{\infty}) \pmod{H} = \prod_{\tau \in \text{Gal}(L_{n_{\text{tr}}}/\mathbb{Q})} \text{Frob}(\tau \mathfrak{Q}, \tilde{L}/L_{\infty}) = \text{Frob}(\mathfrak{Q}, \tilde{L}/L_{\infty})^{2ds} \in Y$$

where, as *ibid.*, $L_{n_{\text{tr}}}$ is such that $L_{\infty}/L_{n_{\text{tr}}}$ is totally ramified at all primes above $p$. The element

$$x = \prod_{i,j} \text{Frob}(\mathfrak{Q}_j^{(i)}, M_{\infty}/L_{\infty}) \in X$$

has infinite order by Lemma 4.13 and corresponds, via the Artin map of global class field theory, to an element in $\lim_{\leftarrow \leftarrow \Pi_n^+}$. This shows $\lambda_{\Pi^+} \ge 1$, finishing the proof. $\square$

*Remark 4.19.* In the proof of (4.34), the hypothesis that $L_{\infty}/\mathbb{Q}$ is Galois is only needed to ensure that all primes in $F$ above $p$ ramify completely in $L_{\infty}/\mathbb{Q}$, in order to apply Proposition 4.12-(c). In particular, the inequality holds also under this weaker assumption.
One condition that guarantees that all assumptions of Corollary 4.18 are satisfied is that the group Gal($F/Q$) is abelian with exponent divisible by ($p-1$): this is shown in [CK82, Theorem 1], together with the result that both the Gross and the Leopoldt conjectures hold for abelian number fields.

**4.4. The structure of the limits of class groups.** As in the previous section, fix a fake $Z_p$-extension of dihedral type $K_\infty/k$ and adopt the conventions introduced in §4.1. Let $B = (B_{L_n}) \in \{O_{L_\bullet}^\times, U_{L_\bullet}, Q_{L_\bullet}, C_{L_\bullet}, A_{L_\bullet}\}$ be any of the five normic systems occurring either in the left column or in the bottom row of (3.1). As in §3.2, set $B_{L_\infty} = \lim_{\to} B_{L_n}$ (resp. $B_{K_\infty} = \lim_{\to} B_{K_n}$, letting $B_{K_n} = B_{L_n}^+$) where the limit is taken with respect to $\iota_{L_m/L_n}$ (resp. with respect to $\iota_{K_m/K_n}$).

In this section we make no use of double systems: only direct and inverse systems will appear. Indeed, the focus will be on the structure of direct and inverse limits, which are coarser invariants. Moreover, one easily realises that the direct system of class groups with ideals extension as morphisms satisfies neither (Inj) nor (Gal), and the assumptions of Proposition 2.11 are not fulfilled, so its $G_n$-cohomology does not give rise to a double system fitting in a cohomological long exact sequence. In fact, the analysis of the direct and inverse limits of class groups relies on our previous work [CN20], and we need to consider morphisms between 0th Tate cohomology groups which are different from the map $\hat{H}^0(\iota_{L_m/L_n})$ defined in §2.2.2, because of the factor $p^{m-n}$ appearing there. To define these morphisms, observe that, for $m \ge n \gg 0$, $N_{L_m/L_n}: A_{L_m} \to A_{L_n}$ is surjective and so, by the arguments of Remark 2.8, we have maps

$$ \hat{\iota}_{L_m/L_n}: \hat{H}^0(G_n, A_{L_n}) \to \hat{H}^0(G_m, A_{L_m}) \quad \text{and} \quad \hat{\iota}_{L_m/L_n}^+ : \hat{H}^0(D_n, A_{L_n}) \to \hat{H}^0(D_m, A_{L_m}). $$

We can therefore take direct limits with respect to these maps, and define the groups$^6$

$$ \hat{H}^0(\Gamma, A_{L_\infty}) := \lim_{\underset{\hat{\iota}_{L_m/L_n}}{\longrightarrow}} \hat{H}^0(G_n, A_{L_n}) \quad \text{and} \quad \hat{H}^0(\mathcal{D}, A_{L_\infty}) := \lim_{\underset{\hat{\iota}_{L_m/L_n}^+}{\longrightarrow}} \hat{H}^0(D_n, A_{L_n}). $$

