# Generalised class groups in dihedral and fake Z_p-extensions

Filippo Alberto Edoardo Nuccio Mortarino Majno di Capriglio

► To cite this version:

Filippo Alberto Edoardo Nuccio Mortarino Majno di Capriglio. Generalised class groups in dihedral and fake Z_p-extensions. Mathematica Slovaca, 2025, 75 (3), pp.537-554. 10.1515/ms-2025-0039. ujm-0470211v4

HAL Id: ujm-0470211

https://ujm.hal.science/ujm-0470211v4

Submitted on 7 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
GENERALISED CLASS GROUPS IN DIHEDRAL AND FAKE
$Z_p$-EXTENSIONS

FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO

ABSTRACT. We study generalised class groups in dihedral extensions of number fields, focusing both on its *p*-adic and *ℓ*-adic properties along fake $Z_p$-extensions, where *p* and *ℓ* are distinct odd primes. Our results generalise previous works in the (pro)-dihedral setting, with applications to the study of more general fake Iwasawa theories beyond the classical case of standard class groups.

# 1. INTRODUCTION

Dihedral extensions of number fields have long been a key object of interest, not only because they represent the “first step” beyond abelian extensions, whose arithmetic is controlled by class field theory, but also in light of the role that they play in the theory of complex multiplication: for a very explicit example of the connection between class field theory for imaginary quadratic fields and dihedral Galois representations arising from CM modular forms, a reader might consult [BN18, Théorème 1.1]. It is therefore natural to extend to the dihedral setting the analysis of “generalised class groups” that is normally performed in the abelian case through class field theory.

The aim of Section 2 is to establish a formula relating the orders of the *S*-class groups in a dihedral extension, generalising the main result of [CN20] (see Theorem 2.12 below). Although some arguments can be borrowed from *ibid.*, one needs to take into accounts that class field theory for the regular class group and for the *S*-class group $Cl_S$ are different, because the local cohomology of the units at the ramified places that lie in *S* is different in the two cases. Beyond analysing these local contributions in detail, we also devise a general strategy different from the one in [CN20] that allows us to replace global cohomology arguments with local ones altogether. Beside its intrinsic interest, this generalisation is motivated by applications to Iwasawa theory. Indeed, studying Iwasawa-theoretic properties in geometric settings (as for elliptic curves or, more generally, modular forms), typically requires to focus on Galois groups with restricted ramification: this is discussed in detail in by Coates–Sujatha in [CS10], and specific examples where this is needed are in the definition of the fine Selmer groups in [NS23, Theorem 4.20] or in [NOR23, Theorem 4.3].

In Section 3, we consider the setting of a fake $Z_p$-extension, as first defined in [CN23] (see Definition 3.1 below for the precise definition). Prototypical examples of fake $Z_p$-extensions are provided by anticyclotomic $Z_p$-extensions: given an imaginary quadratic field $F$, the anticyclotomic $Z_p$-extensions $L_\infty/F$ is the unique $Z_p$-extension of $F$ that is Galois but non abelian over $\mathbb{Q}$, and all its layers $L_n$ are normal extensions of the rationals with $\text{Gal}(L_n/\mathbb{Q}) \cong D_{p^n}$. It gives rise to a tower $\{K_n/\mathbb{Q}\}_{n \ge 0}$ of extensions of degree $p^n$ that are not Galois, being the fixed fields of the non normal subgroup of order 2 in $D_{p^n}$: this tower is a fake $Z_p$-extension. While it is a classical result that many *p*-adic arithmetic properties of $F$ are encoded in the behaviour of class groups along the anticyclotomic $Z_p$-extension (we refer the reader to the fundamental work [Rub88] and references therein), recent advances show the importance of analysing primes above from *p*, as in [KNS24,

<footnote>Date: March 7th, 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 11R29, 11R20; Secondary 11R34, 11R37.</footnote>
**Theorem 3.1**, or in [KL23, Theorem A] (see also the introduction of [BL23] for a thorough literature review). The aim of this section is to generalise some results obtained in [CN23] about fake $\mathbb{Z}_p$-extensions of dihedral type, even beyond the anticyclotomic cases, both for the prime $p$ and for primes $\ell$ different from $p$.

We conclude this introduction by observing that a full formalization of the results in this paper in the proof assistant Lean<fnref n="1" /> through its mathematical library Mathlib (see [mC20]) should be within reach. The definition of the ideal class group, together with its finiteness for number fields, has recently been obtained (see [BDNN21] and [BDNN22] for a longer version); the full formalization of the topological group of the idèles and of its class group is discussed in [dFF22]; in the recent work [dFFN24] de Frutós-Fernandez and the author implemented the basics of the arithmetic of local fields; moreover, Livingston has developed in [Liv23] the foundations of group cohomology in Lean, thus providing the full technical machinery to implement results about dihedral extensions and their class field theory in Lean; finally, the work [Rio25] has paved the route to several formalizations of homological results (see for instance [ABK+24]), and this will allow to efficiently formalise in Lean the arguments about quotient categories needed in §3.1.

## ACKNOWLEDGEMENTS

This work is licensed under a **CC-BY 4.0** licence. I am grateful to the two referees whose comments and suggestions helped improve the presentation. I am also deeply indebted to Luca Caputo for his comments on a first draft of the manuscript, in particular for pointing out an error in the first version of Proposition 2.11.

I take this opportunity to correct two typos introduced by the copy-editor while preparing the publisher version of my joint work [CN20]:

**p. 5, line -5:** ...$c_\delta$ is multiplied by $-1$ on $H^2(G, \mathbb{Z})$... should be replaced with ...$c_\delta$ is multiplication by $-1$ on $H^2(G, \mathbb{Z})$...

**p. 11, line 1:** ...since $N_{L/K} \circ \iota_{L/K}$ is multiplied by $2 = [L:K]$ ... should be replaced with ... since $N_{L/K} \circ \iota_{L/K}$ is multiplication by $2 = [L:K]$ ...

## 2. A FORMULA FOR S-CLASS GROUPS IN DIHEDRAL EXTENSIONS

The main result of [CN20] is a formula expressing the ratio of the class numbers in dihedral extensions of number fields (we refer the reader to the notation introduced *ibid*. or to Notation 2.2 below for the precise notation):

**Theorem 2.1** ([CN20, Theorem 3.14]). Let $L/k$ be a Galois extension of number fields whose Galois group $D = \text{Gal}(L/k)$ is dihedral of order $2q$ with $q$ odd. Let $\Sigma \subseteq D$ be a subgroup of order $2$ and let $G \subseteq D$ be the subgroup of order $q$. Set $K = L^\Sigma$ and $F = L^G$. Then

$$ \frac{|Cl_L| \cdot |Cl_k|^2}{|Cl_F| \cdot |Cl_K|^2} = \frac{\hat{h}^0(D, \mathcal{O}_L^\times[\frac{1}{2}])}{\hat{h}^{-1}(D, \mathcal{O}_L^\times[\frac{1}{2}])} = \frac{\hat{h}^0(D, \mathcal{O}_L^\times)}{\hat{h}^{-1}(D, \mathcal{O}_L^\times)} \frac{\hat{h}^{-1}(\Sigma, \mathcal{O}_L^\times)}{\hat{h}^0(\Sigma, \mathcal{O}_L^\times)}, $$

where, for any group $\Gamma$ and any $\Gamma$-module $B$, $\hat{h}^i(\Gamma, B)$ denotes the order of the Tate cohomology group $\hat{H}^i(\Gamma, B)$, and $B[\frac{1}{2}]$ denotes $B \otimes \mathbb{Z}[\frac{1}{2}]$.

Given any number field<fnref n="2" /> $M$ and any finite set $S$ of finite primes of $M$, we introduce the following notation

<footnote n="1">¹see https://lean-lang.org/.</footnote>

<footnote n="2">²In this paper, by “number field” we mean a *finite field extension of $\mathbb{Q}$*.</footnote>
**Notation 2.2.** $A_M^\times$ is the group of idèles of $M$, and $\mathcal{C}_M = A_M^\times/M^\times$ is its idèle class group. The group of prime-to-$S$ fractional ideals — isomorphic to the free abelian group generated by the primes not in $S$ — is denoted by $\text{Id}_M^S$ and $\text{Pr}_M^S$ is its subgroup of prime-to-$S$ principal fractional ideals. The $S$-class group of $M$ is the quotient $\text{Id}_M^S/\text{Pr}_M^S$ and is denoted $Cl_M^S$. The group of $S$-units $E_M^S$ is defined as

$$E_M^S = \{x \in M^\times \text{ such that } v_p(x) = 0 \text{ for all finite primes } p \notin S\},$$

while the symbol $\mathcal{U}_M^S$ denotes the group of idèles of valuation 0 at every finite place not in $S$,

$$\mathcal{U}_M^S = \prod_{\mathfrak{p} \in S} M_{\mathfrak{p}}^{\times} \times \prod_{\mathfrak{p} \notin S} \mathcal{O}_{M_{\mathfrak{p}}}^{\times} \times \prod_{v|\infty} M_v^{\times}.$$

The $S$-class group of $M$ appears in the bottom-right corner of the following commutative diagram, analogous to [CN20, Diagram (3.6)],

<figure><img src="image_2.png" /></figure>

where $Q_M^S$ is defined by the exactness of the diagram. It is easy to see that $Cl_M^S$ is the quotient of $Cl_M$ modulo the subgroup generated by the classes of the primes in $S$. Moreover, given a finite Galois extension of number fields $M'/M$, all groups appearing in (2.1) are Gal$(M'/M)$-modules, and the diagram is exact and commutative in the category of Gal$(M'/M)$-modules. We refer the reader to [Tat67] for the main results of class field theory that we use.

The following is an adaptation of the notation introduced in [CN20, §3]:

**Notation 2.3.** For an extension $M_2/M_1$ of number fields, denote by $\iota_{M_2/M_1}^S: Cl_{M_1}^S \to Cl_{M_2}^S$ and by $N_{M_2/M_1}^S: Cl_{M_2}^S \to Cl_{M_1}^S$, respectively, the map induced by extending ideals from $M_1$ to $M_2$ and by taking the norm of ideals from $M_2$ to $M_1$.

