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 Izvestiya: Mathematics, 2022, Volume 86, Issue 6, Pages 1143–1161DOI: https://doi.org/10.4213/im9241e (Mi im9241)

Arithmetic of certain $\ell$-extensions ramified at three places. III

L. V. Kuz'min

National Research Centre "Kurchatov Institute", Moscow

### § 1. Introduction

Let $\ell$ be a regular odd prime number, $k=\mathbb Q(\zeta_0)$, where $\zeta_0$ is a primitive root of unity of degree $\ell$ and $K=k(\sqrt[\ell]{a}\,)$, where $a$ is a natural number of the form (3.1) such that its prime divisors $p_1$, $p_2$, $p_3$ remain prime in the cyclotomic $\mathbb Z_\ell$-extension $k_\infty$ of $K$. The arithmetic of $K$, which has many interesting features, was a subject of our investigation in [1] and [2]. So, it appeared that the $\ell$-component of the class group $\operatorname{Cl}(K)_\ell$ of $K$ is non-zero. It has an order at most $\ell^{\ell-1}$, and its period is $\ell$. In the simplest case $\ell=3$ there were found in empirical way the fields $K$, such that their places over $\ell$, generate a subgroup of order $\ell$ in the class group $\operatorname{Cl}(K)_\ell$ (the fields of the type A1 in terminology of [2]) and the fields such that all their places over $\ell$ are principal divisors (the fields of the type A2). Essentially nothing was known in the case $\ell>3$, though it was proved in [1] that in the case of infinite Tate module $T_\ell(K_\infty)$ of $K_\infty$ the generator $\gamma_0$ of the Galois group $\Gamma=G(K_\infty/K)$, which acts on the roots of unity of the $\ell$-primary degree by the rule $\gamma_0(\zeta_n)=\zeta_n^{\varkappa(\gamma_0)}$, where $\varkappa(\gamma_0)\in \mathbb Z_\ell$, acts on $T_\ell(K_\infty)$ by multiplication by $\sqrt{\varkappa(\gamma_0)}$ [1], Theorem 5.1.

It was proved in [2] that in the case of finite Tate module $T_\ell(K_\infty)$ there is some greater group $E'(K_\infty)$, which contains $T_\ell(K_\infty)$, such that $\gamma_0$ acts on it also by multiplication by $\sqrt{\varkappa(\gamma_0)}$.

In the present paper we use a new method to study the arithmetic of $K$. Namely, simultaneously with $K$ we consider another field $L$ of the form (3.2) such that there are only two places $p_1$ and $p_2$ ramified in $L/k$. The arithmetic of $L$ is simpler than that of $K$. Moreover, we can fix the field $L$ and change a prime number $p_3$. Thus, we consider a family of the fields $K=K(p_3)$, and this consideration leads to a series of interesting consequences.

In § 3 we consider the Galois cohomologies groups $H^i(G(L/k),U(L))$ and $H^i(G(K/k), U(K))$. We define some critically important subgroup $\overline U_2(L)$ or $\overline U_2(K)$ of the group of units $U(L)$ or $U(K)$ and prove that $\overline U_2(L)$ is a cohomologically trivial module. Concerning the field $K$, in the present moment we cannot say whether the module $\overline U_2(K)$ may be not cohomologically trivial. In the present paper we use only results that take place for $L$, but since the proofs for $L$ and for $K$ are just the same, we give both proofs, thou the results for $K$ will be used only in the next our paper.

In § 4 we consider some general properties of $G$-modules, where $G=G(K/\mathbb Q)$ and $G=\widetilde G=G(L/\mathbb Q)$. We apply these results for characterization of the Galois module $\mathcal{A}(L)/\mathcal{A}(L)^\ell$ (Theorem 4.1), where $\mathcal{A}(L)=\prod_{v|\ell}U^{(1)}(L_v)$ and $U^{(1)}(L_v)$ is the group of principal units of the local field $L_v$.

In § 5 we apply the obtained results for characterization of some Abelian extensions of the field $L$. Namely, let $N=N(p_3)$ be the maximal Abelian $\ell$-extension of $L$ such that $N\supset K$, $N/K$ is unramified, and the Galois group $G(N/L)$ is of period $\ell$. Using the obtained information about the group $\mathcal{A}(L)/\mathcal{A}(L)^\ell$, we can calculate the Kummer group of the extension $N/L$, and thus, determine its degree. It appears that always we have $[N:L]=\ell^r$ with an even $r$ and $2\leqslant r\leqslant \ell-1$. At last, using the Chebotarev density theorem, we prove that any possible value of $r$ realize for infinitely many values of $p_3$.

If $r<\ell-1$ then all the places over $\ell$ split completely in the extension $N/L$. In the case $r=\ell-1$, using the Chebotarev theorem, we prove that there are infinitely many $p_3$ such that $r=\ell-1$ and all places over $\ell$ split completely in $N/L$ and infinitely many $p_3$ such that $r=\ell-1$ and all places over $\ell$ have inertia degree $\ell$ in $N/L$.

In § 6 we apply the results obtained for the extensions $N/L$ to the extensions of $K$. We prove that for any prime $p_3$ the order of the $\ell$-class group of $K=K(p_3)$ is at least $\ell^2$ (Theorem 6.1). In the case $\ell=3$ we prove that there are infinitely many $p_3$ such that $K(p_3)$ is of the type A1 and infinitely many $p_3$ such that $K(p_3)$ is of the type A2.

### § 2. Notations and definitions

We try to follow the notations of [1], [2]. For a regular odd prime $\ell$, let $\zeta_n$ be a primitive root of unity of degree $\ell^{n+1}$. Let $k=\mathbb Q(\zeta_0)$, $k_n=\mathbb Q(\zeta_n)$ and $k_\infty=\bigcup_{n=1}^\infty k_n$ be the cyclotomic $\mathbb Z_\ell$-extension of $k$. The prime number $p\neq\ell$ remains prime in the field $k_\infty$ if and only if $p$ is a primitive root modulo $\ell^2$. We denote the set of all such primes $p$ by $\mathbb P_0$. We consider two fields $K$ and $L$ defined by (3.1) and (3.2), respectively. We put $H=G(K/k)$ and $\widetilde H=G(L/k)$, where $G(K/k)$ means the Galois group of $K/k$. Thus, $K$ and $L$ are Galois extensions of degree $\ell(\ell-1)$ with isomorphic Galois groups $G=G(K/\mathbb Q)$ and $\widetilde G=G(L/\mathbb Q)$. The Galois group $\Delta=G(k/\mathbb Q)$ acts on $H$ (or on $\widetilde H$) via the Teichmüller character $\omega\colon\Delta\to(\mathbb Z/\ell\mathbb Z)^\times$.

By groups of cohomologies we mean everywhere the Tate groups if the opposite is not said explicitly. We denote by $S$ the set of all places over $\ell$ of the field under consideration. By $\operatorname{Cl}(K)_\ell$ we denote the $\ell$-component of the class group of $K$, and $\operatorname{Cl}_S(K)_\ell$ means a factorgroup of $\operatorname{Cl}(K)_\ell$ by the subgroup generated by all places in $S$.

Let $\overline{\mathbf N}$ be the maximal Abelian unramified $\ell$-extension of $K_\infty$ and $\mathbf{N}$ the maximal subfield of $\overline{\mathbf N}$ such that all places over $\ell$ split in $\mathbf{N}/K_\infty$. We denote the Galois groups of $\overline{\mathbf N}/K_\infty$ and $\mathbf{N}/K_\infty$ by $\overline T_\ell(K_\infty)$ and $T_\ell(K_\infty)$ respectively. These groups are modules with respect to the action of the Galois group $\Gamma=G(K_\infty/K)$. For a field $K$ or its completion $K_v$, we denote by $U(K)$ and $U(K_v)$ the unit group of the corresponding field. By $\mu_\ell(K)$ we denote the group of all $\ell$-primary roots of unity in $K$. For a local field $K_v$, we denote by $U^{(1)}(K_v)$ the group of principal units in $K_v$. By $D(K)$ we denote the group of divisors of $K$. For an Abelian group $A$, we denote by $A[\ell]$ the pro-$\ell$-completion of $A$. By $\overline U(K)$ we denote the group $U(K)/\mu(K)$, where $\mu(K)$ is the group of all roots of unity in $K$. By $\mathcal{A}(K)$ we denote the group $\prod_{v\in S} U^{(1)}(K_v)$ and by $\overline{\mathcal{A}}(K)$ the group $\prod_{v\in S}\overline U^{\,(1)}(K_v)$, where $\overline U^{\,(1)}(K_v)=U^{(1)}(K_v)/\mu_\ell(K_v)$ and $\mu_\ell(K_v)$ is the $\ell$-component of $\mu(K_v)$.

We shall often use additive notations for multiplication without special reminding, since we do not use addition in this paper.