Being direct limits of finite $D_0$-modules, these are $Z_p$-torsion $Z_p[D_0]$-modules; moreover, via Proposition 4.2, they satisfy $\hat{H}^0(\Gamma, A_{L_\infty})^+ \cong \hat{H}^0(\mathcal{D}, A_{L_\infty})$, extending the isomorphisms *ibid.* for $i \ge 1$. More generally, in this section, we tacitly identify plus parts of all $G_n$-cohomology groups (resp. of all $\Gamma$-cohomology groups) with $D_n$-cohomology (resp. $\mathcal{D}$-cohomology).

**4.4.1. An Euler-Poincaré formula for fake $Z_p$-extensions of dihedral type.** The main result of this section is Theorem 4.22, which can be seen as a relationship between dihedral versions of Herbrand quotients of class groups and of units, at least in the limit. This theorem will also be pivotal in the next section to interpret the invariant $\lambda_{\text{fake}}$ as the dimension of a $\mathbb{Q}_p$-vector space.

We begin by relating $H^1(\mathcal{D}, A_{L_\infty})$ to $H^2(\mathcal{D}, Q_{L_\infty})$ on the one hand, and $\hat{H}^0(\mathcal{D}, A_{L_\infty})$ to $H^1(\mathcal{D}, Q_{L_\infty})$ on the other.

**Proposition 4.20.** There is an isomorphism of $Z_p$-modules

$$ \hat{H}^0(\mathcal{D}, A_{L_\infty}) \cong H^1(\mathcal{D}, Q_{L_\infty}) $$

and an exact sequence

$$ 0 \to H^1(\mathcal{D}, A_{L_\infty}) \to H^2(\mathcal{D}, Q_{L_\infty}) \to H^2(\mathcal{D}, C_{L_\infty}) \to 0 $$

in the category $\text{Mod}_{\mathbb{Z}_p}^{\text{co-f.g.}}/\text{Mod}_{\mathbb{Z}_p}^{\text{fin}}$. Moreover, $H^2(\mathcal{D}, C_{L_\infty}) \cong \mathbb{Q}_p/\mathbb{Z}_p$.

<footnote>$^6$In [NSW08, Definition I.1.9.3], the notation $\hat{H}^0(\mathcal{G}, M)$ denotes a different construction, which we do not need, hence we allow ourselves this inconsistency.</footnote>
**Proof.** By Proposition 2.10, we have a commutative diagram

$$ (4.38) \qquad \begin{array}{ccc} H^0(D_m, A_{L_m}) & \xrightarrow{\delta^{(0)}} & \hat{H}^1(D_m, Q_{L_m}) \\ \bigwedge_{\iota_{L_m/L_n}} & & \bigwedge \hat{H}^1(\iota_{L_m/L_n}) \\ H^0(D_n, A_{L_n}) & \xrightarrow{\delta^{(0)}} & \hat{H}^1(D_n, Q_{L_n}) \end{array} $$

The definition of $\hat{H}^1(\iota_{L_m/L_n})$ given in Remark 2.8 makes it clear that (4.38) induces the commutative diagram

$$ \begin{array}{ccc} \hat{H}^0(D_m, A_{L_m}) & \xrightarrow{\hat{\delta}^{(0)}} & \hat{H}^1(D_m, Q_{L_m}) \\ \bigwedge_{\hat{\iota}_{L_m/L_n}} & & \bigwedge \hat{H}^1(\hat{\iota}_{L_m/L_n}) \\ \hat{H}^0(D_n, A_{L_n}) & \xrightarrow{\hat{\delta}^{(0)}} & \hat{H}^1(D_n, Q_{L_n}) \end{array} $$

From [CN20, Proposition 3.4], the rows of the above diagram are isomorphisms, and so by taking limits we obtain the first statement of the proposition.