Given a finite group $\Gamma$, we denote by $\text{Nm}_\Gamma$ the element

$$\text{Nm}_\Gamma = \sum_{g \in \Gamma} g \in \mathbb{Z}[\Gamma];$$

for a $\Gamma$-module $B$, we let the same symbol denote the map $\text{Nm}_\Gamma: B \to B$ induced by multiplication by $\text{Nm}_\Gamma$. Similarly, we let $B[\text{Nm}_\Gamma]$ denote the kernel of multiplication by $\text{Nm}_\Gamma$. Writing $\hat{H}^i(\Gamma, B)$ (resp. $H^i(\Gamma, B)$) for the Tate (resp. ordinary) $i$-th cohomology group of $\Gamma$ with values in $B$, the symbol $\hat{h}^i(\Gamma, B)$ (resp. $h^i(\Gamma, B)$) denotes the order of $\hat{H}^i(\Gamma, B)$ (resp. of $H^i(\Gamma, B)$). We occasionally write $B^\Gamma$ to denote the group $H^0(\Gamma, B)$.

One of our first results is a generalisation of [Nuc10, Proposition 2.2] to the setting of $S$-class groups.
**Lemma 2.4.** Let $M'/M$ be a Galois extension of number fields, with Galois group $\Gamma = \text{Gal}(M'/M)$. Then there are isomorphisms

$$ \text{Ker}(\iota_{M'/M}^S) \cong \text{Ker}(H^1(\Gamma, E_{M'}^S) \to H^1(\Gamma, \mathcal{U}_{M'}^S)) \quad \text{and} \quad \text{Coker}(\iota_{M'/M}^S) \cong H^1(\Gamma, \mathcal{Q}_{M'}^S). $$

*Proof.* Taking $\Gamma$-cohomology of the bottom row of (2.1) yields the exact sequence

$$ 0 \to (Q_{M'}^S)^\Gamma \to (\mathcal{C}_{M'})^\Gamma \to (Cl_{M'}^S)^\Gamma \to H^1(\Gamma, Q_{M'}^S) \to H^1(\Gamma, \mathcal{C}_{M'}) = 0. \quad (2.2) $$

The sequence (2.2) sits in the second line of the following exact, commutative diagram

$$ \begin{tikzcd} 0 \arrow[r] & Q_M^S \arrow[r] \arrow[d] & \mathcal{C}_M \arrow[r] \arrow[d] & Cl_M^S \arrow[r] \arrow[d] & 0 \\ 0 \arrow[r] & (Q_{M'}^S)^\Gamma \arrow[r] \arrow[d] & (\mathcal{C}_{M'})^\Gamma \arrow[r] \arrow[d] & (Cl_{M'}^S)^\Gamma \arrow[r] \arrow[d] & H^1(\Gamma, Q_{M'}^S) \arrow[r] \arrow[d] & 0 \end{tikzcd} $$

and the snake lemma yields isomorphisms

$$ \text{Ker}(\iota_{M'/M}^S) \cong (Q_{M'}^S)^\Gamma / Q_M^S \quad \text{and} \quad \text{Coker}(\iota_{M'/M}^S) \cong H^1(\Gamma, Q_{M'}^S) \quad (2.3) $$

already establishing the second isomorphism in the statement. Taking $\Gamma$-cohomology of the leftmost column in (2.1) gives rise to the commutative diagram of exact rows

$$ \begin{tikzcd} 0 \arrow[r] & E_M^S \arrow[r] \arrow[d, "\|"] & \mathcal{U}_M^S \arrow[r] \arrow[d, "\|"] & Q_M^S \arrow[r] \arrow[d, "\downarrow"] & 0 \\ 0 \arrow[r] & (E_{M'}^S)^\Gamma \arrow[r] \arrow[d, "\|"] & (\mathcal{U}_{M'}^S)^\Gamma \arrow[r] \arrow[d, "\|"] & (Q_{M'}^S)^\Gamma \arrow[r] \arrow[d, "\downarrow"] & H^1(\Gamma, E_{M'}^S) \arrow[r] \arrow[d, "\|"] & H^1(\Gamma, \mathcal{U}_{M'}^S) \end{tikzcd} $$

By applying the snake lemma again, we get an isomorphism

$$ (Q_{M'}^S)^\Gamma / Q_M^S \cong \text{Ker}(H^1(\Gamma, E_{M'}^S) \to H^1(\Gamma, \mathcal{U}_{M'}^S)) $$

which, combined with the first isomorphism in (2.3), yields the first isomorphism of the statement. $\square$

<figure><img src="image_3.png" /></figure>

From now on in this section, we work in the dihedral setting and we need the following notation:

**Notation 2.5.** Fix a number field $k$ and let $L/k$ be a Galois extension whose Galois group $D = \text{Gal}(L/k)$ is isomorphic to $D_q$ the dihedral group of order $2q$, where $q$ is any odd integer. Write $G \triangleleft D$ for the normal subgroup of order $q$ and fix a subgroup $\Sigma < D$ of order 2. Fix also the choice of a generator $\rho$ of $G$ and $\sigma$ of $\Sigma$, letting $\Sigma'$ denote the subgroup $\langle \sigma\rho \rangle < D$, again of order 2. Let $F/k$ be the subextension of $L/k$ fixed by $G$ and let $K/k$ (resp. $K'/k$) be the subextension of $L/k$ fixed by $\Sigma$ (resp. fixed by $\Sigma'$). Finally, we let $S$ be a fixed, finite set of finite primes in $k$ and we denote by $S_s \subseteq S$ its subset consisting of those primes that split in $F/k$.

The first ingredient for the proof of Theorem 2.12 is a generalisation of [CN20, Proposition 3.2], extending the result loc. cit. to $S$-class groups. Before stating it, observe that there are equalities

$$ (Cl_L^{S[\frac{1}{2}]})^{\Sigma} = \iota_{L/K}^{S}(Cl_K^{S[\frac{1}{2}]}) \quad \text{and} \quad (Cl_L^{S[\frac{1}{2}]})^{\Sigma'} = \iota_{L/K'}^{S}(Cl_{K'}^{S[\frac{1}{2}]}) : \quad (2.4) $$

to see this (we provide the argument for $K$, the one for $K'$ being analogous), observe first that $\iota_{L/K}^S$ is injective when restricted to $Cl_K^{S[\frac{1}{2}]}$, because its composition with $N_{L/K}^S : Cl_L^{S[\frac{1}{2}]} \to Cl_K^{S[\frac{1}{2}]}$ coincides with the multiplication by 2. Moreover, the “algebraic” norm $\text{Nm}_\Sigma : Cl_L^S \to Cl_L^S$ can be factored as

$$ \text{Nm}_\Sigma : Cl_L^S \xrightarrow{N_{L/K}^S} Cl_K^S \xrightarrow{\iota_{L/K}^S} Cl_L^S. $$
Hence, given any class $c \in (Cl_L^S[\frac{1}{2}])^\Sigma$, we have $2c = c + \sigma c = \text{Nm}_\Sigma c = \iota_{L/K}^S(\text{N}_{L/K}^S c)$: since multiplication by 2 is an automorphism, this implies that $\iota_{L/K}^S$ is also surjective, establishing (2.4).

**Proposition 2.6.** In the setting of Notation 2.5, there is an equality

$$ \frac{|Cl_L^S[\frac{1}{2}]|}{|Cl_K^S[\frac{1}{2}]|^2} = \frac{h^0(G, Cl_L^S[\frac{1}{2}])}{h^0(D, Cl_L^S[\frac{1}{2}]) \cdot \hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}]) \cdot |\text{Nm}_D Cl_L^S[\frac{1}{2}]|} $$

*Proof.* Let $\eta^S: Cl_K^S \oplus Cl_{K'}^S \to Cl_L^S$ be the map defined by

$$ \eta^S(c, c') = \iota_{L/K}^S(c) \iota_{L/K'}^S(c') \quad \text{for} \quad c \in Cl_K^S, c' \in Cl_{K'}^S. $$

The proposition will follow from the following two equalities:

$$ (Cl_L^S[\frac{1}{2}] : \text{Im}(\eta^S)[\frac{1}{2}]) = \frac{h^0(Cl_L^S[\frac{1}{2}], )}{\hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}]) \cdot |\text{Nm}_D Cl_L^S[\frac{1}{2}]|} \quad (2.5) $$

and

$$ |\text{Ker } \eta^S[\frac{1}{2}]| = h^0(D, Cl_L^S[\frac{1}{2}]). \quad (2.6) $$

Indeed, comparing orders in the tautological exact sequence

$$ 0 \to \text{Ker } \eta^S[\frac{1}{2}] \to Cl_K^S[\frac{1}{2}] \oplus Cl_{K'}^S[\frac{1}{2}] \to Cl_L^S[\frac{1}{2}] \to Cl_L^S[\frac{1}{2}]/\text{Im } \eta^S[\frac{1}{2}] \to 0 $$

and using (2.5) and (2.6) yields the statement.

We start with the proof of (2.5). Before inverting 2, we can interpret the index in the left-hand side as

$$ (Cl_L^S : \text{Im } \eta^S) = (\text{Nm}_G Cl_L^S : \text{Nm}_G (\text{Im } \eta^S)) \cdot (Cl_L^S[\text{Nm}_G] : \text{Im } \eta^S \cap Cl_L^S[\text{Nm}_G]) \quad (2.7) $$

by applying equation [CN20, (3.4)] with $f = \text{Nm}_G$, $B = Cl_L^S$ and $B' = \text{Im } \eta^S$. Moreover, combining (2.4) with the equality $\text{Im } \eta^S = \iota_{L/K}^S(Cl_K^S) \iota_{L/K}^S(Cl_{K'}^S)$, that follows from the definition of $\eta^S$, yields

$$ \text{Im}(\eta^S[\frac{1}{2}]) = (Cl_L^S[\frac{1}{2}])^\Sigma (Cl_L^S[\frac{1}{2}])^{\Sigma'}. \quad (2.8) $$

Replacing (2.8) in the right-most term of (2.7) (after inverting 2) we obtain

$$ (Cl_L^S[\frac{1}{2}] : \text{Im}(\eta^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (\text{Im} \eta^S[\frac{1}{2}])) \cdot (Cl_L^S[\frac{1}{2}][\text{Nm}_G] : ((Cl_L^S[\frac{1}{2}])^\Sigma (Cl_L^S[\frac{1}{2}]))^{\Sigma'}) \cap Cl_L^S[\text{Nm}_G]). $$