### § 3. Cohomologies of certain groups of units

So, let $p_1,p_2,p_3\in \mathbb P_0$ and $K=k(\sqrt[\ell]{a}\,)$, where

$$$$a=p_1^{r_1}p_2^{r_2}p_3^{r_3},\qquad r_1r_2r_3\not\equiv 0\pmod \ell,\qquad a^{\ell-1}\equiv 1\pmod{\ell^2}.$$ \tag{3.1}$$
Then $K/k$ is an extension described in [1], Proposition 2.1. This extension ramifies at $\mathfrak {p}_1$, $\mathfrak {p}_2$ and $\mathfrak {p}_3$, where $\mathfrak{p}_i=(p_i)$ for $i=1,2,3$, and any place $v$ of $K$ over $\ell$ splits in this extension. The analogous result holds also for any $n$ for the extension $K_n/k_n$, where $K_n=K\cdot k_n$. The places $\mathfrak p_i$ remain prime in $k_n$ and ramify in the extension $K_n/k_n$. In $k_n$ there is the single place $v_n$ over $\ell$, and $v_n$ splits completely in $K_n/k_n$.

In parallel with $K$ we shall consider the field $L\colon =k(\sqrt[\ell]{b}\,)$, where

$$$$b=p_1^{s_1}p_2^{s_2},\qquad s_1s_2\not\equiv 0\pmod{\ell}, \qquad b^{\ell-1}\equiv 1\pmod{\ell^2}.$$ \tag{3.2}$$
The extension $L/k$ ramify in $\mathfrak p_1$, $\mathfrak p_2$, and the place $v$ splits completely in it.

As it was shown in [1], Proposition 2.1, for a given triple of primes $p_1,p_2,p_3\in \mathbb P_0$ there are $\ell-2$ different fields $K$ corresponding to this choice of $r_1$, $r_2$, $r_3$. On the contrary, the numbers $p_1$, $p_2$ define $L$ uniquely. The next assertion is an analogue of Theorem 4.1 of [1].

Proposition 3.1. The following equalities hold: $\overline T_\ell(L_\infty)=T_\ell(L_\infty)=0$, $\operatorname{Cl}(L)_\ell=\operatorname{Cl}_S(L)_\ell=0$.

Proof. Consider an $\widetilde H$-module $\overline T_\ell (L_\infty)$. Let $f$ be the minimal number of its generators as $\widetilde H$-module. Then $f$ equals the minimal number of generators of the Abelian group $C:=\overline T_\ell(L_\infty)/(\widetilde\sigma-1)\overline T_\ell(L_\infty)$. As in [1], this implies that $C$ is isomorphic to $\mathbb F^f_\ell$ as an Abelian group.

Let $\overline N$ be the extension of $L_\infty$, whose Galois group is naturally isomorphic to the group $\overline T_\ell(L_\infty)$. Let $\overline N_0$ be the subfield of $\overline N$ fixed by the action of the Galois group $(\widetilde\sigma-1)\overline T_\ell(L_\infty)$, where $\widetilde \sigma$ is a generator of the group $\widetilde H$. Then, as in [1], we obtain that $G(\overline N_0/k_\infty)$ is an Abelian group of period $\ell$ and of order $\ell^{f+1}$. To calculate $f$, we use the Kummer theory. Let $M$ be a subfield of $\overline N_0$ of degree $\ell$ over $L_\infty$. Then $M$ is of the form $M=k_\infty(\sqrt[\ell]{\alpha}\,)$. Applying to the field $M$ the same arguments that were used in the proof of Theorem 4.1 of [1], we get that the element $\alpha$ is defined uniquely up to raising in some power prime to $\ell$ and multiplication by some $\ell$-th power. It means that $M=\overline N_0$. Then $\overline N_0=L_\infty$ and $f=0$. Therefore, $\overline T_\ell(L_\infty)=0$. Since there are epimorphisms $\overline T_\ell(L_\infty)\to T_\ell(L_\infty)$, $\overline T_\ell(L_\infty)\to\operatorname{Cl}(L)_\ell$ and $\overline T_\ell(L_\infty)\to\operatorname{Cl}_S(L)_\ell$, we get $T_\ell(L_\infty)=0$ and $\operatorname{Cl}(L)_\ell=\operatorname{Cl}_S(L)_\ell=0$. This proves Proposition 3.1.

Remark. We can give more short but less direct proof of Proposition 3.1. Namely, we shall prove that $\overline T_\ell(L_\infty)=0$. Applying the Riemann–Hurwitz formula to the extension $L_\infty/k_\infty$ (see, for example, [1], Theorem 1.1), we obtain $d(L_\infty)=\lambda'(L_\infty)=\lambda''(L_\infty)=r''(L_\infty)=0$. Therefore, the module $\overline T_\ell(L_\infty)$ is at most finite and $D(L_\infty)=0$ (the definition of $D(L_\infty)$ one can find in [1], § 6). Then the module $E'(L_\infty)$ defined in [1], Proposition 6.3, also vanishes. According to [2], Theorem 3.1, the module $E'(L_\infty)$ contains a submodule $E'_2(L_\infty)$, which is isomorphic to $\overline T_\ell(L_\infty)$. Hence, $\overline T_\ell(L_\infty)=0$.

Our nearest goal is to study the groups of cohomologies $H^i(H,U(K))$ and $H^i(\widetilde H,U(L))$, and also the cohomologies of some subgroups and factorgroups of these groups of units. The exact sequence of $H$-modules

$$\begin{equation*} 1\to U(K)\to K^\times\to D_K^0\to 1, \end{equation*} \notag$$
where $D_K^0$ is the group of principal divisors of $K$, induces an exact sequence of cohomologies
$$$$1\to U(k)\to k^\times\to (D_K^0)^H\xrightarrow{\eta} H^1(H,U(K))\to 1.$$ \tag{3.3}$$
By Theorem 90 of Hilbert any element of $H^1(H,U(K))$ may be presented by a cocycle $f\colon H\to U(K)$ of the form $f=f_c$, where $f_c(h)=h(c)/c$, $c\in K^\times$ and the principal divisor $(c)$ is fixed under the action of $H$. Any such $(c)$ is of the form $(c)=a_1a_2$, where $a_1\in D(k)$ and $a_2$ is a product of some prime divisors of $K$ ramified in $K/k$. In particular, the group $H^1(H,U(K))$ contains an element, which is presented by the cocycle $f_{\overline a\,}(h)$, where $\overline a=\sqrt[\ell]{a}$ and $a$ is as in (3.1). Analogously, the group $H^1(\widetilde H, U(L))$ contains an element presented by $f_{\overline b\,}(h)$, where $\overline b=\sqrt[\ell]{b}$ and $b$ is the element in (3.2). These cocycles take their values in the groups $\mu_\ell(K)$ and $\mu_\ell(L)$ respectively.

Proposition 3.2. The inclusion $\mu_\ell(K)\hookrightarrow U(K)$ induces inclusions

$$$$i_1\colon H^1(H,\mu(K))\hookrightarrow H^1(H,U(K)),\qquad i_2\colon H^0(H,\mu_\ell(K))\hookrightarrow H^0(H,U(K)).$$ \tag{3.4}$$
Analogously, the inclusion $\mu_\ell(L)\hookrightarrow U(L)$ induces inclusions
$$$$j_1\colon H^1(\widetilde H,\mu_\ell(L))\hookrightarrow H^1(\widetilde H,U(L)),\qquad j_2\colon H^0(\widetilde H,\mu_\ell(L))\hookrightarrow H^0(\widetilde H,U(L)).$$ \tag{3.5}$$

Proof. The class $\operatorname{cls}(f_{\overline a\,}(h))$ of the cocycle $f_{\overline a\,}(h)$ generates the group $H^1(H,\mu_\ell(K))$. Suppose that $i_1(\operatorname{cls}(f_{\overline a\,}(h)))=0$. It means that there is a unit $u\in U(K)$ such that $h(u)/u=h(\overline a)/\overline a$ for any $h\in H$. Then $u/\overline a\in k^\times$, $u_1:=u^\ell\in k$ and $K=k(\sqrt[\ell]{u}\,)$, but this is impossible. Indeed, the place $v$ of $K$ over $\ell$ splits completely in $K/k$, but any unit $u_1$ of $K$, which is an $\ell$-th power in the local field $k_v$, is also an $\ell$-th power in $K$ (otherwise $k(\sqrt[\ell]{u_1}\,)/k$ would be an unramified extension of degree $\ell$). This proves that the map $i_1$ in (3.4) is an inclusion. The monomorphic character of $j_1$ in (3.5) may be proved in the same way.

To prove the monomorphic character of $i_2$ in (3.4), it is enough to check that $\zeta_0$ is not a norm in the extension $K/k$. To do this, we note that $\mathfrak p_1=(p_1)$ ramifies in $K/k$ and also in the extension of local fields $K_{\mathfrak P_1}/k_{\mathfrak p_1}$, where $\mathfrak P_1$ is a prime divisor of $\mathfrak p_1$ in $K$. By local class field theory it means that

$$$$\bigl(U(k_{\mathfrak p_1}):N(U(K_{\mathfrak P_1}))\bigr)=\ell,$$ \tag{3.6}$$
where $N$ means the norm map from $K_{\mathfrak P_1}$ into $k_{\mathfrak p_1}$. Since $\mathfrak p_1$ remains prime in $k_\infty$, the field $k_{\mathfrak p_1}$ does not contain the primitive root of unity $\zeta_1$ of degree $\ell^2$, that is, the $\ell$-component of $U(k_{\mathfrak p_1})$ is generated by $\zeta_0$. Then (3.6) means that $\zeta_0$ is not a norm in $K_{\mathfrak P_1}/k_{\mathfrak p_1}$ and, moreover, in $K/k$. This proves the monomorphic character of $j_1$ in (3.5). The monomorphic character of $j_2$ may be proved in the same way. This proves Proposition 3.2.