Arguing as in the proof of [CN20, Proposition 3.4], we get an exact sequence

$$ 0 \to \hat{H}^1(G_n, A_{L_n}) \to \hat{H}^2(G_n, Q_{L_n}) \to \hat{H}^2(G_n, C_{L_n}) \to \hat{H}^2(G_n, A_{L_n}) \to \hat{H}^3(G_n, Q_{L_n}) \to 0 $$
with its plus counterpart

$$ 0 \to \hat{H}^1(D_n, A_{L_n}) \to \hat{H}^2(D_n, Q_{L_n}) \to \hat{H}^2(D_n, C_{L_n}) \to \hat{H}^2(D_n, A_{L_n}) \to \hat{H}^3(D_n, Q_{L_n}) \to 0. $$
By Proposition 4.2, this exact sequence can be shortened as

$$ (4.39) \qquad 0 \to \hat{H}^1(D_n, A_{L_n}) \to \hat{H}^2(D_n, Q_{L_n}) \to \hat{H}^2(D_n, C_{L_n}) \to \mathrm{Ker}(\hat{\delta}_n^{(0)})^- \to 0 $$

where $\hat{\delta}_n^{(0)}: \hat{H}^0(G_n, A_{L_n}) \to \hat{H}^1(G_n, Q_{L_n})$ is the morphism analysed in Remark 4.11. As shown ibid., $\mathrm{Ker}(\hat{\delta}_n^{(0)})$ is bounded independently of $n$, and taking direct limits of (4.39) we obtain the exact sequence of the statement.

Concerning the final isomorphism, an argument similar to the one used to prove the commutativity of the left-hand diagram of Lemma 3.5 shows that the direct system $(\hat{H}^2(D_n, C_{L_n}), \hat{H}^2(\iota_{L_m/L_n}))$ is isomorphic to $(\hat{H}^0(D_n, \mathbb{Z}_p), \hat{H}^0(j_{n,m})) \cong (\frac{1}{p^n}\mathbb{Z}_p/\mathbb{Z}_p, \mathrm{incl})$ (where incl denotes the inclusion $\frac{1}{p^n}\mathbb{Z} \hookrightarrow \frac{1}{p^m}\mathbb{Z}$), finishing the proof. $\square$

**Remark 4.21.** The proof of the above proposition actually shows that there is an exact sequence

$$ 0 \to H^1(\mathcal{D}, A_{L_\infty}) \to H^2(\mathcal{D}, Q_{L_\infty}) \to H^2(\mathcal{D}, C_{L_\infty}) \to V \to 0 $$

in the category $\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{co-f.g.}}$ with $V$ in $\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{fin}}$. Since we are only interested in comparing coranks of divisible $\mathbb{Z}_p$-modules, here and in what follows we content ourselves to work in $\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{co-f.g.}}/\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{fin}}$.

We can now proceed to the proof of the main result of this section, that we regard as an Euler-Poincaré formula:

**Theorem 4.22.** Given a fake $\mathbb{Z}_p$-extension of dihedral type $K_\infty/k$ with normal closure $L_\infty/k$, we have

$$ \mathrm{cork} \, \hat{H}^0(\mathcal{D}, A_{L_\infty}) - \mathrm{cork} \, H^1(\mathcal{D}, A_{L_\infty}) = \mathrm{cork} \, H^2(\mathcal{D}, \mathcal{O}_{L_\infty}^\times) - \mathrm{cork} \, H^1(\mathcal{D}, \mathcal{O}_{L_\infty}^\times) + 1. $$
**Proof.** Taking direct limits in (3.28) we obtain the exact sequence in $\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{co-f.g.}}/\mathrm{Mod}_{\mathbb{Z}_p}^{\mathrm{fin}}$

$$0 \dashrightarrow H^1(\Gamma, \mathcal{O}_{L_\infty}^\times) \dashrightarrow H^1(\Gamma, \mathcal{U}_{L_\infty}) \dashrightarrow H^1(\Gamma, Q_{L_\infty}) \dashrightarrow H^2(\Gamma, \mathcal{O}_{L_\infty}^\times) \xrightarrow{\bar{\nu}_\infty} H^2(\Gamma, \mathcal{U}_{L_\infty}).$$