The second isomorphism of [CN20, Lemma 2.5] (with $B = Cl_L^S[\frac{1}{2}]$) allows to simplify the right-hand side of the above equation to

$$ (Cl_L^S[\frac{1}{2}] : \text{Im}(\eta^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (\text{Im} \eta^S[\frac{1}{2}])) \cdot h^1(D, Cl_L^S[\frac{1}{2}]). \quad (2.9) $$

Replacing again (2.8), this time in the first factor of the right-hand side of (2.9), gives the equality

$$ (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (\text{Im} \eta^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G ((Cl_L^S[\frac{1}{2}])^\Sigma (Cl_L^S[\frac{1}{2}])^{\Sigma'})) $$

and since $\text{Nm}_G(Cl_L^S[\frac{1}{2}])^\Sigma = \text{Nm}_G(Cl_L^S[\frac{1}{2}])^{\Sigma'}$, this further simplifies to

$$ (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (\text{Im} \eta^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (Cl_L^S[\frac{1}{2}])^\Sigma). $$

Applying again the equality $\text{Nm}_\Sigma Cl_L^S[\frac{1}{2}] = (Cl_L^S[\frac{1}{2}])^\Sigma$, we can continue the substitutions to obtain

$$ (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G (\text{Im} \eta^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_G \text{Nm}_\Sigma (Cl_L^S[\frac{1}{2}])) = (\text{Nm}_G Cl_L^S[\frac{1}{2}] : \text{Nm}_D (Cl_L^S[\frac{1}{2}])) $$
and inserting this back into (2.9) gives

$$ (Cl_L^S[\frac{1}{2}]: \mathrm{Im}(\eta^S[\frac{1}{2}])) = (\mathrm{Nm}_GCl_L^S[\frac{1}{2}]: \mathrm{Nm}_D(Cl_L^S[\frac{1}{2}])) \cdot h^1(D, Cl_L^S[\frac{1}{2}]) = \frac{|\mathrm{Nm}_GCl_L^S[\frac{1}{2}]| \cdot h^1(D, Cl_L^S[\frac{1}{2}])}{|\mathrm{Nm}_D(Cl_L^S[\frac{1}{2}])|} \quad (2.10) $$

By [CN20, Proposition 2.1] there is an equality

$$ h^1(D, Cl_L^S[\frac{1}{2}]) = \frac{h^1(G, Cl_L^S[\frac{1}{2}])}{\hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}])} $$

whereas, by definition of Tate cohomology and thanks to the vanishing of the Herbrand quotient of a finite module,

$$ |\mathrm{Nm}_GCl_L^S[\frac{1}{2}]| = \frac{Cl_L^S[\frac{1}{2}]^G}{\hat{h}^0(G, Cl_L^S[\frac{1}{2}])} = \frac{Cl_L^S[\frac{1}{2}]^G}{h^1(G, Cl_L^S[\frac{1}{2}])} : $$

thus we can rewrite the numerator in the right-most term of (2.10) as

$$ |\mathrm{Nm}_GCl_L^S[\frac{1}{2}]| \cdot h^1(D, Cl_L^S[\frac{1}{2}]) = \frac{h^0(G, Cl_L^S[\frac{1}{2}])}{h^1(G, Cl_L^S[\frac{1}{2}])} \cdot \frac{h^1(G, Cl_L^S[\frac{1}{2}])}{\hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}])} = \frac{h^0(G, Cl_L^S[\frac{1}{2}])}{\hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}])} $$

Inserting the above equality in (2.10) finally yields (2.5).

Passing now to (2.6), we are actually going to show that the arrow

$$ \theta: \mathrm{Ker} \eta^S[\frac{1}{2}] \to H^0(D, Cl_L^S[\frac{1}{2}]) $$

defined by $\theta(c, c') = \iota_{L/K}^S(c)$ is an isomorphism, as in the end of the proof of [CN20, Proposition 3.2].

That $\theta$ is well-defined follows from the equality $\iota_{L/K}^S c = (\iota_{L/K'}^S c')^{-1} \in Cl_L^S$ (by definition of $\eta^S$), and therefore both $\Sigma$ and $\Sigma'$ fix $\iota_{L/K}(c)$, so this class is actually invariant under the whole $D$. That $\theta$ is injective has already been observed during the proof of (2.4), when we established that the restriction of $\iota_{L/K}^S$ to the prime-to-2 component is injective. To show that it is surjective, let $b \in H^0(D, Cl_L^S)$: in particular, $b$ is fixed by both $\Sigma$ and $\Sigma'$ and we can write

$$ b = \iota_{L/K}^S(c) = \iota_{L/K}^S(c') $$

for some $c \in Cl_K^S[\frac{1}{2}]$, $c' \in Cl_{K'}^S[\frac{1}{2}]$, thanks to (2.8). It follows that the element $(c, c')$ is in Ker $\eta^S[\frac{1}{2}]$ and $b = \theta(c, c')$, concluding the proof that $\theta$ is an isomorphism and establishing (2.6). $\square$

Before stating the next result, we introduce one more piece of notation:

**Notation 2.7.** Given any abelian group $B$, write $B^{(2)}$ to denote its 2-Sylow subgroup.

**Remark 2.8.** The restriction of the morphism $\iota_{L/F}^S$ to $(Cl_F^S)^{(2)}$ is injective: indeed, the composition $N_{L/F}^S \circ \iota_{L/F}^S$ coincides with raising to the $q$-th power, that is an automorphism of the 2-Sylow subgroup. The same argument shows that the restrictions of $\iota_{K/k}^S$ and $\iota_{K'/k}^S$ to $(Cl_k^S)^{(2)}$ are injective: we thus identify the 2-Sylow subgroup $(\iota_{L/F}^S Cl_F^S)^{(2)}$ (resp. $(\iota_{K/k}^S Cl_k^S)^{(2)}$, resp. $(\iota_{K'/k}^S Cl_k^S)^{(2)}$) with $(Cl_F^S)^{(2)}$ (resp. with $(Cl_k^S)^{(2)}$).

**Lemma 2.9.** The 2-adic valuations of the S-class groups in the dihedral extension $L/k$ satisfy the relation

$$ v_2 \left( \frac{|Cl_L^S|}{|Cl_F^S|} \right) = 2 \cdot v_2 \left( \frac{|Cl_K^S|}{|Cl_k^S|} \right). $$
*Proof.* It follows from Walter's isomorphism (see [Wal77, Lemma 5.2]) that

$$ \left| \frac{(Cl_L^S)^{(2)}/H^0(G, Cl_L^S)^{(2)}}{H^0(\Sigma, Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)}} \right| \cdot \left| \frac{H^0(\Sigma', Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)}}{H^0(G, Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)}} \right| \quad (2.11) $$

We claim that there are isomorphisms

$$ \begin{aligned} H^0(\Sigma, Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)} &\cong (Cl_K^S/Cl_k^S)^{(2)} \\ H^0(\Sigma', Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)} &\cong (Cl_{K'}^S/Cl_k^S)^{(2)}, \end{aligned} \quad (2.12) $$

as well as

$$ H^0(G, Cl_L^S)^{(2)} \cong (Cl_F^S)^{(2)}. \quad (2.13) $$

Granting (2.12) and (2.13), the lemma follows from (2.11).

Concerning (2.13), it is enough by Remark 2.8 to show that every class in $(Cl_L^S)^{(2)}$ fixed by $G$ is in the image of $\iota_{L/F}^S$: identifying $H^0(G, Cl_L^S)^{(2)}$ with $H^0(G, (Cl_L^S)^{(2)})$ and then with $\mathrm{Nm}_G(Cl_L^S)^{(2)}$ (because $\hat{H}^0(G, (Cl_L^S)^{(2)}) \cong H^2(G, (Cl_L^S)^{(2)}) = 0$), it is enough to show that every class of the form $\mathrm{Nm}_G x$ is in the image of $\iota_{L/F}^S$. This follows from the factorization $\mathrm{Nm}_G = \iota_{L/F}^S \circ N_{L/F}^S$.

To prove (2.12) (we prove the relation for $K$, the proof for $K'$ being analogous), consider the commutative diagram with exact rows

$$ \begin{array}{ccc} 0 & \to & (Cl_k^S)^{(2)} \to (Cl_K^S)^{(2)} \to (Cl_K^S/Cl_k^S)^{(2)} \to 0 \\ \Big\downarrow_{\iota_{L/k}^S} & & \Big\downarrow_{\iota_{L/K}^S} & \Big\downarrow \\ 0 & \to & H^0(D, Cl_L^S)^{(2)} \to H^0(\Sigma, Cl_L^S)^{(2)} \to H^0(\Sigma, Cl_L^S)^{(2)}/H^0(D, Cl_L^S)^{(2)} \to 0 \end{array} $$

The isomorphisms of Lemma 2.4 combined with [CN20, Lemma 2.9] show that the kernels (resp. the cokernels) of the first and the second vertical arrow are isomorphic. The snake lemma yields the result. $\square$

As remarked before Proposition 2.6, and as the proof of Lemma 2.4 somewhat exemplifies, the cohomological arguments in [CN20] generalise easily to the case of $S$-class groups, since they rely on the study of the commutative square (2.1), analogous to [CN20, (3.6)]. In particular, the proof of the two equalities in the following lemma is analogous to the corresponding ones *ibid.* and we simply sketch it.

**Lemma 2.10.** Let $H \subseteq D$ be any subgroup and let $M = L^H$ be the subfield fixed by $H$. The equalities

$$ |Cl_M^{S[\frac{1}{2}]}| = \frac{h^0(H, Cl_L^{S[\frac{1}{2}]} \cdot h^1(H, E_L^{S[\frac{1}{2}]}))}{h^1(H, \mathcal{U}_L^{S[\frac{1}{2}]} \cdot h^1(H, \mathrm{Pr}_L^{S[\frac{1}{2}]}))} $$

and

$$ |Cl_k^{S[\frac{1}{2}]}| = \frac{\hat{h}^{-1}(D, Cl_L^{S[\frac{1}{2}]} \cdot |\mathrm{Nm}_D Cl_L^{S[\frac{1}{2}]}| \cdot \hat{h}^0(D, E_L^{S[\frac{1}{2}]}))}{\hat{h}^0(D, \mathcal{U}_L^{S[\frac{1}{2}]} \cdot \hat{h}^{-1}(D, \mathrm{Pr}_L^{S[\frac{1}{2}]}))} $$

hold.