For local or global algebraic number field $F$ we shall denote by $\overline U(F)$ the factor group $U(F)/\mu(F)$.

Proposition 3.3. For $i=0,1$ the equalities hold

$$$$|H^i(H,\overline U(K))| =\ell^{-1}|H^i(H,U(K))|,$$ \tag{3.7}$$
$$$$|H^i(\widetilde H,\overline U(L)| =\ell^{-1}|H^i(\widetilde H,U(L))|.$$ \tag{3.8}$$

Proof. The exact sequence of $H$-modules
$$\begin{equation*} 1\to \mu(K)\to U(K)\to \overline U(K)\to 1 \end{equation*} \notag$$
induces an exact sequence of cohomologies
\begin{aligned} \, &\cdots\to H^0(H,\mu(K))\xrightarrow{\alpha}H^0(H,U(K))\to H^0(H,\overline U(K))\to H^1(H,\mu(K)) \nonumber \\ &\qquad\xrightarrow{\beta}H^1(H,U(K))\to H^1(H,\overline U(K))\xrightarrow{\gamma} H^2(H,\mu(K))\to\cdots\,. \end{aligned} \tag{3.9}
Since $H^i(H,\mu(K))=H^i(H,\mu_\ell(K))$ for any $i$ it follows from Proposition 3.2 that $\alpha$ and $\beta$ are injections. So, for any $i$ and any $H$-module $A$ there is a natural isomorphism $H^i(H,A)\cong H^{i+2}(H,A)$. Hence the sequence (3.9) is periodic with period $2$, and injectivity of $\alpha$ yields $\gamma=0$. Therefore, (3.9) induces short exact sequences
$$\begin{equation*} \begin{gathered} \, 0\to H^0(H,\mu(K))\xrightarrow{\alpha}H^0(H,U(K))\to H^0(H,\overline U(K))\to 0, \\ 0\to H^1(H,\mu(K))\xrightarrow{\beta} H^1(H,U(K))\to H^1(H,\overline U(K))\to 0, \end{gathered} \end{equation*} \notag$$
whence it follows the assertion of the proposition for $K$. The proof for $L$ is just the same. This proves Proposition 3.3.

Let $F$ be an arbitrary algebraic number field, $F_v$ the completion of $F$ relative to some place $v$ over $\ell$ and $F_{v,\infty}$ the cyclotomic $\mathbb Z_\ell$-extension of $F_v$. Put $\Gamma_v=G(F_{v,\infty}/F_v)$.

If the extension $F_{v,\infty}/F_v$ is purely ramified then by the local class field theory we have the canonical epimorphism $U^{(1)}(F_v)\to \Gamma_v$, whose kernel coincides with the group $\mathcal{U}(F_v)$ of universal norms from all groups $U^{(1)}(F_{v,n})$ in the extension $F_{v,\infty}/F_v$, that is, $\mathcal{U}(F_v)=\cap_n N_n(U^{(1)}(F_{v,n}))$, where $N_n$ means the norm map from $F_{v,n}$ into $F_v$.

As in [2], § 3, we denote by $P(F)$ the kernel of the natural map $\chi_F\colon \!\! \prod_{v\in S}\Gamma_v\to \Gamma$, where $\Gamma=G(F_\infty/F)$, and $\Gamma_v$ inserts in $\Gamma$ as a decomposition subgroup of $v$. Thus, if all places over $\ell$ purely ramify in $F_\infty/F$ then there are exact sequences

$$$$1\to P(F)\to \prod_{v\in S}\Gamma_v\xrightarrow{\chi_F}\Gamma\to 1,$$ \tag{3.10}$$
$$$$1\to \prod_{v\in S}\mathcal{U}(F_v)\to \prod_{v\in S}U^{(1)}(F_v)\xrightarrow{\pi_F}\prod_{v\in S}\Gamma_v\to 1.$$ \tag{3.11}$$
If $F$ is normal over $\mathbb Q$ then all groups entering (3.10) and (3.11) are $G(F/\mathbb Q)$-modules.

The diagonal inclusion $U(F)[\ell]\hookrightarrow P(F)$ combined with (3.11) induce a map $\varphi_F\colon U(F)[\ell]\to P(F)$, whose kernel, which we denote by $U_2(F)$, – is the subgroup of all local universal norms from $F_\infty$ in the group $U(F)[\ell]$. In particular, in the cases $F=K$ or $F=L$ we have the maps $\varphi_K$, $\varphi_L$ and the groups $U_2(K)$, $U_2(L)$.

Proposition 3.4. The maps $\varphi_K$ and $\varphi_L$ are epimorphisms. In other words, there are exact sequences

$$$$1\to U_2(K)\to U(K)[\ell]\to P(K)\to 1,$$ \tag{3.12}$$
$$$$1\to U_2(L)\to U(L)[\ell]\to P(L)\to 1.$$ \tag{3.13}$$

Proof. For $K$ the desired assertion was already proved in [2], Proposition 4.2. For $L$ the proof is even simpler. Let $M$ be the maximal Abelian $\ell$-extension of $L$ unramified out of $\ell$, and such that for any $v\in S$ the field $M_v$ contains in the cyclotomic $\mathbb Z_\ell$-extension of $L_w$, where $w$ is a place of $L$ under $v$. Then, taking into account, that $\operatorname{Cl}(L)_\ell=0$ by Proposition 3.1 and applying the class field theory, we get that $G(M/L_\infty)\cong P(L)/\operatorname{Im}\varphi_L$, but according to Proposition 3.1 we have $\overline T_\ell(L_\infty)=0$, whence it follows $M=L_\infty$. This proves Proposition 3.4.

In each of the extensions $K/k$ and $L/k$ the place $v\in S$ of $K$ splits completely, hence (3.10) yields

$$$$P(K)\cong I_H,\qquad P(L)\cong I_{\widetilde H},$$ \tag{3.14}$$
where $I_H$ and $I_{\widetilde H}$ are the ideals of augmentation of the group rings $\mathbb Z_\ell[H]$ and $\mathbb Z_\ell[\widetilde H]$ respectively. In particular, $H^0(H,P(K))=0$ and $H^{-1}(H,P(K))\cong \mathbb Z_\ell/\ell \mathbb Z_\ell$. The analogous result holds for $P(L)$.

We put $\overline U_2(K)=U_2(K)/\mu_\ell(K)$ and $\overline U_2(L)=U_2(L)/\mu_\ell(L)$.

Proposition 3.5. Let $\mathfrak P_1$ be a prime divisor in $K$ (or in $L$) of the divisor $\mathfrak p_1=(p_1)$, where $p_1$ is a prime number entering in $a$ in (3.1) (or in $b$ in (3.2)). Let the order of $\mathfrak P_1$ in the group $\operatorname{Cl}(K)$ (respectively, in the group $\operatorname{Cl}(L))$ is prime to $\ell$. (Note that the last condition always holds for $L$ because of Proposition 3.1.) Then the following equalities hold

\begin{equation*} \begin{alignedat}{2} |H^1(H,\overline U_2(K))|&=\ell^{-1}|H^1(H,\overline U(K))|,&\quad |H^0(H,\overline U_2(K))|&=|H^0(H,\overline U(K))|, \\ |H^1(\widetilde H,\overline U_2(L))|&=\ell^{-1}|H^1(\widetilde H,\overline U(L))|=1,&\quad |H^0(\widetilde H,\overline U_2(L))|&= |H^0(\widetilde H,\overline U(L))|=1. \end{alignedat} \end{equation*} \notag

Proof. Let $x\in K^\times$ be such an element that $(x)=\mathfrak P_1^h$ for some $h$ prime to $\ell$. The element $x$ is defined up to multiplication by an arbitrary unit in $U(K)$. Then $x$, as well as $p_1$, is a unit in the local field $K_v$ for any $v$ over $\ell$. Therefore, under the diagonal inclusion $K^\times\hookrightarrow \prod_{v\in S}K_v^\times$ the elements $x$ and $p_1$ go into $\prod_{v\in S} U(K_v)$, so, we can consider they as elements of the group $\prod_{v\in S}U(K_v)[\ell]=\prod_{v\in S}U^{(1)}(K_v)$. Obviously, $N_{K/k}(x)=p_1^hu$, where $u$ is a unit in $U(K)$. Let $x_1=\pi_k(x)$, where $\pi_k$ is the map from (3.11) for $K$.