By construction, the rightmost morphism $\bar{\nu}_\infty: H^2(\Gamma, \mathcal{O}_{L_\infty}^\times) \dashrightarrow H^2(\Gamma, \mathcal{U}_{L_\infty})$ is the image, in the quotient category, of the direct limit of the morphisms $H^2(G_n, \mathcal{O}_{L_n}^\times) \to H^2(G_n, \mathcal{U}_{L_n})$. It follows that we can complete the above sequence as

$$0 \dashrightarrow H^1(\Gamma, \mathcal{O}_{L_\infty}^\times) \dashrightarrow H^1(\Gamma, \mathcal{U}_{L_\infty}) \dashrightarrow H^1(\Gamma, Q_{L_\infty}) \dashrightarrow H^2(\Gamma, \mathcal{O}_{L_\infty}^\times) \dashrightarrow H^2(\Gamma, \mathcal{U}_{L_\infty}) \dashrightarrow H^2(\Gamma, Q_{L_\infty}) \dashrightarrow 0$$

where the final 0 is a consequence of $\Gamma$ having strict cohomological dimension equal to 2. By taking plus parts we get, through Corollary 4.4-(4.3),

$$\begin{align*}
\mathrm{cork} H^1(\mathcal{D}, Q_{L_\infty}) - \mathrm{cork} H^2(\mathcal{D}, Q_{L_\infty}) &= \mathrm{cork} H^1(\mathcal{D}, \mathcal{U}_{L_\infty}) - \mathrm{cork} H^2(\mathcal{D}, \mathcal{U}_{L_\infty}) + \mathrm{cork} H^2(\mathcal{D}, \mathcal{O}_{L_\infty}^\times) \\
&\qquad - \mathrm{cork} H^1(\mathcal{D}, \mathcal{O}_{L_\infty}^\times) \\
&= \mathrm{cork} H^2(\mathcal{D}, \mathcal{O}_{L_\infty}^\times) - \mathrm{cork} H^1(\mathcal{D}, \mathcal{O}_{L_\infty}^\times).
\end{align*}$$

Moreover, by Proposition 4.20,

$$\mathrm{cork} H^2(\mathcal{D}, Q_{L_\infty}) = \mathrm{cork} H^1(\mathcal{D}, A_{L_\infty}) + 1$$

and

$$\mathrm{cork} H^1(\mathcal{D}, Q_{L_\infty}) = \mathrm{cork} \hat{H}^0(\mathcal{D}, A_{L_\infty}),$$

finishing the proof.

□

**4.4.2. Interpretation of $\lambda_{\mathrm{fake}}$.** Let $\Lambda = \mathbb{Z}_p[\Gamma]$ denote the Iwasawa algebra of the group $\Gamma = \mathrm{Gal}(L_\infty/F)$. In this section we show that the invariant $\lambda_{\mathrm{fake}}$ obtained in Theorem 4.6 is also the dimension of the $\mathbb{Q}_p$-vector space $X_{\mathrm{fake}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$, where

$$X_{\mathrm{fake}} = \varinjlim A_{K_n},$$

the inverse limit being taken with respect to norms. The analogous result for $X_{\mathrm{Iw}}$ (with $\lambda_{\mathrm{Iw}}$ replacing $\lambda_{\mathrm{fake}}$) is well-known and comes from the structure theorem for finitely generated torsion $\Lambda$-modules together with a classical descent argument (see [Iwa73, §1.2 and Theorem 7]). Since $X_{\mathrm{fake}}$ is not stabilized by $\Gamma$, and hence it is not a $\Lambda$-submodule of $X_{\mathrm{Iw}}$, we argue in a different way. Our approach is to work with direct, rather than inverse, limits because they are easier to analyse in our setting. We then recover the result about inverse limits via a purely algebraic argument due to Yamashita.