*Proof.* The proof of the first equality is the same as the proof of [CN20, Proposition 3.5], replacing Diagram (3.6) *ibid.* with (2.1). Similarly, the second equality is obtained through the proof of [CN20, Proposition 3.7], invoking our Lemma 2.4 instead of equation (3.14) *ibid.* $\square$
Unlike Lemma 2.10, the following result requires some adaptation to take into account the primes in the set S. Recall that we introduced in Notation 2.5 the symbol $S_s$ to denote the primes in S that split in $F/k$.

**Proposition 2.11.** *There is an equality*

$$h^1(G, \mathcal{U}_L^S[\frac{1}{2}]) = h^1(D, \mathcal{U}_L^S[\frac{1}{2}]) \cdot \hat{h}^0(D, \mathcal{U}_L^S[\frac{1}{2}]) \cdot \left( \prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}} \right)^{-1}$$

where $d_{\mathfrak{p}}(L/k) = d_{\mathfrak{p}}$ is the local degree of a prime $\mathfrak{p}$ in the extension $L/k$.

*Proof.* Just for this proof, write $S$ to denote the set of primes ideal in $O_F$ dividing the primes in $S$. Accordingly, write $\text{Ram}(L/F)$ (resp. $\text{Ram}(L/k)$) for the set of primes in $F$ (resp. in $k$) that ramify in $L/F$ (resp. in $L/k$), and let $\text{Ram}(L/F) \setminus S$ (resp. $\text{Ram}(L/k) \setminus S$) denote the set difference $\text{Ram}(L/F) \setminus (S \cap \text{Ram}(L/F))$ (resp. $\text{Ram}(L/k) \setminus (S \cap \text{Ram}(L/k))$).

Given any prime ideal $\mathfrak{p}$ in $k$, choose and fix a prime ideal $\mathfrak{P}$ in $L$ diving it, and write $G(\mathfrak{P})$ (resp. $D(\mathfrak{P})$) to denote the decomposition group of $\mathfrak{P}$ in the extension $L/F$ (resp. $L/k$). Whenever $\mathfrak{p}$ splits in $F/k$ we also fix a conjugate ideal $\tilde{\mathfrak{P}}$ in $L$, dividing $\mathfrak{p}$ but not dividing $\mathfrak{P} \cap F$. Accordingly, given any prime $\mathfrak{l}$ in $F$, denote by $\mathfrak{L}$ the prime in $L$ lying above $\mathfrak{l} \cap k$ that has been fixed with the previous choice. Finally, denote by $O_{\mathfrak{P}}^{\times}$ — instead of $O_{L_{\mathfrak{l}}}^{\times}$ — the local units at a prime $\mathfrak{P}$.

We begin with the left-hand side in the statement. The description in [Tat67, Proposition 7.2] of the Galois structure of the idèles yields an identification

$$H^1(G, \mathcal{U}_L^S) = \prod_{\mathfrak{l} \in S} H^1(G(\mathfrak{L}), L_{\mathfrak{L}}^\times) \times \prod_{\mathfrak{l} \notin S} H^1(G(\mathfrak{L}), O_{\mathfrak{L}}^\times) \times \prod_{v|\infty} H^1(G(v), L_v^\times) = \prod_{\mathfrak{l} \notin S} H^1(G(\mathfrak{L}), O_{\mathfrak{L}}^\times)$$

where the second equality follows from Hilbert 90. Since the cohomology of local units is trivial in unramified extension, the product above simplifies as

$$H^1(G, \mathcal{U}_L^S) = \prod_{\mathfrak{l} \in \text{Ram}(L/F) \setminus S} H^1(G(\mathfrak{L}), O_{\mathfrak{L}}^\times) \cong \prod_{\mathfrak{l} \in \text{Ram}(L/F) \setminus S} \mathbb{Z}/e_{\mathfrak{l}} \quad (2.14)$$

where $e_{\mathfrak{l}}$ is the ramification degree of $\mathfrak{l}$ in the extension $L/F$, and the isomorphism follows from local class field theory. Given a prime $\mathfrak{l}$ occurring in the above product, let $\mathfrak{p} = \mathfrak{l} \cap k$; it lies in $\text{Ram}(L/k) \setminus S$ and its ramification index $e_{\mathfrak{p}}$ in $L/k$ satisfies

$$e_{\mathfrak{p}} = \begin{cases} e_{\mathfrak{l}} & \text{if } \mathfrak{p} \text{ splits or is inert in } F/k; \\ 2e_{\mathfrak{l}} & \text{if } \mathfrak{p} \text{ is ramified in } F/k. \end{cases}$$

In particular,

$$h^1(G, \mathcal{U}_L^S) = \prod_{\mathfrak{l} \in \text{Ram}(L/F) \setminus S} e_{\mathfrak{l}} = \prod_{\mathfrak{p} \in \text{Ram}(L/k)_s \setminus S} e_{\mathfrak{p}}^2 \cdot \prod_{\mathfrak{p} \in \text{Ram}(L/k)_i \setminus S} e_{\mathfrak{p}} \cdot \prod_{\mathfrak{p} \in \text{Ram}(L/k)_r \setminus S} \frac{e_{\mathfrak{p}}}{2} \quad (2.15)$$

where $\text{Ram}(L/k)_s$ (resp. $\text{Ram}(L/k)_i$, $\text{Ram}(L/k)_r$) denotes the subset of $\text{Ram}(L/k)$ consisting of primes that split (resp. that are inert, that ramify) in $F/k$.

Consider now to the right-hand side: when $i = 1$, the same argument as above shows that

$$\hat{H}^1(D, \mathcal{U}_L^S) = \prod_{\mathfrak{p} \in \text{Ram}(L/k) \setminus S} \hat{H}^1(D(\mathfrak{P}), O_{\mathfrak{P}}^\times).$$

and therefore

$$h^1(D, \mathcal{U}_L^S) = \prod_{\mathfrak{p} \in \text{Ram}(L/k) \setminus S} e_{\mathfrak{p}} = \prod_{\mathfrak{p} \in \text{Ram}(L/k)_s \setminus S} e_{\mathfrak{p}} \cdot \prod_{\mathfrak{p} \in \text{Ram}(L/k)_i \setminus S} e_{\mathfrak{p}} \cdot \prod_{\mathfrak{p} \in \text{Ram}(L/k)_r \setminus S} e_{\mathfrak{p}} \quad (2.16)$$
When $i = 0$ we need to distinguish between places in $S$ and not in $S$: using again that local units have trivial cohomology at unramified places together with the main statement of local class field theory ([Ser67, §2, Theorem 1]), we find

$$ \hat{H}^0(D, \mathcal{U}_L^S) = \prod_{\mathfrak{p} \notin S} \hat{H}^0(D(\mathfrak{P}), \mathcal{O}_{\mathfrak{P}}^\times) \times \prod_{\mathfrak{p} \in S} \hat{H}^0(D(\mathfrak{P}), L_{\mathfrak{P}}^\times) = \\ \qquad \prod_{\mathfrak{p} \in \mathrm{Ram}(L/k) \setminus S} \hat{H}^0(D(\mathfrak{P}), \mathcal{O}_{\mathfrak{P}}^\times) \times \prod_{\mathfrak{p} \in S} D(\mathfrak{P})^{\mathrm{ab}}. \qquad (2.17) $$

Consider now a prime $\mathfrak{p} \in S$: if it lies in $S_s$, the decomposition subgroup $D(\mathfrak{P})$ is cyclic of odd order (dividing $q$), and $D(\mathfrak{P})^{\mathrm{ab}} \cong \mathbb{Z}/d_{\mathfrak{p}}$; when $\mathfrak{p} \notin S_s$, the decomposition group is either cyclic of order 2 or dihedral, and in both cases $D(\mathfrak{P})^{\mathrm{ab}} \cong \mathbb{Z}/2$. Combining this argument with [CN20, Lemma 3.9] we find

$$ \hat{h}^0(D, \mathcal{U}_L^S) = \prod_{\mathfrak{p} \in \mathrm{Ram}(L/k)_r \setminus S} 2 \cdot \prod_{\mathfrak{p} \in \mathrm{Ram}(L/k)_s \setminus S} e_{\mathfrak{p}} \cdot \prod_{\mathfrak{p} \in S \setminus S_s} 2 \cdot \prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}}. \qquad (2.18) $$

Comparing (2.15) with (2.16) and (2.18) gives

$$ h^1(G, \mathcal{U}_L^S) = h^1(D, \mathcal{U}_L^S) \cdot \hat{h}^0(D, \mathcal{U}_L^S) \cdot 4^{-|\mathrm{Ram}(L/k)_r|} \cdot 2^{-|S \setminus S_s|} \cdot \left( \prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}} \right)^{-1} \qquad (2.19) $$

and after inverting 2 in (2.19) (taking into account that $d_{\mathfrak{p}}$ is odd for $\mathfrak{p} \in S_s$) we obtain the proposition. $\square$

We are now ready to state the main result of this section:

**Theorem 2.12.** With the notation introduced in Notation 2.5, the following equality holds

$$ \frac{|Cl_L^S| \cdot |Cl_k^S|^2}{|Cl_F^S| \cdot |Cl_K^S|^2} = \left( \prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}} \right)^{-1} \cdot \frac{\hat{h}^0(D, E_L^S[\frac{1}{2}])}{\hat{h}^{-1}(D, E_L^S[\frac{1}{2}])}, $$

where $d_{\mathfrak{p}}(L/k) = d_{\mathfrak{p}}$ is the local degree of a prime $\mathfrak{p}$ in the extension $L/k$.

*Proof.* We proceed by first proving that the 2-parts of the left- and of the right-hand side coincide, and we eventually analyse the prime-to-2 parts. The cohomology groups appearing in the right-hand side have trivial 2-part, because $E_L^S[\frac{1}{2}]$ is a 2-divisible $D$-module; and all $d_{\mathfrak{p}}$ appearing in the product are odd, as observed in the proof of Proposition 2.11, after (2.19). Hence the 2-adic valuation of the right-hand side is trivial, as is the one of the left-hand side by Lemma 2.9.