Since $\mathfrak P_1^\ell=\mathfrak p_1=(p_1)$ we obtain that in the group $\Gamma$ in (3.10) holds the relation $\chi_K(\pi_K(x)^\ell))=\chi_K(\pi_K(p_1^h))$, but $\chi_K(\pi_K(p_1^h))$ generates the group $\Gamma^\ell$. Hence the element $y=\chi_K(\pi_K(x))$ generates the group $\Gamma$. If $y'$ is some lifting of $y$ in $\prod_{v\in S}\Gamma_v$ then $y'$ generates this group as an $H$-module.

Let $z$ be the image of $(\sigma-1)(x)$ in $\overline U(K)$. Then $z$ defines some element of the group $H^{-1}(H,\overline U(K))$, moreover, $\pi_K(z)$ generates the group $H^{-1}(H,P(K))\cong H^1(H,P(K))$. Therefore, the natural map $H^{-1}(H,\overline U(K)[\ell])\to H^{-1}(H,P(K))$, induced by the map $\varphi_K$ of (3.11), is an epimorphism. Then the exact cohomological sequence for (3.12) yields exact sequences

$$\begin{equation*} \begin{gathered} \, 0=H^{-2}(H,P(K))\to H^{-1}(H,\overline U_2(K))\to H^{-1}(H,\overline U(K)[\ell]\to H^{-1}(H,P(K))\,{\to}\, 0, \\ 0\to H^0(H,\overline U_2(K))\to H^0(H,\overline U(K)[\ell])\to H^0(H,P(K))=0. \end{gathered} \end{equation*} \notag$$
This proves the proposition for $K$. The proof for $L$ is quite analogous (with using the exact sequence (3.13)). This proves Proposition 3.5.

Theorem 3.1. There are two possibilities for $K$.

(A) All three divisors $\mathfrak{P}_1$, $\mathfrak{P}_2$, $\mathfrak{P}_3$ have an order prime to $\ell$ in the group $\operatorname{Cl}(K)$. In this case

$$\begin{equation*} |H^0(H,\overline U_2(K))|=|H^1(H,\overline U_2(K))|=\ell. \end{equation*} \notag$$
This case always take place if $|{\operatorname{Cl}_\ell(K)}|<\ell^{\ell-1}$.

(B) At least, one of the divisors $\mathfrak{P}_1$, $\mathfrak{P}_2$, $\mathfrak{P}_3$ present non-zero element in $\operatorname{Cl}(K)_\ell$. Then

$$\begin{equation*} |H^0(H,\overline U_2(K))|=|H^1(H,\overline U_2(K))|=1\quad\textit{and}\quad |{\operatorname{Cl}(K)_\ell}|=\ell^{\ell-1}. \end{equation*} \notag$$
For the field $L$ we always have
$$\begin{equation*} H^0(\widetilde H,\overline U_2(L))=H^1(\widetilde H,\overline U_2(L))=0. \end{equation*} \notag$$

Proof. Let $\mathfrak A=\mathfrak P_1^{m_1}\mathfrak P_2^{m_2}\mathfrak P_3^{m_3}$ be a product of prime divisors, which ramify in $K/k$. If $\mathfrak A$ is a principal divisor then by (3.3) the element $\mathfrak A\in (D_K^0)^H$ determines some element $\eta(\mathfrak A)$ in the group $H^1(H,U(K))$. Put
$$\begin{equation*} \mathcal{A} = (\mathbb Z\mathfrak P_1\oplus \mathbb Z\mathfrak P_2\oplus\mathbb Z\mathfrak P_3)/(\mathbb Z\mathfrak p_1\oplus\mathbb Z\mathfrak p_2\oplus\mathbb Z\mathfrak p_3)\cong (\mathbb Z/\ell\mathbb Z)^3. \end{equation*} \notag$$
Then there is an exact sequence
$$$$0\to (D_K^0)^H/D_k\to \mathcal{A}\xrightarrow{\psi}\operatorname{Cl}(K)_\ell^H.$$ \tag{3.15}$$
It follows from [1], Theorem 4.1, that $\operatorname{Cl}(K)_\ell$ is a cyclic $H$-module, that vanishes under the norm map $N_H$, therefore, by [1], Lemma 3.1, the group $\operatorname{Cl}(K)_\ell^H$ is of order $\ell$.

So, it follows from (3.15) that the group $H^1(H,U(K))=(D_K^0)^H/D_k$ is of order $\ell^2$ if $\psi\neq 0$, or of order $\ell^3$ if $\psi=0$.

In the case (A) we have $\psi=0$, that is, $|H^1(H,U(K))|=\ell^3$. Calculating the Herbrand index $h(U(K))$ via Dirichlet theorem, we obtain

$$\begin{equation*} h(U(K))=|H^0(H,U(K))|/|H^1(H,U(K))|=\ell^{-1}, \text{ that is, } |H^0(H,U(K))|=\ell^2. \end{equation*} \notag$$
Then by Proposition 3.2 we get $|H^1(H,\overline U(K))|=\ell^2$ and $|H^0(H,\overline U(K))|=\ell$. The condition $\psi=0$ means that the assumptions of Proposition 3.5 hold, that is,
$$\begin{equation*} |H^1(H,U_2(K))|=\ell \quad\text{and}\quad |H^0(H,U_2(K))|=\ell. \end{equation*} \notag$$
Consider the lower central series of the $H$-module $\operatorname{Cl}(K)_\ell$. The first factor of this series is isomorphic to $\mathbb F_\ell(1)$ as a $\Delta$-module. Then [1], Lemma 3.2, yields that in the case $|{\operatorname{Cl}(K)_\ell}|<\ell^{\ell-1}$ this central series has no factors isomorphic to $\mathbb F_\ell(0)$ as a $\Delta$-module. This is the situation of the case (A).

Now we turn to the case (B). If one of the divisors $\mathfrak P_i$, $\mathfrak P_1$ for example, presents a non-zero element of $\operatorname{Cl}(K)$ then this element is fixed under the action of $G$. Then, applying Lemma 3.2 of [1] again and taking into account that $\operatorname{Cl}(K)_\ell/(\sigma- 1)\operatorname{Cl}(K)_\ell\cong \mathbb F_\ell(1)$ as a $\Delta$-module, we obtain that the length of the lower central series of the $H$-module $\operatorname{Cl}(K)_\ell$ is at least $\ell-1$, but its length cannot be bigger, since the group $\operatorname{Cl}(K)_\ell$ is of period $\ell$ by Corollary 3.1 of [2]. Therefore, in this case we have $|{\operatorname{Cl}(K)_\ell}|=\ell^{\ell-1}$.

Since $\psi\neq 0$ in (3.15) we get

$$\begin{equation*} |H^1(H,U(K))|=\ell^2\quad\text{and}\quad |H^0(H,U(K))|=\ell. \end{equation*} \notag$$
Now, it follows from Proposition 3.2 that $|H^1(H, \overline U(K))|\,{=}\,\ell$ and $|H^0(H,\overline U(K))|\,{=}\, 1$. This completes the proof of the theorem for $K$.

The proof for $L$ is analogous. We can write down an analogue of Formula (3.15) for the field $L$ again, but in this case we have $\operatorname{Cl}(L)_\ell^H=0$ by Proposition 3.1, and $\mathcal{A}\cong (\mathbb Z/\ell\mathbb Z)^2$, so, $|H^1(\widetilde H,U(L))|=\ell^2$ and $|H^0(\widetilde H,U(L))|=\ell$. By virtue of Proposition 3.2 we have $|H^1(\widetilde H,\overline U(L))|=\ell$ and $|H^0(\widetilde H,\overline U(L))|=1$. Then Proposition 3.5 yields that $H^1(\widetilde H, U_2(L))=H^0(\widetilde H, U_2(L))=0$. This proves Theorem 3.1.

### § 4. Structure of certain groups of units of the field $L$

Let $\mathcal{A}(L)=\prod_{v\in S}U^{(1)}(L_v)$ and $\overline{\mathcal{A}}(L)=\prod_{v\in S}\overline U^{\,(1)}(L_v)$ be the maximal $\mathbb Z_\ell$-free factor module of $\mathcal{A}(L)$. Our nearest goal is to characterize $\mathcal{A}(L)$ as a $\mathbb Z_\ell[\widetilde G]$-module. To do this, we need some general results on $\mathbb Z_\ell[\widetilde G]$-modules, which are analogous to those in [1], § 3. Since the groups $G=G(K/\mathbb Q)$ and $\widetilde G=G(L/\mathbb Q)$ are isomorphic we shall formulate these general results for the case of $G$-modules.

We shall consider a Noetherian $G$-module $A$ that is free as a $\mathbb Z_\ell[H]$-module. These conditions on $A$ are equivalent to simultaneous fulfilment of the following two conditions:

(i) $A$ is Noetherian and has no torsion over $\mathbb Z_\ell$;

(ii) $A$ is a cohomologically trivial $\mathbb Z_\ell[H]$-module.

As in [1], we assume that some section $f\colon \Delta \to G$ is fixed. Let $\delta'=f(\delta)$ be a fixed generator of $f(\Delta)$ and $e_i$ the idempotent of $\mathbb Z_\ell[f(\Delta)]$ corresponding to the character $\omega^i$, where $\omega$ is the Teichmüller character. For $a\in A$ we denote by $a[i]$ the element $e_ia$. Let $\sigma$ be a fixed generator of $H$.