Tensoring the exact sequence [CN20, (3.1)] with $\mathbb{Z}_p$, we obtain

$$(4.40) \qquad 1 \to \mathrm{Ker} \eta_n \to A_{K_n} \oplus A_{K'_n} \xrightarrow{\eta_n} A_{L_n} \to \mathrm{Coker} \eta_n \to 1$$

where the map $\eta_n$ is defined via the formula

$$\eta_n(c, c') = \iota_{L_n/K_n}(c) \iota_{L_n/K'_n}(c') \quad \text{for } c \in A_{K_n}, c' \in A_{K'_n}.$$

The explicit definition of $\eta_n$ makes it evident that, for $m \ge n$, the relations

$$(4.41) \qquad \eta_m \circ (\iota_{K_m/K_n} \oplus \iota_{K'_m/K'_n}) = \iota_{L_m/L_n} \circ \eta_n$$

hold. We can therefore take direct limits with respect to the extension maps to obtain an exact sequence

$$(4.42) \qquad 1 \to \mathrm{Ker} \eta_\infty \to A_{K_\infty} \oplus A_{K'_\infty} \xrightarrow{\eta_\infty} A_{L_\infty} \to \mathrm{Coker} \eta_\infty \to 1$$

where $\eta_\infty = \varinjlim \eta_n$.
By [CN20, (2.7)] and by definition of $\hat{H}^{-1}(\iota_{L_m/L_n})$ (see §2.2.1), this can be rewritten as

<figure><img src="image_27.png" /></figure>

Moreover, the cup product with $\kappa_n$ induces an isomorphism $\hat{H}^{-1}(G_n, A_{L_n})^{-} \cong \hat{H}^{1}(D_n, A_{L_n})$, and by definition of $\hat{H}^{1}(\iota_{L_m/L_n})$, together with Lemma 2.7, the diagram becomes

<figure><img src="image_28.png" /></figure>

The groups on the right have bounded order since $|Nm_{G_n}A_{L_n}| \le |A_F|$, by the same argument as at the end of the proof of Lemma 4.23. Taking direct limits we obtain the statement. $\square$

Combined with Theorem 4.22, the above lemmas yield the following

**Corollary 4.25.** We have

$$2 \operatorname{cork} A_{K_{\infty}} - \operatorname{cork} A_{L_{\infty}} = \operatorname{cork} \operatorname{Ker} \eta_{\infty} - \operatorname{cork} \operatorname{Coker} \eta_{\infty} = \lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} + 1$$

where $\lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+}$ and $\lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+}$ are the invariants defined in Corollary 4.4-ii).

**Proof.** From the exact sequence (4.42), we obtain

$$2 \operatorname{cork} A_{K_{\infty}} - \operatorname{cork} A_{L_{\infty}} = \operatorname{cork} \operatorname{Ker} \eta_{\infty} - \operatorname{cork} \operatorname{Coker} \eta_{\infty}$$

$$(\text{Lemmas 4.23 and 4.24}) = \operatorname{cork} \hat{H}^0(\mathcal{D}, A_{L_{\infty}}) - \operatorname{cork} H^1(\mathcal{D}, A_{L_{\infty}})$$

$$(\text{Theorem 4.22}) = \operatorname{cork} H^2(\mathcal{D}, \mathcal{O}_{L_{\infty}}^{\times}) - \operatorname{cork} H^1(\mathcal{D}, \mathcal{O}_{L_{\infty}}^{\times}) + 1$$

$$(\text{Corollary 2.32 and Corollary 4.4-ii}) = \lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} + 1. \quad \square$$

We are now in shape to relate the invariant $\lambda_{\text{fake}}$ to the $\mathbb{Z}_p$-module structure of $X_{\text{fake}}$, but we first need a final lemma:

**Lemma 4.26.** The equalities

$$\operatorname{cork} A_{L_{\infty}} = \dim_{\mathbb{Q}_p} X_{Iw} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p \quad \text{and} \quad \operatorname{cork} A_{K_{\infty}} = \dim_{\mathbb{Q}_p} X_{\text{fake}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$$

hold.