We now move on to prove the equality for the prime-to-2 parts, replacing all $S$-class groups $Cl_S^\bullet$ by their prime-to-2 components $Cl_S^\bullet[\frac{1}{2}]$ in what follows. Applying Proposition 2.6, we just need to verify that

$$ \frac{|Cl_k^S[\frac{1}{2}]|^2}{|Cl_F^S[\frac{1}{2}]|} \stackrel{?}{=} \left(\prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}}\right)^{-1} \cdot \frac{\hat{h}^0(D, E_L^S[\frac{1}{2}])}{\hat{h}^{-1}(D, E_L^S[\frac{1}{2}])} \cdot \frac{h^0(D, Cl_L^S[\frac{1}{2}]) \cdot \hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}]) \cdot |\mathrm{Nm}_D Cl_L^S[\frac{1}{2}]|}{h^0(G, Cl_L^S[\frac{1}{2}])} \qquad (2.20) $$

Invoking the first equality of Lemma 2.10 twice (once for $H = G$ and once for $H = D$), the left-hand side of (2.20) satisfies the equality

$$ \frac{|Cl_k^S[\frac{1}{2}]|^2}{|Cl_F^S[\frac{1}{2}]|} = \frac{h^1(G, \mathcal{U}_L^S[\frac{1}{2}]) \cdot h^1(G, \mathrm{Pr}_L^S[\frac{1}{2}])}{h^0(G, Cl_L^S[\frac{1}{2}]) \cdot h^1(G, E_L^S[\frac{1}{2}])} \cdot \frac{h^0(D, Cl_L^S[\frac{1}{2}]) \cdot h^1(D, E_L^S[\frac{1}{2}])}{h^1(D, \mathcal{U}_L^S[\frac{1}{2}]) \cdot h^1(D, \mathrm{Pr}_L^S[\frac{1}{2}])} \cdot |Cl_k^S[\frac{1}{2}]|; $$
through the second equality established in Lemma 2.10, this becomes

$$ \frac{|Cl_k^S[\frac{1}{2}]|^2}{|Cl_F^S[\frac{1}{2}]|} = \frac{h^1(G, \mathcal{U}_L^S[\frac{1}{2}]) \cdot h^1(G, \mathrm{Pr}_L^S[\frac{1}{2}])}{h^0(G, Cl_L^S[\frac{1}{2}]) \cdot h^1(G, E_L^S[\frac{1}{2}])} \cdot \frac{h^0(D, Cl_L^S[\frac{1}{2}]) \cdot h^1(D, E_L^S[\frac{1}{2}])}{h^1(D, \mathcal{U}_L^S[\frac{1}{2}]) \cdot h^1(D, \mathrm{Pr}_L^S[\frac{1}{2}])} \cdot \frac{\hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}]) \cdot |\mathrm{Nm}_D Cl_L^S[\frac{1}{2}]| \cdot \hat{h}^0(D, E_L^S[\frac{1}{2}])}{\hat{h}^0(D, \mathcal{U}_L^S[\frac{1}{2}]) \cdot \hat{h}^{-1}(D, \mathrm{Pr}_L^S[\frac{1}{2}])} $$

Upon reordering, (2.20) can thus be restated as

$$ \frac{|Cl_k^S[\frac{1}{2}]|^2}{|Cl_F^S[\frac{1}{2}]|} = \frac{\hat{h}^0(D, E_L^S[\frac{1}{2}]) \cdot h^0(D, Cl_L^S[\frac{1}{2}])}{h^0(G, Cl_L^S[\frac{1}{2}])} \cdot \frac{h^1(D, E_L^S[\frac{1}{2}])}{h^1(G, E_L^S[\frac{1}{2}])} \cdot \frac{h^1(G, \mathcal{U}_L^S[\frac{1}{2}])}{h^1(D, \mathcal{U}_L^S[\frac{1}{2}])} \cdot \hat{h}^0(D, \mathcal{U}_L^S[\frac{1}{2}]) \\ \cdot \frac{h^1(G, \mathrm{Pr}_L^S[\frac{1}{2}])}{h^1(D, \mathrm{Pr}_L^S[\frac{1}{2}])} \cdot \hat{h}^{-1}(D, \mathrm{Pr}_L^S[\frac{1}{2}]) \cdot |\mathrm{Nm}_D Cl_L^S[\frac{1}{2}]| $$

The third factor of the right-hand side is equal to $(\prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}})^{-1}$, by Proposition 2.11. Furthermore, [CN20, Proposition 2.1] implies that the fourth factor is equal to 1 whereas the second equals $\hat{h}^{-1}(D, E_L^S[\frac{1}{2}])^{-1}$: hence the above equality simplifies to

$$ \frac{|Cl_k^S[\frac{1}{2}]|^2}{|Cl_F^S[\frac{1}{2}]|} = \frac{\hat{h}^0(D, E_L^S[\frac{1}{2}]) \cdot h^0(D, Cl_L^S[\frac{1}{2}])}{h^0(G, Cl_L^S[\frac{1}{2}])} \cdot \frac{(\prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}})^{-1}}{\hat{h}^{-1}(D, E_L^S[\frac{1}{2}])} \cdot \hat{h}^{-1}(D, Cl_L^S[\frac{1}{2}]) \cdot |\mathrm{Nm}_D Cl_L^S[\frac{1}{2}]| $$

which is precisely (2.20). This concludes the proof of the theorem. □

**Corollary 2.13.** In the same setting as in Theorem 2.12,

$$ \frac{|Cl_L^S| \cdot |Cl_k^S|^2}{|Cl_F^S| \cdot |Cl_K^S|^2} = \left( \prod_{\mathfrak{p} \in S_s} d_{\mathfrak{p}} \right)^{-1} \cdot \frac{\hat{h}^0(D, E_L^S)}{\hat{h}^{-1}(D, E_L^S)} \cdot \frac{\hat{h}^{-1}(\Sigma, E_L^S)}{\hat{h}^0(\Sigma, E_L^S)} $$

*Proof.* Observe that, for all $i \in \mathbb{Z}$, there is an isomorphism

$$ \hat{H}^i(D, E_L^S[\frac{1}{2}]) \cong \hat{H}^i(D, E_L^S)/\hat{H}^i(D, E_L^S)^{(2)}. $$

The corollary follows by combining Theorem 2.12 with [CN20, Lemma 2.9]. □

### 3. GROWTH OF S-CLASS GROUPS IN FAKE $\mathbb{Z}_p$-EXTENSIONS OF DIHEDRAL TYPE

Fix a number field $F$, let $p \ge 3$ be an odd prime and let $L_\infty/F$ be a $\mathbb{Z}_p$-extension, with layers $L_n$ such that $[L_n : F] = p^n$ (so that $L_0 = F$). The goal of this section is to study the behaviour of S-class groups in fake $\mathbb{Z}_p$-extensions. We begin with the definition of this concept, taken from [CN23, Definition 4.1]:

**Definition 3.1.** Suppose that there exists a subfield $k \subseteq F$ such that, for every $n \ge 0$, $L_n/k$ is Galois, with dihedral Galois group $D_n := \mathrm{Gal}(L_n/k) \cong D_{p^n}$ of order $2p^n$. Denote by $K_n$ a subfield of index 2 in $L_n/k$, chosen so that $K_n \supseteq K_{n-1}$ for all $n \ge 1$, and put $K_\infty = \bigcup K_n$. The extension $K_\infty/k$ is said to be a fake $\mathbb{Z}_p$-extensions of dihedral type, the extension $L_\infty/k$ is the Galois closure of the fake $\mathbb{Z}_p$-extension and $F/k$ is said to be the normalizing quadratic extension of $K_\infty/k$.

In this setting, we extend Notation 2.5 for a 2-divisible $D_n$-module $B$, letting $B^\pm$ denote the eigenspace of eigenvalue $\pm$ for the action of $D_0$ (denoted $\Sigma \mathrm{ loc. cit.}$).
Fix from now on a finite set of finite primes $S$ containing the primes above $p^3$ of $k$ as well as a fake $Z_p$-extension of dihedral type $K_\infty/k$: with the notation as in Definition 3.1, fix the normalizing quadratic extension $F/k$, and denote by $\Gamma$ (resp. by $\mathcal{D}$) the Galois group $\Gamma = \text{Gal}(L_\infty/F)$ (resp. the Galois group $\mathcal{D} = \text{Gal}(L_\infty/F)$); finally, let $G_n \subsetneq D_n$ be the subgroup of index 2, isomorphic to $\text{Gal}(L_n/F)$. The corresponding diagram of fields is the following:

<figure><img src="image_4.png" /></figure>

The tower of extensions $K_\infty = \bigcup K_n/k$ bears strong similarities with a $Z_p$-extension, with the
crucial difference that its layers are all non-Galois extensions. In particular, the usual approach of
obtaining useful information by analysing Galois actions (pioneered by Iwasawa in [Iwa73] for the
$p$-adic behaviour, and recently developed by Bandini-Longhi in [BL23] for the $\ell$-adic counterpart,
where $\ell \neq p$) breaks down completely for a fake $Z_p$-extension. In [CN23] we developed other
techniques, based on the analysis of Tate cohomology of normic systems, in order to address these
questions in the non-Galois setting, and we apply them to analyse the $p$-adic behaviour of $S$-class
groups along fake $Z_p$-extensions in §3.1. In §3.2 we rather rely on [BL23] to study the $\ell$-adic
behaviour of these groups.

3.1. **The $p$-adic behaviour.** The results in this section heavily rely on the theory of normic and double system introduced in [CN23]. We refer the reader to our previous work, and more specifically to [CN23, §2.1, §2.3 and §3.2] for both the notation, the main definitions and the details: in order to ease the comparison, we systematically adopt the same notation as *ibid*. Notions that will be particularly crucial are the construction of the quotient category $\mathbf{DS}_D^{\text{co-f.g.}} / \mathbf{B}_D^{\text{co-f.g.}}$ (see [CN23, Proposition 2.31]), that of the endofunctor $\mathcal{L} : \mathbf{DS}_D^{\text{co-f.g.}} / \mathbf{B}_D^{\text{co-f.g.}} \to \mathbf{DS}_D^{\text{co-f.g.}} / \mathbf{B}_D^{\text{co-f.g.}}$ (see [CN23, equations (2.17)-(2.18)]), and the double systems $\hat{\mathcal{H}}^i(\mathcal{U}_{L\bullet}^S)$, $\hat{\mathcal{H}}^i(E_{L\bullet}^S)$ of Tate cohomology of the normic systems of idelic and global units (see [CN23, §3.2], here adapted to idelic $S$-units and $S$-units). Finally, we refer to [CN23, Notation 3.1] for our choices concerning the notation for prime ideals in $k$, and the corresponding ones in $F$ and in $L_\infty$ that divide those in $k$.