As in [1], let $F_1=\mathbb Z_\ell[G]$ be a free $G$-module of rank $1$. Then we have

$$\begin{equation*} F_1/(\ell,(\sigma- 1))F_1\cong\bigoplus_{i=0}^{\ell-2}\mathbb F_\ell(i). \end{equation*} \notag$$
If $a\in F_1$ is a generator of $G$-module $F_1$ then $a=\sum_{i-0}^{\ell-2}a[i]$ and thus
$$$$F_1=Q(0)+Q(1)+\dots +Q(\ell-2), \ \text{ where } Q(i)=\mathbb Z_\ell[H]a[i].$$ \tag{4.1}$$
Any of $Q(i)$ is a $G$-module and has the rank at most $\ell$ as a $\mathbb Z_\ell$-module. Then it follows from (4.1) that the rank is exactly $\ell$, and (4.1) defines a decomposition of $F_1$ into a direct sum of $G$-modules $Q(i)$ for $i=0,1,\dots,\ell-2$. Therefore, each of the modules $Q(i)$ is projective as a $G$-module.

Proposition 4.1. Let $A$ be a Noetherian $G$-module, which is free as a $\mathbb Z_\ell[H]$-module. Then there is a (non-canonical) isomorphism of $G$-modules

$$$$A\cong\bigoplus_{i=0}^{\ell-2}Q(i)^{r_i},$$ \tag{4.2}$$
where the exponents $r_i$ are defined uniquely by the module $A$. In particular, if $A$ is a cyclic $H$-module then $A\cong Q(i)$ for some $i$.

Proof The proof of this proposition is completely analogous to that of Proposition 3.1 of [1].

Corollary 4.1. Suppose that we have an exact sequence of Noetherian $\mathbb Z_\ell[G]$-modules

$$\begin{equation*} 0\to A\to B\to C\to 0, \end{equation*} \notag$$
such that $A,B,C$ are free as $\mathbb Z_\ell$-modules. If any two modules of this sequence are projective then so is the third one.

So, if $A$ is a free $\mathbb Z_\ell[H]$-module of rank $1$ then by Proposition 4.1we have $A\cong Q(i)$ for some $i$. In this case $A/(\ell,(\sigma -1))A\cong \mathbb F_\ell(i)$. In general, if $A/(\ell,(\sigma-1))A\cong \mathbb F_\ell(i)$ then we say that $A$ begins at $\mathbb F_\ell(i)$. Under the same assumption, if $A$ is finite and $A^H\cong \mathbb F_\ell(k)$ then we say that $A$ ends at $\mathbb F_\ell(k)$.

Put $A\cong Q(i)$ and $B=A/\ell A$. Then $B$ is a cohomologically trivial $\mathbb F_\ell[H]$-module, and there is a lower central series

$$\begin{equation*} B=B_0\supset B_1\dots \supset B_\ell=0, \quad \text{where}\quad B_j/B_{j+1}\cong \mathbb F_\ell(i+j) \end{equation*} \notag$$
for $j=0,1,\dots,\ell-1$, as it follows easily from Lemma 3.2 of [1]. The next Proposition is an analogue of Proposition 4.1 for the case $\mathbb F_\ell[G]$-modules and may be proved by the same arguments.

Proposition 4.2. Let $\mathscr A$ be a Noether $G$-module, which is free as a $\mathbb F_\ell[H]$-module. Then there is a (non-canonical) isomorphism of $G$-modules

$$$$\mathscr A\cong \bigoplus_{i=0}^{\ell-2}\mathscr Q(i)^{r_i},$$ \tag{4.3}$$
where $\mathscr Q(i)=Q(i)/\ell Q(i)$ and the exponents $r_i$ are defined uniquely by $\mathscr A$. In particular, if $\mathscr A$ is a cyclic $H$-module then $\mathscr A\cong \mathscr Q(i)$ for some $i$.

Corollary 4.2. Let an exact sequence of $\mathbb F_\ell[G]$-modules be given

$$\begin{equation*} 0\to\mathscr A\to\mathscr B\to \mathscr C\to 0. \end{equation*} \notag$$

If any two modules of this sequence are projective in the category of Noetherian $\mathbb F_\ell[G]$-modules then so is the third one.

Proposition 4.3. Let an exact sequence of Noetherian $\mathbb Z_\ell[G]$-modules be given

$$\begin{equation*} 0\to A\to B\to C\to 0, \end{equation*} \notag$$
and these modules are free as $\mathbb Z_\ell$-modules. If the $G$-module $A$ is projective then there is a decomposition into a direct sum $B\cong A'\oplus C'$, where $A\cong A'$ and $C\cong C'$.

Proof. Put $A^0=\operatorname{Hom}_{\mathbb Z_\ell}(A,\mathbb Z_\ell)$ and let $B^0$ and $C^0$ be of analogous meaning. Let $A^0$, $B^0$ and $C^0$ has a standard structure of left $G$-modules, as it is explained in [3]. Then there is an exact sequence of $G$-modules
$$\begin{equation*} 0\to C^0\to B^0\to A^0\to 0 \end{equation*} \notag$$
with projective $A^0$. Therefore, it splits, that is, $B^0\cong C^0\oplus \widetilde A^0$, where $\widetilde A^0\cong A^0$. Thus $B\cong (B^0)^0 \cong(C^0\oplus \widetilde A^0)^0\cong C^{00}\oplus \widetilde A^{00}\cong C\oplus \widetilde A\cong A\oplus C$. This proves Proposition 4.3.

Let $M(L)$ be the kernel of the natural map $\mathcal{A}(L)\to\overline{\mathcal{A}}(L)$. Hence, $M(L)=\prod_{v\in S}\mu_\ell(L_v)$.

Proposition 4.4. The module $M(L)$ is isomorphic to $(\mathbb Z/\ell\mathbb Z)^\ell$ as an Abelian group. It is cyclic and cohomologically trivial as a $\widetilde H$-module, where $\widetilde H=G(L/k)$. The module $M(L)$ begins at $\mathbb F_\ell(1)$ and ends at $\mathbb F_\ell(1)$. So, taking into account the notions introduced above, we get an isomorphism of $\widetilde G$-modules $M(L)\cong \mathscr Q(1)$.

Proof. The first assertion is obvious. The group $\widetilde H$ substitutes the factor $\mu_\ell(L_v)$, hence $M(L)$ is an induced $\widetilde H$-module. Therefore, it is cohomologically trivial. The norm map $N_{L/k}$ is a $\Delta$-homomorphism that maps $M(L)$ onto $M(k)=\mu_\ell(k)\cong \mathbb F_\ell(1)$. Therefore, $M(L)$ begins at $\mathbb F_\ell(1)$. This proves Proposition 4.4.

We see that the $\widetilde G$-module $\overline{\mathcal{A}}(L)$ is induced as a $\widetilde H$-module. Therefore it is cohomologically trivial as a $\widetilde H$-module, and by Proposition 4.1 it has a decomposition of the form (4.2). Note that every direct summand $Q(i)$ enters this decomposition with multiplicity $1$, since $N_{L/k}$ induces an isomorphism

$$$$\overline{\mathcal{A}}(L)/(\widetilde\sigma-1)\overline{\mathcal{A}}(L)\cong \overline{\mathcal{A}}(k)\cong \bigoplus_{i=0}^{\ell-2}\mathbb Z_\ell(i),$$ \tag{4.4}$$
where $\widetilde\sigma$ is a generator of $\widetilde H$ and $\mathbb Z_\ell(i)$ denotes a $\widetilde G$-module, which is isomorphic to $\mathbb Z_\ell$ as an Abelian group with trivial action of $\widetilde H$, while $\Delta$ acts on it by multiplication by $\omega^i$.

In sequel we need the decomposition of $\mathcal{A}(L)$ with some additional properties.

Proposition 4.5. $\widetilde G$-module $\mathcal{A}(L)$ has a decomposition into the direct sum of $\widetilde G$-modules $\mathcal{A}(L)=\bigoplus_{i=0}^{\ell-2}\mathcal{A}[i]$, where $\mathcal{A}[i]\cong Q(i)$ as a $\widetilde G$-module, and the following conditions hold:

1) $\mathcal{A}[2i]\subset U_2(L)$ for $i=1,\dots,(\ell-3)/2$;

2) the direct summand $\mathcal{A}[0]$ has a generator $\mathbf{a}_0$ such that $\Delta$ acts trivially on $\mathbf{a}_0$ and $(\widetilde\sigma-1)\mathbf{a}_0\in \overline U(L)$.

Proof. The cohomological triviality of $\overline U_2(L)$ (see Theorem of 3.1) yields that $\overline U_2(L)/(\widetilde\sigma-1)\overline U_2(L)\cong \overline U(k)$, hence
$$$$\overline U_2(L)\cong \bigoplus_{i=1}^{(\ell-3)/2} Q(2i).$$ \tag{4.5}$$
Then $\widetilde G$-module $\overline{\mathcal{A}}(L)/\overline U_2(L)$ is projective by Corollary 4.1, hence there is a decomposition into the direct sum $\overline{\mathcal{A}}(L)\cong (\overline{\mathcal{A}}(L)/\overline U_2(L))\oplus\overline U_2(L)$. Then we put $\overline{\mathcal{A}}[2i]=Q(2i)$, where $Q(2i)$ are the components of $\overline U_2(L)$ in the decomposition (4.5).