**Proof.** The first equality can be deduced from [Iwa73, Theorem 11]. Here we give an alternative proof, following that of [Yam84, Lemma 2], which also yields the second equality.

Since $X_{Iw}$ is a finitely generated torsion $\Lambda$-module, we know, by the structure theorem of such modules, that there exists $\beta \in \mathbb{N}$ such that $p^{\beta}X_{Iw}$ is a finitely generated torsion $\Lambda$-module with trivial $\mu$-invariant. Since $p^{\beta}X_{Iw} = \lim_{\leftarrow\leftarrow} p^{\beta}A_{L_n}$, we deduce that the $\mathbb{Z}_p$-ranks of $p^{\beta}A_{L_n}$ are bounded and thus so are those of $p^{\beta}A_{K_n}$. As arithmetic norms are eventually surjective on $A_{L_n}$, their restriction to $p^{\beta}A_{L_n}$ is also eventually surjective, and likewise when restricting to $p^{\beta}A_{K_n}$. Similarly,
the kernel of the extension map restricted to $p^\beta A_{L_n}$ has bounded order and, again, the same holds when restricting to $p^\beta A_{K_n}$. We can therefore apply [Yam84, Lemma 1] to obtain that

$$ \mathrm{cork}(p^\beta A_{L_\infty}) = \dim_{\mathbb{Z}_p}(p^\beta X_{Iw} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p) \quad \text{and} \quad \mathrm{cork}(p^\beta A_{K_\infty}) = \dim_{\mathbb{Z}_p}(p^\beta X_{\mathrm{fake}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p). $$

This concludes the proof because, for every $\mathbb{Z}_p$-module $M$, $\dim_{\mathbb{Z}_p}(p^\beta M \otimes_{\mathbb{Z}_p} \mathbb{Q}_p) = \dim_{\mathbb{Z}_p}(M \otimes_{\mathbb{Z}_p} \mathbb{Q}_p)$ and $\mathrm{cork}(p^\beta M) = \mathrm{cork} M$. $\square$

Finally, we obtain the main result of this section.

**Theorem 4.27.** Given a fake $\mathbb{Z}_p$-extension of dihedral type $K_\infty/k$, we have $\lambda_{\mathrm{fake}} = \dim_{\mathbb{Z}_p} X_{\mathrm{fake}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$.

**Proof.** By the structure theorem of finitely generated $\Lambda$-modules, we know that

$$ (4.44) \qquad \dim_{\mathbb{Z}_p} X_{Iw} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p = \lambda_{Iw}. $$

Now, using Theorem 4.6, we deduce

$$ \begin{align*} 2\lambda_{\mathrm{fake}} &= \lambda_{Iw} + \lambda(-1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(0, \mathcal{O}_{L_{\bullet}}^{\times})^{+} \\ \text{(Corollary 4.4-(9) and -(4.4))} &= \lambda_{Iw} + \lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} + 1 \\ \text{(by (4.44))} &= \dim_{\mathbb{Z}_p} X_{Iw} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p + \lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} + 1 \\ \text{(Lemma 4.26)} &= \mathrm{cork} \, A_{L_{\infty}} + \lambda(2, \mathcal{O}_{L_{\bullet}}^{\times})^{+} - \lambda(1, \mathcal{O}_{L_{\bullet}}^{\times})^{+} + 1 \\ \text{(Corollary 4.25)} &= 2 \mathrm{cork} \, A_{K_{\infty}} \\ \text{(Lemma 4.26)} &= 2 \dim_{\mathbb{Z}_p} (X_{\mathrm{fake}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p). \end{align*} \square $$

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UNIVERSITÉ JEAN MONNET SAINT-ÉTIENNE, CNRS UMR 5208, INSTITUT CAMILLE JORDAN, F-42023 SAINT-ÉTIENNE, FRANCE