**Lemma 3.2.** For $i = 1, 2$ the double system $\hat{\mathcal{H}}^i(\mathcal{U}_{L\bullet}^S)$ is $\mathbf{B}_D$-isomorphic to $\mathcal{L}(\hat{\mathcal{H}}^i(\mathcal{U}_{L\bullet}^S))$.

*Proof.* To establish the lemma, it suffices to prove that $\hat{\mathcal{H}}^i(\mathcal{U}_{L\bullet}^S)$ is isomorphic to a certain number of copies of the double system $\mathcal{Z}(\Gamma)$ introduced in [CN23, Example 2.35], because this double

<footnote n="3">³This requirement is only needed at two points: in the proof of Proposition 3.3-c) to extend Ozaki’s result [Oza95, Proposition] to $S$-class groups; and in the proof of Theorem 2.12 to obtain an Iwasawa formula for the growth of $S$-class groups along classical $Z_p$-extensions. Explicit references for these results in the literature can only be found under the assumption that $S$ contains the primes above $p$ (that it contains infinite primes is not needed since we are assuming $p \neq 2$), which motivates our requirement.</footnote>
system is itself $\mathbf{B}_D$-isomorphic to its image $\mathcal{L}(\mathcal{Z}(\Gamma))$ and the endofunctor $\mathcal{L}$ is additive (both claims are discussed loc. cit. ).

As in the proof of [CN23, Corollary 3.7] we begin by first observing that [CN23, Proposition 3.4] combined with (2.14) yields an isomorphism

$$
\hat{\mathcal{H}}^1(\mathcal{U}_{L_{\bullet}}^S) \cong \bigoplus_{\bar{p} \in \mathrm{Ram}(L_{\infty}/F) \setminus S} \hat{\mathcal{H}}^1(\mathcal{O}_{\mathfrak{P}, L_{\bullet}}^{\times})
$$

where $\mathcal{O}_{\mathfrak{P},L_{\bullet}}^{\times}$ denotes the normic system $(\mathcal{O}_{\mathfrak{P}^n, \iota_{\mathfrak{P}^m/\mathfrak{P}^n}, \mathfrak{N}_{\mathfrak{P}^m/\mathfrak{P}^n}})_{m \ge n \ge 0}$ of the local units at a sequence of primes $(\mathfrak{P}n)_{n \ge 0}$ above $\bar{p} = \mathfrak{P}_{\infty} \cap F$, analogous to the one studied in [CN23, §3.2]. Similarly, it follows from [CN23, Proposition 3.4] and the computations performed in the proof of Proposition 2.11 (analogous to those leading to (2.17) but replacing $D_n$-cohomology by $G_n$-cohomology) that

$$
\hat{\mathcal{H}}^2(\mathcal{U}_{L_{\bullet}}^S) \cong \bigoplus_{\bar{p} \in \mathrm{Ram}(L_{\infty}/F) \setminus S} \hat{\mathcal{H}}^2(\mathcal{O}_{\mathfrak{P}, L_{\bullet}}^{\times}) \oplus \bigoplus_{\bar{p} \in S} \hat{\mathcal{H}}^2(L_{\bullet}^{\times})
$$

where $L_{\mathfrak{P}, \bullet}^{\times}$ is the normic system $(L_{\mathfrak{P}, n}^{\times}, \iota_{\mathfrak{P}, m/\mathfrak{P}, n}, \mathfrak{N}_{\mathfrak{P}, m/\mathfrak{P}, n})_{m \ge n \ge 0}$. By [CN23, eq. (3.25)] and [CN23, Remark 3.6], there are $\mathbf{B}_D$-isomorphisms

$$
\hat{\mathcal{H}}^i(\mathcal{O}_{\mathfrak{P}, L, \bullet}^{\times}) \cong_{\mathbf{B}_D} \mathcal{Z}(\Gamma) \quad \text{and} \quad \hat{\mathcal{H}}^2(L_{\mathfrak{P}, \bullet}^{\times}) \cong_{\mathbf{B}_D} \mathcal{Z}(\Gamma)
$$

and therefore $\hat{\mathcal{H}}^i(\mathcal{U}_{L_{\bullet}}^S)$ is $\mathbf{B}_D$-isomorphic (for $i=1,2$) to the sum of finitely many copies of $\mathcal{Z}(\Gamma)$ (in detail, $|\mathrm{Ram}(L_{\infty}/F) \setminus S|$ copies when $i=1$ and $|\mathrm{Ram}(L_{\infty}/F)|$ when $i=2$), as needed to establish the lemma.
$\square$

As in [CN23, §4] working in the quotient category $\mathbf{D}\mathbf{S}_D^{\mathrm{co-f.g.}}/\mathbf{B}_D^{\mathrm{co-f.g.}}$ allows us to apply standard techniques in homological algebra in order to deduce, from the above lemma, the following

**Proposition 3.3.** For $\pm \in \{+, -\}$ and $i \in \{1, 2\}$ there are $\mathbf{B}_D$-isomorphisms $\hat{\mathcal{H}}^i(E_{L_{\bullet}}^S)^{\pm} \cong \mathcal{L}(\hat{\mathcal{H}}^i(E_{L_{\bullet}}^S))^{\pm}$. In particular, there exists integers $\lambda(0, E_{L_{\bullet}}^S)^{+}$ and $\lambda(1, E_{L_{\bullet}}^S)^{-}$ such that

$$
\hat{h}^0(D_n, E_{L_n}^S) \propto p^{n \cdot \lambda(0, E_{L_n}^S)^+} \quad \text{and} \quad \hat{h}^{-1}(D_n, E_{L_n}^S) \propto p^{n \cdot \lambda(1, E_{L_n}^S)^-},
$$

where $\propto$ is the relation of being eventually proportional introduced in [CN23, Definition 2.27].

*Proof.* The key point of the proof is to establish the existence of $\mathbf{B}_D$-isomorphisms

$$
\hat{\mathcal{H}}^i(E_{L_{\bullet}}^S) \cong_{\mathbf{B}_D} \mathcal{L}(\hat{\mathcal{H}}^i(E_{L_{\bullet}}^S)). \qquad (3.1)
$$

The existence of such isomorphisms follows, as in the proof of [CN23, Corollary 3.11], from [CN23, Lemma 3.10] once we establish the existence of a commutative diagram with exact rows

$$
\begin{tikzcd}
0 \arrow[r] & \hat{\mathcal{H}}^1(E_{L_{\bullet}}^S) \arrow[r] \arrow[d] & \hat{\mathcal{H}}^1(\mathcal{U}_{L_{\bullet}}^S) \arrow[r] \arrow[d] & \hat{\mathcal{H}}^1(Q_{L_{\bullet}}^S) \arrow[r] \arrow[d] & \hat{\mathcal{H}}^2(E_{L_{\bullet}}^S) \arrow[r] \arrow[d] & \hat{\mathcal{H}}^2(\mathcal{U}_{L_{\bullet}}^S) \\
\phantom{0} \arrow[r, "\|"'] & \phantom{\hat{\mathcal{H}}^1(E_{L_{\bullet}}^S)} & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} \\
\phantom{0} \arrow[r, "\|"'] & \phantom{\hat{\mathcal{H}}^1(E_{L_{\bullet}}^S)} & \arrow[r, "\cong"'] & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} \\
\phantom{0} \arrow[r, "\|"'] & \arrow[r, "\cong"'] & \arrow[r, "\|"'] & \arrow[r, "\cong"'] & \phantom{\arrow[r, "\|"'] \arrow[d]} & \phantom{\arrow[r, "\|"'] \arrow[d]} \\
\phantom{0} \arrow[r, "\|"'] & \phantom{\hat{\mathcal{H}}^1(E_{L_{\bullet}}^S)} & \phantom{\arrow[r, "\|"'] \arrow[d]} & \arrow[r, "\|"'] & \arrow[r, "\|"'] & \phantom{\arrow[r, "\|"'] \arrow[d]} \\
0 \arrow[r] & \mathcal{L}(\hat{\mathcal{H}}^1(E_{L_{\infty}}^S)) \arrow[r] & \mathcal{L}(\hat{\mathcal{H}}^1(\mathcal{U}_{L_{\infty}}^S)) \arrow[r] & \mathcal{L}(\hat{\mathcal{H}}^1(Q_{L_{\infty}}^S)) \arrow[r] & \mathcal{L}(\hat{\mathcal{H}}^2(E_{L_{\infty}}^S)) \arrow[r] & \mathcal{L}(\hat{\mathcal{H}}^2(\mathcal{U}_{L_{\infty}}^S))
\end{tikzcd}
$$

The existence of the diagram, as well as its exactness, was established in [CN23, Theorem 3.8] because it follows formally from taking cohomology in the commutative diagram (2.1); what required an arithmetic argument ibid., and we still need to prove, is that

a) the first arrow $\hat{\mathcal{H}}^1(E_{L_{\bullet}}^S) \dashrightarrow \hat{\mathcal{H}}^1(\mathcal{U}_{L_{\bullet}}^S)$ of the first row is $\mathbf{B}_D$-injective (which will, in turn, ensure that the first arrow $\mathcal{L}(\hat{\mathcal{H}}^1(E_{L_{\bullet}}^S)) \dashrightarrow \mathcal{L}(\hat{\mathcal{H}}^1(\mathcal{U}_{L_{\bullet}}^S))$ in the second row is $\mathbf{B}_D$-injective as well because the functor $\mathcal{L}$ is left-exact);
b) the second and last vertical arrows are isomorphisms;

c) the double system $(\mathrm{Ker}(Cl_{L_n}^S \to Cl_{L_\infty}^S), \iota_{L_m/L_n}^S, N_{L_m/L_n}^S)_{m \ge n \ge 0}$ belongs to the thick sub-category $\mathbf{B}_D$, ensuring the $\mathbf{B}_D$-injectivity of the third vertical arrow.

Point a) is obtained by replacing [Nuc10, Proposition 2.2] with our Lemma 2.4 in the argument *ibid.*
Point b) is the consequence of Lemma 3.2, while point c) follows by replacing Ozaki's original
result [Oza95, Proposition] with its generalisation [NSW08, Proposition 11.1.9] to S-class groups.