Let $\overline{\mathcal{A}}_2(L)$ be the subgroup of local universal norms (of $L_\infty$) in the group $\overline{\mathcal{A}}(L)$, so, that there is a natural isomorphism

$$$$\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\cong\prod_{v\in S}\Gamma_v,$$ \tag{4.6}$$
where $\prod_{v\in S}\Gamma_v$ has the same meaning, as in (3.9) and(3.10). Thus $\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)$ is a cohomologically trivial $\widetilde H$-module of $\mathbb Z_\ell$ rank $\ell$. This module admits a natural $\widetilde G$-epimorphism $\varphi\colon\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\to \Gamma=G(L_\infty/L)$. Therefore by Proposition 4.2 we have $\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\cong Q(0)$.

Now consider a $\widetilde G$-module $\overline{\mathcal{A}}(L)/\overline U_2(L)$. By Proposition 4.2 and formulae (4.3), (4.5) we have an isomorphism $B:=\mathcal{A}(L)/\overline U_2(L)\cong\bigoplus_{i=0}^{(\ell-3)/2} Q(2i+1)\oplus Q(0)$. Let $C=\mathcal{A}_2(L)/U_2(L)$ and $P(L)$ be of the same meaning as in (3.12). Then we can consider $C$ and $P(L)$ as submodules of $B$, where $C\cong \bigoplus_{i=0}^{(\ell-3)}Q(2i+1)$. Taking into account that $C$ and $P(L)$ have zero intersection in $B$, we obtain an exact sequence

$$$$0\to C\oplus P(L)\to B\to \Gamma\to 0$$ \tag{4.7}$$
with a projective $\widetilde G$-module $C$, whence, taking factor-module with respect to $P(L)$, we get an exact sequence
$$\begin{equation*} 0\to C\to B/P(L)\to \Gamma\to 0 \end{equation*} \notag$$
with projective $C$. Applying Proposition 4.3 to this sequence, we see that $B/P(L)$ contains a submodule $\Gamma'$, which is isomorphic to $\Gamma$ as a $\widetilde G$-module, that is, $\Gamma'$ is isomorphic to $\mathbb Z_\ell$ as an Abelian group and $\widetilde G$ acts trivially on $\Gamma'$. Thus, we have an exact sequence
$$\begin{equation*} 0\to\Gamma'\xrightarrow{\chi}B/P(L)\to C'\to 0, \end{equation*} \notag$$
where $C'\cong C$. Let $a'$ be a generator of $\Gamma'$. Then, lifting $a'$ up to some element $a''\in B$, we obtain an element $a''\in B$ such that $b:=(\widetilde \sigma-1)a''\in P(L)$. If a $\widetilde H$-submodule generated by $b$ would be a proper submodule of $P(L)$ then , after replacing $b$ by some $b'=b+x$ for a suitable $x\in P(L)$, we should obtain $\widetilde\sigma (b')=b'$. It means that $B\cong C\oplus P(L)\oplus \mathbb Z_\ell b'$ as an $\widetilde H$-module. But this contradicts to projectivity of $B$.

Finally, we put $\mathbf{a}=a''[0]$. Obviously, $a''$ and $\mathbf{a}$ are equal modulo $P(L)$, $\mathbf{a}$ generates a submodule of $B$, which is isomorphic to $Q(0)$, and $\mathbf{a}$ is fixed under the action of $\Delta$. This completes the proof of Proposition 4.5.

Using Proposition 4.5, we shall obtain some special decomposition for $\mathscr B(L):=\overline{\mathcal{A}}(L)/\ell\overline{\mathcal{A}}(L)$ that we need in the next section. We shall construct a decomposition into the direct sum $\mathscr B(L)=\bigoplus_{i=0}^{\ell-2}\mathscr B[i]$, where for $i\neq 1$ we put $\mathscr B[i]=\mathcal{A}[i]/\ell\mathcal{A}[i]$ and for $i=1$ we define the component $\mathscr B[1]$ as follows.

Let $L_v$ be the completion of $L$ at the place $v$ over $\ell$. The group $U(L_v)$ contains $\ell$-primary elements, that is, such elements $u_v$, that $L_v(\sqrt[\ell]{u_v}\,)$ is an unramified extension of $L_v$ of degree $\ell$. We shall consider $u_v$ as an element of the group $\overline U(L_v)/(\overline U(L_v))^\ell$. Thus, there are $\ell-1$ primary elements, which together with the unity form a subgroup denoted by $\mathcal{P}_v$ of the group $\overline U(L_v)/\overline U(L_v)^\ell$. We put $\mathscr B[1]=\prod_{v\in S}\mathcal{P}_v$ and consider this group as a subgroup of $\mathscr B$. Obviously, $\mathscr B[1]$ is a Galois submodule in $\mathscr B(L)$.

The group $\widetilde H$ substitutes the components $\mathcal{P}_v$ of $\mathscr B[1]$, so, $\mathscr B[1]\cong \mathbb F_\ell[\widetilde H]$ as an $\widetilde H$-module.

Proposition 4.6. We have an isomorphism of the Galois modules

$$\begin{equation*} \mathscr B[1]\cong \mathscr Q(1). \end{equation*} \notag$$

Proof. Put $G_v^{\mathrm{un}}=G(L_v(\sqrt[\ell]{u_v}\,)/L_v)$. Then by the local class field theory there is a canonical isomorphism $G_v^{\mathrm{un}}\cong L^\times_v/(L_v^\times)^\ell U(L_v)$. Thus, if $D_S$ is the group of divisors of $L$ with supports in $S$ then $D:=\prod_{v\in S}G_v^{\mathrm{un}}\cong D_S/D_S^\ell$. This defines a structure of Galois module on $D$. Since there is an epimorphism of Galois modules $D\to \mathbb F_\ell(0)$ induced by the norm map, we get that $D\cong \mathscr B[0]$, that is $D$ starts with $\mathbb F_\ell(0)$ and ends with $\mathbb F_\ell(0)$. On the other hand, by Kummer theory there is a non-degenerate pairing between $D$ and $\mathscr B[1]$, hence $\mathscr B[1]$ starts with $\mathbb F_\ell(1)$ and ends with $\mathbb F_\ell(1)$. This proves Proposition 4.6.

Proposition 4.7. The Galois module $\mathscr B(L)$ has a decomposition into the direct sum

$$$$\mathscr B(L)=\bigoplus_{i=0}^{I=\ell-2} \mathscr B[i],$$ \tag{4.8}$$
where if $i\neq 1$ then $\mathscr B[i]=\mathcal{A}[i]$ and $\mathcal{A}[i]$ has the same meaning as in Proposition 4.5, and if $i=1$ then $\mathscr B[1]$ has the same meaning as in Proposition 4.6.

Proof. Put
$$$$\mathscr B'(L)=\bigoplus_{i=0,\,i\neq 1}^{i=\ell-2}\mathscr B[i].$$ \tag{4.9}$$
To prove the formula (4.8) it is enough to check that in the group $\mathscr B(L)$ we have $C:=\mathscr B'(L)\cap\mathscr B[1]=0$. To do this, it is enough to check that $C^{\widetilde H}=0$. Obviously, $C^{\widetilde H}=\mathscr B'(L)^{\widetilde H}\cap \mathscr B[1]^{\widetilde H}$, but we have $\mathscr B'(L)^{\widetilde H}\cong \bigoplus_{i=0,\, i\neq 1}^{\ell-2}\mathscr B[i]^{\widetilde H}\cong \bigoplus_{i=0,\, i\neq 1}^{i=\ell-2}\mathbb F_\ell(i)$, while $\mathscr B[1]^{\widetilde H}\cong\mathbb F_\ell(1)$. This proves the existence of the decomposition (4.8).

The main results of this section we can formulate as a following theorem.

Theorem 4.1. For the field $L$, the Galois module $\mathcal{A}(L)/\mathcal{A}(L)^\ell$ contains in the split exact sequence of Galois modules

$$$$0\to M(L)\to\mathcal{A}(L)/\mathcal{A}(L)^\ell\to\mathscr B(L)\to 0,$$ \tag{4.10}$$
where the structure of $M(L)$ is given in Proposition 4.4, and the structure of $\mathscr B(L)$ is given in Proposition 4.7.

This result may be presented in another form, taking into account that $M(L)$ and $\mathscr B[1]$ are defined canonically as submodules of $\mathcal{A}(L)/\mathcal{A}(L)^\ell$. Namely, there is a split exact sequence of Galois modules

$$$$0\to \mathscr B[1]\oplus M(L)\to \mathcal{A}(L)/\mathcal{A}(L)^\ell \to \mathscr B'(L)\to 0,$$ \tag{4.11}$$
where $\mathscr B'(L)$ was defined in the proof of Proposition 4.7.

### § 5. Certain $\ell$-extensions of the field $L$

In this section we assume that the field $L=k(\sqrt[\ell]{b})$ defined as in (3.2) is fixed. In particular, we fix the pair of primes $p_1,p_2\,{\in}\,\mathbb P_0$ such that $b=p_1^{s_1}p_2^{s_2}$. Let $p_3\in \mathbb P_0$, where $p_3$ has the same meaning as in (3.1). Let $N=N(p_3)$ be the maximal Abelian $\ell$-extension of $L$ such that the Galois group $G(N/L)$ is of period $\ell$, and only prime divisors of $p_3$ may ramify in the extension $N/L$. We wish to obtain the spectrum of all possible values of degree $[N:L]$. Note that we have always $[N:L]>1$ since $N\supseteq K\cdot L$. To avoid plenty of indices, we shall denote $p_3$ by $q$ in this section. By $(\mathfrak q)$ we denote the principal divisor of $q$, which is a prime divisor in $\mathbb Q$ or in $K$.

Proposition 5.1. If $q\neq p_1,p_2$ then $\mathfrak q$ splits completely in the extension $L/k$.

Proof. If $\mathfrak q$ remains prime in $L/k$ then $\mathfrak q$ remains prime in $L/\mathbb Q$. It means that $\widetilde G=G(L/\mathbb Q)$ coincides with the decomposition subgroup of $\mathfrak q$. But $\mathfrak q$ does not ramify in $L/\mathbb Q$, hence its decomposition subgroup must be cyclic hence it cannot coincide with $\widetilde G$. This concludes the proof.

Let, as in the beginning of § 3 some section $f\colon \Delta\to \widetilde G$ is fixed. Since there are exactly $\ell$ subgroups of order $\ell-1$ in $\widetilde G$ and all these subgroups are mutually conjugate, we may assume that the prime divisors $\mathfrak q_1,\dots,\mathfrak q_\ell$ are enumerated in such a way that $f(\Delta)$ coincides with the decomposition subgroup of $\mathfrak q_1$.

Since the Galois group $G(N/L)$ is of period $\ell$ we can use the Kummer theory for description of $N$. Suppose that $L(\sqrt[\ell]{\alpha}\,)\subseteq N$. This means that all divisors prime to $q$ enter $\alpha$ with exponents divisible by $\ell$.

Let $h$ be the class number of $L$, which is prime to $\ell$ by Proposition 3.1, and $\bf h$ is such number that $\mathbf{h}\equiv 0\pmod h$ and $\mathbf{h}\equiv 1\pmod\ell$. Then, putting $\alpha'=\alpha^{\mathbf{h}}$, we get $L(\sqrt[\ell]{\alpha}\,)=L(\sqrt[\ell]{\alpha'}\,)$.

If $\alpha$ contains a prime divisor distinct from $\mathfrak q_1,\dots,\mathfrak q_\ell$, the divisor $\mathfrak p^{r_\mathfrak p}$, for example, then $r_{\mathfrak p}\equiv 0\pmod\ell$ and $(\alpha')$ contains $\mathfrak p$ with exponent $r_\mathfrak p\mathbf{h}$. But the divisor $\mathfrak p^{\mathbf{h}}$ is principal, hence, multiplying $\alpha'$ by some $\ell$-th power if necessary, we may assume that $\alpha'$ contains only the prime divisors $\mathfrak q_1,\dots, \mathfrak q_\ell$.

Thus, we have to characterize all the elements $\alpha'\in L$ such that $(\alpha')$ contains only prime divisors $\mathfrak q_1,\dots,\mathfrak q_\ell$ while all the places $v\in S$ split or remain unramified in $L(\sqrt[\ell]{\alpha'}\,)/L$.

Proposition 5.2. Let $\alpha_1,\alpha_2\in L$ be such elements that $L_1:=L(\sqrt[\ell]{\alpha_1}\,)\subseteq N$ and $L_2:=L(\sqrt[\ell]{\alpha_2}\,)\subseteq N$. Then the equality of the divisors $(\alpha_1)=(\alpha_2)$ yields that $L_1=L_2$ and $\alpha_1=\alpha_2u^\ell$, where $u$ is a unit of $L$.

Proof. Put $u_1=\alpha_1\alpha_2^{-1}$. Then $u_1$ is a unit of $L$ and $L(\sqrt[\ell]{u_1}\,)\subseteq N$. The extension $L(\sqrt[\ell]{u_1}\,)/L$ cannot have ramification out of $S$, but it cannot have ramification in $S$ by definition of $N$. Therefore, the extension $L(\sqrt[\ell]{u_1}\,)/L$ is unramified, but then it follows from Proposition 3.1 that $L(\sqrt[\ell]{u_1}\,)=L$, that is, $u_1=u^\ell$ for some unit $u\in U(L)$. This proves Proposition 5.2.

Let $\beta_1\in L^\times$ be such an element that $(\beta_1)=\mathfrak q_1^{\mathbf{h}}$. In this case the element $u:=\beta^{1-\delta}$ is a unit. Since $H^{-1}(\Delta,U(L)[\ell])=0$ we get that there are units $u_1$ and $u_2$ such that $u=u_1^{1-\delta}u_2^\ell$. Then, putting $\beta=\beta_1u^{-1}$, we obtain such an element $\beta$ that

$$$$(\beta)=\mathfrak q_1^{\mathbf{h}} \text{ and } \beta^{1-\delta}=u_2^\ell\quad \text{for some } u_2\in U(L)[\ell].$$ \tag{5.1}$$
Any divisor with support in $\mathfrak q_1,\dots,\mathfrak q_\ell$ may be written (in additive notation) in the form $\eta(\beta)$ for some $\eta\in \mathbb Z_\ell[\widetilde H]$. The element $\eta\beta$ is a unit in the local field $L_v$ for all $v\in S$, so, we can consider $\eta\beta$ as an element of $\mathcal{A}(L)$. By $\pi(\eta\beta)$ we denote the image of $\eta\beta$ in $\mathcal{A}(L)/\ell\mathcal{A}(L)$.

Using the exact sequence (4.11), we can transform $\pi(\eta\beta)$ into the element $\pi'(\eta\beta)\in\mathscr B'(L)$. Then it follows from (4.9) that $\pi'(\eta\beta)=\bigoplus_{i=0,\,i\neq 1}^{i=\ell-2}\pi_i(\eta\beta)$, where $\pi_i(\eta\beta)\in\mathscr B[i]$. Formula 4.5 and the definition of the component $\mathscr B[i]$ yield that

$$\begin{equation*} \bigoplus_{i=1}^{(\ell-3)/2}\mathscr B[2i]\cong \overline U_2(L)/\ell\overline U_2(L), \end{equation*} \notag$$
so, after multiplying $\beta$ by some suitable unit in the group $\overline U_2(L)$, we may assume and we shall assume in further that $\pi_i(\beta)=0$ for $i=2,4,\dots,\ell\,{-}\,3$. Then the same property have the elements $\pi(\eta\beta)$ for any $\eta$ and $i=2,4,\dots,\ell-3$.

So, we can assume that some element $\beta\in L^\times$ is chosen such that $\pi(\beta)^\delta=\pi(\beta)$, moreover, $\pi_i(\beta)=0$ for $i=2,4,\dots,\ell-3$. We shall consider a system $\eta_1,\dots,\eta_{\ell^\ell}\in \mathbb Z_\ell[\widetilde H]$ that presents all the elements of $\mathbb F_\ell[\widetilde H]$, and we have to determine all $\eta_j$ such that $L(\sqrt[\ell]{\eta_j\beta}\,)\subseteq N$. The elements $\eta_j\beta$ must satisfy the following condition

$$$$\pi_i(\eta_j\beta)=0 \text{ for } i=2,\dots,\ell-2\quad\text{and}\quad \pi_M(\eta_j\beta)=0,$$ \tag{5.2}$$
where the projection $\pi_M\colon\mathcal{A}(L)/\ell\mathcal{A}(L)\to M(L)$ is defined by the split exact sequence (4.10). Herein we can assume that the conditions (5.2) hold already for even $i>0$. The conditions that appear in the case $i=0$, we shall discuss below (see Proposition 5.3).

Obviously, the images of those representatives of $\eta_1,\dots,\eta_{\ell^\ell}$ that satisfy the condition (5.2) form an ideal $I$ of $\mathbb F_\ell[\widetilde H]$, and $I=I(q)$ is of the form $I=I_{\mathrm{max}}^i$, where $I_{\mathrm{max}}$ is the maximal ideal of $\mathbb F_\ell[\widetilde H]$.

Proposition 5.3. For any $q$ we have $I\neq 0$.

Proof. Indeed, there is always the extension $KL/L$, that corresponds to $\eta=\sum_{h\in \widetilde H}h$, hence $I$ always contains the ideal $\mathbb F_\ell(\sum_{h\in \widetilde H}h)$.

Proposition 5.4. For the field $L$ we have always a strict inclusion $I\subset \mathbb F_\ell[\widetilde H]$.