Once the isomorphism (3.1) is at our disposal, we can take $\pm$-parts to obtain $\mathbf{B}_D$-isomorphisms

$$ \hat{\mathcal{H}}^i(E_{\bullet, \cdot}^S)^{\pm} \cong_{\mathbf{B}_D} \mathcal{L}(\hat{\mathcal{H}}^i(E_{\bullet, \cdot}^S)^{\pm}), \quad (3.2) $$

as in [CN23, Corollary 4.4]. As a consequence of [CN23, Corollary 2.32], the isomorphisms in (3.2)
grant the existence of the constants $\lambda(i, E_{\bullet, \cdot}^S)^{\pm}$ appearing in the statement. $\square$

By combining Theorem 2.12 with the above proposition, we obtain the following

**Theorem 3.4.** There exist constants $\mu^{\mathrm{fk}}$, $\nu^{\mathrm{fk}} \in \mathbb{Z}[\frac{1}{2}]$ and $\lambda^{\mathrm{fk}} \in \mathbb{Z}$ such that the $p$-adic valuation of the order of $Cl_{K_n}^S$ satisfies

$$ v_p(|Cl_{K_n}^S|) = \mu^{\mathrm{fk}}p^n + \lambda^{\mathrm{fk}}n + \nu^{\mathrm{fk}} \quad \text{for all } n \gg 0. $$

*Proof.* Since $\Sigma$-cohomology groups have trivial $p$-Sylow, we can take $p$-adic valuations in Corollary 2.13 to obtain, through [CN20, Proposition 2.1], the equality

$$ 2v_p(|Cl_{K_n}^S|) = v_p(|Cl_{K_n}^S|) + \sum_{\mathfrak{p} \in S_s} v_p(d_{\mathfrak{p}}(L_n/k)) - v_p(\hat{h}^0(G_n, E_{L_n}^S)^+) + v_p(\hat{h}^1(G_n, E_{L_n}^S)^-) + \nu_0 $$

valid for all $n \ge 0$, where $\nu_0$ is a constant independent of $n$ — incorporating the contributions
coming from the $S$-class groups of $F$ and $k$ — and where $S_s$ is the set of primes in $S$ that split in
$F/k$. In light of Proposition 3.3 we can asymptotically replace the terms involving the cohomology
of the units with $n \cdot \lambda(i, E_{\bullet, \cdot}^S)^{\pm}$ (for the relevant value $i \in 1, 2$ and $\pm \in \{+, -\}$), obtaining

$$ 2v_p(|Cl_{K_n}^S|) = v_p(|Cl_{K_n}^S|) + \sum_{\mathfrak{p} \in S_s} v_p(d_{\mathfrak{p}}(L_n/k)) - \lambda(0, E_{\bullet, \cdot}^S)^+ \cdot n + \lambda(1, E_{\bullet, \cdot}^S)^- \cdot n + \nu_1 \quad \text{for } n \gg 0 \quad (3.3) $$

for a potentially new constant $\nu_1 \in \mathbb{Z}$. Given a prime $\mathfrak{p} \in S$, the local degree of the extension $L_\infty/k$ satisfies

$$ d_{\mathfrak{p}}(L_n/k) \propto \begin{cases} 1 & \text{if there are infinitely many primes in } L_\infty \text{ above } \mathfrak{p} \\ p^n & \text{else} \end{cases} $$

and therefore

$$ \sum_{\mathfrak{p} \in S_s} v_p(d_{\mathfrak{p}}(L_n/k)) \propto \tau \cdot n \quad (3.4) $$

where $\tau$ is the number of ideals such that $d_{\mathfrak{p}}(L_n/k) \propto p^n$. Finally, the classical $S$-generalised Iwasawa formula (see [Iwa73, Theorem 8] or<fnref n="4" /> [NSW08, p. 728, Remark 1]) for the growth of $S$-class numbers in the Galois $\mathbb{Z}_p$-extension $L_\infty/L$ provides three integers $\mu^{\mathrm{Iw}}, \lambda^{\mathrm{Iw}}, \nu^{\mathrm{Iw}}$ such that

$$ v_p(|Cl_{K_n}^S|) = \mu^{\mathrm{Iw}} \cdot p^n + \lambda^{\mathrm{Iw}} \cdot n + \nu^{\mathrm{Iw}} \quad \text{for all } n \gg 0. \quad (3.5) $$

<footnote n="4">⁴As a matter of fact, both references only contain a proof when $S$ coincides with the primes above $p$, but neither provides a full proof, only observing that the argument for regular class groups generalises because $p$-class groups are a quotient of the regular ones; and this is still true for $Cl_{\bullet, \cdot}^S$ independently of $S$. The cautious reader might want to stick to the case where $S$ is reduced to the primes dividing $p$.</footnote>
Combining (3.3), (3.4), (3.5) and setting $\mu^{\text{fk}} = \mu^{\text{Iw}}/2$, $\lambda^{\text{fk}} = (\lambda^{\text{Iw}} + \tau - \lambda(0, E_{L\bullet}^S))^+ + \lambda(1, E_{L\bullet}^S)^-)/2$, we obtain

$$v_p(|Cl_{K_n}^S|) = \mu^{\text{fk}}p^n + \lambda^{\text{fk}}n + \nu^{\text{fk}} \quad \text{for all } n \gg 0 \quad (3.6)$$

for a suitable $\nu^{\text{fk}} \in \mathbb{Z}[\frac{1}{2}]$. The argument to show that $\lambda^{\text{fk}}$ is actually integer is analogous to that of [CN23, Theorem 4.6]: pick $n$ large enough so that the formula in (3.6) holds, and consider the difference

$$v_p(|Cl_{K_{n+1}}^S|) - v_p(|Cl_{K_n}^S|):$$

it is clearly an integer and, by (3.6), equal to

$$\mu^{\text{fk}}p^n(p-1) + \lambda^{\text{fk}}.$$

Since $(p-1)$ is even, the summand $\mu^{\text{fk}}p^n(p-1)$ is an integer, so $\lambda^{\text{fk}} \in \mathbb{Z}$. $\square$

**3.2. The $\ell$-adic behaviour.** Fix in this section a prime number $\ell$ different from $p$. In order to study the $\ell$-adic behaviour of the orders $|Cl_{K_n}^S|$, we following [BL23, § 4.2], and we start by defining the following sequence of natural numbers (for $n \ge 0$):

$$f_n(\ell, p) = f_n = \text{order of } \ell \text{ in } (\mathbb{Z}/p^n)^\times.$$

As remarked *ibid.*, setting $a = v_p(\ell^{f_1} - 1)$, it is easy to check that these values satisfy

$$f_n = [\mathbb{Q}_\ell(\mu_{p^n}) : \mathbb{Q}_\ell] = \begin{cases} 1 & \text{if } n=0 \\ f_1 & \text{if } 1 \le n \le a \\ p^{n-a}f_1 & \text{if } n > a \end{cases} \quad (3.7)$$

where $\mu_{p^n}$ denotes the group of $p^n$-th roots of unity.

To define a second sequence of natural numbers that occur in the statement of Theorem 3.7 below we need some preliminaries.

**Notation 3.5.** For any field $\bullet \in \{k, F, L_n, K_n\}_{n \ge 0}$ let $A_{\bullet}^S$ be the $\ell$-Sylow $(Cl_{\bullet}^S)^{(\ell)}$ of the $S$-class group of $\bullet$. Denote by $e_{[n]}$, for $n \ge 0$, the idempotents in the topological $\mathbb{Z}_\ell$-algebra $\Lambda = \mathbb{Z}_\ell[\text{Gal}(L_\infty/F)]$ defined in [BL23, Theorem 2.1]; letting $X = \lim_{\leftarrow} A_{L_n}^S$ (where the transition morphisms are induced by the norm), write $X_{[n]}$ for the projection $X_{[n]} = e_{[n]}X$. We set

$$t_{[n]}^{\text{Iw}} = \frac{v_{\ell}(|X_{[n]}|)}{f_n} \quad (\text{for } n \ge 0) \quad \text{and} \quad t_{[n]}^{\text{fk}} = \begin{cases} \frac{v_{\ell}(|A_{K_n}^S / A_{K_{n-1}}^S|)}{f_n} & \text{for } n \ge 1 \\ \frac{t_{[0]}^{\text{Iw}}}{2} & \text{for } n = 0. \end{cases}$$

*Remark 3.6.* Thanks to [BL23, eq. (10)], one sees that $t_{[m]}^{\text{Iw}} \in \mathbb{Z}$ for all $m \ge 0$. In particular, the above values satisfy $f_m t_{[m]}^{\text{fk}} \in \mathbb{Z}$ for $m \ge 1$ and $f_0 t_{[0]}^{\text{fk}} = t_{[0]}^{\text{fk}} \in \mathbb{Z}[\frac{1}{2}]$.

Our main result is the following

**Theorem 3.7.** In the setting of Notation 3.5, for all $n \ge 0$ the $\ell$-adic valuation of the order $|A_{K_n}^S|$ satisfies

$$v_{\ell}(|A_{K_n}^S|) = \nu + \sum_{m=0}^{n} f_m t_{[m]}^{\text{fk}} = \nu + t_{[0]}^{\text{fk}} + f_1 \left( \sum_{m=1}^{a} t_{[m]}^{\text{fk}} + \sum_{m=a+1}^{n} p^{m-a} t_{[m]}^{\text{fk}} \right).$$

where $a = v_p(\ell^{f_1} - 1)$ and where $\nu \in \mathbb{Z}[\frac{1}{2}]$ is a constant independent of $n$. Moreover, the values $t_{[m]}^{\text{fk}}$ belong to $\mathbb{Z}[\frac{1}{2}]$ for every $m \ge 0$.
*Proof.* The second equality follows from the first via the relations in (3.7). Concerning the first equality, [BL23, Theorem 4.6] shows<fnref n="5" /> that, for all $n \ge 0$,

$$v_{\ell}(|A_{L_n}^S|) = \sum_{m=0}^{n} f_m t_{[m]}^{\text{Iw}}. \qquad (3.8)$$

The key ingredient is our Theorem 2.12: since for every prime $p$ in $S_s$, the local degree $p$ is a power of $p$, the $\ell$-adic valuation $v_{\ell}(d_p)$ is trivial and Theorem 2.12 implies that for all $n \ge 0$,

$$\begin{aligned} v_{\ell}(|A_{L_n}^S|) &= 2 \cdot v_{\ell}(|A_{K_n}^S|) + v_{\ell}(|A_F^S|) - 2 \cdot v_{\ell}(|A_k^S|) + v_{\ell}(\hat{h}^0(D_n, E_{L_n}^S)) - v_{\ell}(\hat{h}^{-1}(D_n, E_{L_n}^S)) \\ &= 2v_{\ell}(|A_{K_n}^S|) - 2\nu \end{aligned} \qquad (3.9)$$

where $\nu = v_{\ell}(|A_k^S|) - v_{\ell}(|A_F^S|)/2$ is independent of $n$; and where the two terms $\hat{h}^i(D_n, E_{L_n}^S)$ have trivial $\ell$-adic valuation since they are orders of cohomology groups of $D_n$ and thus are only divisible by 2 and $p$. Moreover, [BL23, Theorem 4.3] yields isomorphisms<fnref n="5" /> $X_{[m+1]} \cong A_{L_{m+1}}^S / A_{L_m}^S$ for all $m \ge 0$, and $X_{[0]} \cong A_F^S$; hence, for all $m \ge 0$,

$$\begin{aligned} t_{[m+1]}^{\text{Iw}} &= \frac{v_{\ell}(|X_{[m+1]}|)}{f_{m+1}} \\ &= \frac{v_{\ell}(|A_{L_{m+1}}^S|) - v_{\ell}(|A_{L_m}^S|)}{f_{m+1}} \\ &= \frac{2v_{\ell}(|A_{K_{m+1}}^S|) - 2\nu - 2v_{\ell}(|A_{K_m}^S|) + 2\nu}{f_{m+1}} \qquad (3.10) \\ &= \frac{2v_{\ell}(|A_{K_{m+1}}^S / A_{K_m}^S|)}{f_{m+1}} \\ &= 2t_{[m+1]}^{\text{fk}} \end{aligned}$$

Combining (3.9) with (3.8) and (3.10) (and recalling that, by definition, $t_{[0]}^{\text{Iw}} = 2t_{[0]}^{\text{fk}}$) one obtains that, for all $n \ge 0$,

$$\begin{aligned} 2v_{\ell}(|A_{K_n}^S|) &= v_{\ell}(|A_{L_n}^S|) + 2\nu \\ &= 2\nu + t_{[0]}^{\text{Iw}} + \sum_{m=1}^{n} f_m t_{[m]}^{\text{Iw}} \\ &= 2\nu + 2t_{[0]}^{\text{fk}} + \sum_{m=1}^{n} 2f_m t_{[m]}^{\text{fk}}. \end{aligned}$$

Dividing both members of the above equalities by 2 establishes the main equation in the statement of the theorem. That $t_{[m]}^{\text{fk}} \in \mathbb{Z}[\frac{1}{2}]$ follows from Remark 3.6 for $m = 0$ and from (3.10) for general $m \ge 1$. $\square$

### 3.3. p-adic convergence.
Building upon Theorem 3.4 and Theorem 3.7, we can obtain the following corollary, that is an analogue of [BL23, Theorem 4.8]:

**Corollary 3.8.** For every odd prime $q$, the sequence $|Cl_{K_n}^S|^{(q)}|^2$ of the squares of the orders of the $q$-Sylow subgroups of $Cl_{K_n}^S$ converges p-adically.

<footnote n="5">⁵ The main result *loc. cit.* is proved for $S = \emptyset$ but, as observed in [BL23, §5], the proof relies on general algebraic properties that extend straightforwardly to $S$-class groups for any finite $S$.</footnote>
*Proof.* For $q = p$ this follows from the formula for the growth of $p$-Sylow subgroups established in Theorem 3.4 (that actually proves that the sequence of the orders, and not only of their squares, converges $p$-adically). When $q \neq p$, we can apply Theorem 3.7 with $\ell = q$ (so $Cl_{K_n}^{S(q)} = A_{L_n}^S$) to find that for all $n \ge a$,

$$|A_{K_{n+1}}^S|^2 - |A_{K_n}^S|^2 = |A_{K_n}^S|^2 (\ell^{2 \cdot f_1 p^{n+1-a}} \ell_{[n+1]}^{f_k} - 1) = |A_{K_n}^S|^2 (\ell^{f_{n+1} \cdot 2 \ell_{[n+1]}^{f_k}} - 1). \quad (3.11)$$

As proven in Theorem 3.7, the term $2\ell_{[n+1]}^{f_k}$ is an integer. Since $\ell^{f_{n+1}} \equiv 1 \pmod{p^{n+1}}$ by definition, it follows that the right-hand side of (3.11) tends to 0 $p$-adically: therefore, the sequence of $|A_{K_n}^S|^2$ is Cauchy and its limit exists, finishing the proof of the corollary. $\square$

## REFERENCES

[ABK$^{+}$24] Dagur Asgeirson, Riccardo Brasca, Nikolas Kuhn, Filippo A. E. Nuccio Mortarino Majno di Capriglio, and Adam Topaz, *Categorical foundations of formalized condensed mathematics*, The Journal of Symbolic Logic (2024), 1–28.

[BDNN21] Anne Baanen, Sander R. Dahmen, Ashvni Narayanan, and Filippo A. E. Nuccio Mortarino Majno di Capriglio, *A Formalization of Dedekind Domains and Class Groups of Global Fields*, 12th International Conference on Interactive Theorem Proving (ITP 2021) (Dagstuhl, Germany) (Liron Cohen and Cezary Kaliszyk, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 193, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2021, pp. 5:1–5:19.

[BDNN22] ——, *A formalization of Dedekind domains and class groups of global fields*, J. Automat. Reason. **66** (2022), no. 4, 611–637.

[BL23] Andrea Bandini and Ignazio Longhi, *The algebra $Z_\ell[[Z_p^d]]$ and applications to Iwasawa theory*, arXiv:2312.04666v1, 2023, version 1.

[BN18] Nicolas Billerey and Filippo A. E. Nuccio Mortarino Majno di Capriglio, *Représentations galoisiennes diédrales et formes à multiplication complexe*, J. Théor. Nombres Bordeaux **30** (2018), no. 2, 651–670.

[CN20] Luca Caputo and Filippo A. E. Nuccio Mortarino Majno di Capriglio, *Class number formula for dihedral extensions*, Glasgow Math. J. **62** (2020), no. 2, 323–353.

[CN23] ——, *Cohomology of normic systems and fake $Z_p$-extensions*, New York J. Math. **29** (2023), 1196–1272.

[CS10] John Coates and Ramdorai Sujatha, *Galois cohomology of elliptic curves*, second ed., Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2010.

[dFF22] María Inés de Frutos-Fernández, *Formalizing the Ring of Adèles of a Global Field*, 13th International Conference on Interactive Theorem Proving (ITP 2022) (Dagstuhl, Germany) (June Andronick and Leonardo de Moura, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 237, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022, pp. 14:1–14:18.

[dFFN24] María Inés de Frutos-Fernández and Filippo A. E. Nuccio Mortarino Majno di Capriglio, *A formalization of complete discrete valuation rings and local fields*, Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs (New York, NY, USA), CPP 2024, Association for Computing Machinery, 2024, p. 190–204.

[Iwa73] Kenkichi Iwasawa, *On $Z_l$-extensions of algebraic number fields*, Ann. Math. (1973), 246–326.

[KL23] Debanjana Kundu and Antonio Lei, *Growth of $p$-parts of ideal class groups and fine Selmer groups in $Z_q$-extensions with $p \neq q$, Acta Arith. **207** (2023), no. 4, 297–313.

[KNS24] Debanjana Kundu, Filippo A. E. Nuccio Mortarino Majno di Capriglio, and Ramdorai Sujatha, *Structure of fine Selmer groups in abelian $p$-adic Lie extensions*, Osaka Journal of Mathematics **61** (2024), no. 1, 121–146.

[Liv23] Amelia Livingston, *Group Cohomology in the Lean Community Library*, 14th International Conference on Interactive Theorem Proving (ITP 2023) (Dagstuhl, Germany) (Adam Naumowicz and René Thiemann, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 268, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2023, pp. 22:1–22:17.

[mC20] The mathlib Community, *The Lean mathematical library*, Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, POPL '20, ACM, January 2020.

[NOR23] Filippo A. E. Nuccio Mortarino Majno di Capriglio, Tadashi Ochiai, and Jishnu Ray, *A formal model of Coleman families and applications to Iwasawa invariants*, Annales Mathematiques du Quebec (2023), doi.10.1007/s40316-023-00217-0.
[NS23] Filippo A. E. Nuccio Mortarino Majno di Capriglio and Ramdorai Sujatha, *Residual supersingular Iwasawa theory and signed Iwasawa invariants*, Rend. Semin. Mat. Univ. Padova **149** (2023), 83–129.

[NSW08] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, *Cohomology of number fields*, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer-Verlag, Berlin, 2008.

[Nuc10] Filippo A. E. Nuccio Mortarino Majno di Capriglio, *Cyclotomic units and class groups in $Z_p$-extensions of real abelian fields*, Math. Proc. Cambridge Philos. Soc. **148** (2010), no. 1, 93–106.

[Oza95] Manabu Ozaki, *A note on the capitulation in $Z_p$-extensions*, Proc. Japan Acad. Ser. A Math. Sci. **71** (1995), no. 9, 218–219.

[Rio25] Joël Riou, *Formalization of derived categories in Lean/mathlib*, Annals of Formalized Mathematics (2025), to appear, https://hal.science/hal-04546712v4.

[Rub88] Karl Rubin, *On the main cojecture of Iwasawa theory for imaginary quadratic fields.*, Inventiones mathematicae **93** (1988), no. 3, 701–714.

[Ser67] Jean-Pierre Serre, *Local class field theory*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Academic Press, London, 1967, pp. 128–161.

[Tat67] John Tate, *Global class field theory*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162–203.

[Wal77] Colin D. Walter, *A class number relation in Frobenius extensions of number fields*, Mathematika **24** (1977), no. 2, 216–225.

Email address: filippo.nuccio@univ-st-etienne.fr

UNIVERSITÉ JEAN MONNET, CNRS, ECOLE CENTRALE DE LYON, INSA LYON, UNIVERSITE CLAUDE BERNARD LYON 1, ICJ UMR5208, 42023 SAINT-ETIENNE, FRANCE.