Proof. We suppose that $(q)$ remains prime in the extension $k_\infty/k$. This means that the Artin symbol of divisor $(q)$ is a generator of $\Gamma=G(k_\infty/k)$. The map of restriction of automorphisms from $L_\infty$ to $k_\infty$ maps isomorphically $\Gamma_1:=G(L_\infty/L)$ onto $\Gamma$. This map agree with the norm map $N_{L/k}$, which sends $\mathfrak q_1$ into $(q)$. It means that for any $\beta\in L^\times$ such that $(\beta)=\mathfrak q_1^{\mathbf{h}}$, there is a place $v\in S$ of $L$ such that the inclusion $v\colon L\hookrightarrow L_v$ sends the element $\beta$ into a generator of $\Gamma_v$, where $\Gamma_v$ is of the same meaning as in (3.10). It is known by local class field theory that the extension $L(\sqrt[\ell]{\beta}\,)/L$ is ramified for such $\beta$, hence, for example, $1\notin I$. This proves Proposition 5.4.

Remark. Consider an element $\eta\beta$ such that $\eta\in I_{\mathrm{max}}$. Then $\pi_0(\eta\beta)$ belongs to $\pi_0(P(L))$, where $P(L)$ has the same meaning as in (3.10). This means that, multiplying $\beta$ by some unit in $P(L)$, we always may assume that $\pi_0(\eta\beta)=0$ (under assumption that $\eta\in I_{\mathrm{max}}$).

Now we consider the restrictions that yields the equality $\pi_i(\eta\beta)=0$ for some odd $i=3,5,\dots,\ell-2$. Remind that we use additive notation for multiplication. The lower central series of the Galois module $\mathscr B[i]$ starts with $\mathbb F_\ell(i)$. Meanwhile, the element $\beta$ is fixed under the action of $\Delta$. If $\pi_i(\beta)=0$, then there appear no obstructions connected with the projection $\pi_i$. If $\pi_i(\beta)\neq 0$ then, applying Lemma 3.2 of [1] in this case, we obtain that $\pi_i$ sends $\beta$ into a generator of $\mathscr B[i]^{(\widetilde\sigma-1)^{\ell-1-i}}$ ($\Delta$ acts identically on $\beta$ and the module $\mathscr B[i]^{(\widetilde\sigma-1)^{\ell-1-i}}$ starts with $\mathbb F_\ell(0)$). Then the element $\pi_i(\beta)$ generates in $\mathscr B[i]$ a submodule of order $\ell^{i+1}$. Thus, in this case we have $\pi_i(\eta\beta)=0$ if and only if $\eta\in I_{\mathrm{max}}^{i+1}$.

We have to consider two particular cases else: the behavior of $\pi_1(\beta)$ and the behavior of the element $\pi_M(\beta)$. In the case $i=1$ we, repeating all preceding considerations, obtain that $\pi_1(\beta)$ generates a submodule of order $\ell^2$ in $\mathscr B[1]$. But since $I$ is a proper ideal, we have either $\pi_1(\eta\beta)=0$, which corresponds to the extension $L(\sqrt[\ell]{\eta\beta}\,)/L$, where all places over $\ell$ completely split, or $\pi_1(\eta\beta)\neq 0$, and this corresponds to the case when all places over $\ell$ of the field $L$ remain unramified in the extension $L(\sqrt[\ell]{\eta\beta}\,)/L$.

At last, let us consider $\pi_M(\beta)$. Since $M$ starts by $\mathbb F_\ell(1)$ and ends by $\mathbb F_\ell(1)$, we obtain, as in the case $i=1$, that $\pi_M(\eta\beta)\in M^{\widetilde H}=\mu_\ell(L)$, hence, after multiplying the element $\eta\beta$ by a suitable root of unity in $\mu_\ell(L)$, we can assume that $\pi_M(\eta\beta)=0$.

We can formulate the results of our study in the form of the following theorem.

Theorem 5.1. Let $X:=3,5,\dots, \ell-2$ and $X_1$ be the subset of those $i\in X$, for which $\pi_i(\beta)\neq 0$. Let $i_0$ be the maximal index in $X_1$. Then

$$$$I(q)= I_{\mathrm{max}}^{i_0+1}.$$ \tag{5.3}$$
If $X_1$ is empty then $I(q)=I_{\mathrm{max}}$. Here either all places over $\ell$ split completely in $N/L$, or each of them has inertia degree $\ell$.

Proof. Any odd index $i>1$, that is, any $i\in X_1$ must satisfy the condition $I(q)\subseteq I_{\mathrm{max}}^{i+1}$, whence it follows (5.3). If $X_1$ is empty then any element $\eta\beta$ is admissible, that is $I(q)=I_{\mathrm{max}}$. If $\pi_1(\eta\beta)=0$ for any $\eta$ then all places over $\ell$ split in the extension $N/L$. Otherwise, all places over $\ell$ have non-unit inertia degree, which, as man can check easily is equal to $\ell$. This proves Theorem 5.1.

Corollary 5.1. We have always $[N:L]=\ell^{r(q)}$ with even $r(q)$.

Indeed, $|(I(q))|$ is the order of the Kummer group of the extension $N/L$, which coincides with $I(q)$. The index $i_0$ is odd by definition, $|(I_{\mathrm{max}})|=\ell^{\ell-1}$, so it follows from (5.3) that $|I(q)|$ is an even power of $\ell$. Then the degree $[N:L]$ is equal to $\ell^{r(q)}$, where $r(q)=\ell-i_0-2$ is even. In the case $\ell=3$ the set $X_1$ is empty, therefore, $[N:L]=9$, and all the places over $\ell$ either split completely (if $\pi_1(\beta)= 0$), or have inertia degree $3$ (if $\pi_1(\beta)\neq 0$).

Till now we studied how the properties of the field $N(q)$, in particular, its degree over $L$ depend on $q$. Now we shall treat the inverse problem: to prove the existence of prime $q$ such that the field $N(q)$ has a given properties. Namely, the following theorem holds.

Theorem 5.2. Assume that for the set of indices $i=1,2,\dots,\ell-2$ a collection of elements $\alpha_i\in\mathscr B[i]$ is given such that $\alpha_i$ are fixed under the action of $f(\Delta)$. Then there are infinitely many prime divisors $(q)$ of $\mathbb Q$ such that $(q)$ remains prime in $k_\infty$, $(q)$ splits completely in $L/k$ and $(q)$ has the following additional property: Let $\mathfrak q_1$ be a prime divisor of $(q)$ in $L$ fixed by the action of $f(\Delta)$. Then there is $\beta=\beta(\mathfrak q_1)$ such that the principal divisor of $\beta$ is equal to $\mathfrak q_1$, and the following properties hold:

1) $\pi_0(\beta(\mathfrak q_1))\notin P(L)/\ell P(L)$;

2) $\pi_i(\beta(\mathfrak q_1)) =\alpha_i$ for $i=1,2,\dots,\ell-2$.

Proof. Consider the finite Galois module $\mathscr B(L)=\mathcal{A}/\ell\mathcal{A}$. Let $\widetilde U(L)$ be an image of $U(L)/\ell U(L)$ in the group $\mathscr B(L)$. As it was shown in Propositions 4.5 and 4.6, in $\widetilde U(L)$ there is a subgroup $U_2(L)/\ell U_2(L)$ and there is a subgroup $P(L)/\ell P(L)$ that contains in the component $\mathscr B_0$.

According to the global class field theory, man can interpret the group $R(L):=\mathscr B(L)/\widetilde U(L)$ as a Galois group of some Abelian extension $F_1(L)/L$. Put $F(L)=F_1(L)^{M(L)}$. Then the element $\varphi:=\prod_{i=1}^{\ell-2}\alpha_i^{-1}$ may be considered as an element of the group $R(L)$, that is, as an automorphism $\varphi$ of $F(L)/L$. By the Chebotarev density theorem there are infinitely many prime divisors $\mathfrak q_1'$ of $L$ such that $\varphi$ coincides with Frobenius automorphism that corresponds to $\mathfrak q_1'$.

The divisor $\mathfrak q_1'$, that we have constructed, has the desired projections $\alpha_i^{-1}$, but we have to check if the divisor $(q)$ under $\mathfrak q_1$ remains prime in $k_\infty$? This condition is equivalent to the claim that $(q)$ remains prime simultaneously in the extension $\mathbb Q_\infty/\mathbb Q$ and in $k/\mathbb Q$. The fact that $(q')$ remains prime in $\mathbb Q_\infty/\mathbb Q$ follows from the fact that $\pi_0(\beta(q'_1))$ does not contain in the group $P(L)/\ell P(L)\subset \mathscr B[0]$. We have already discussed this situation in detail during the proof of Proposition 5.4.

Consider a tower of fields $F(L)\supset L\supset \mathbb Q$. The group $G(L/\mathbb Q)$ contains the subgroup $f(\Delta)$, which fixes the divisor $\mathfrak q'_1$. Since $G(L/\mathbb Q)$ acts on $F(L)$ by conjugation, we obtain that $\varphi\in G(F(L)/L)$ commutes with the generator $\widetilde\delta$ of $f(\Delta)$. Then the product of these two commuting elements $\varphi$ and $\widetilde\delta$ generates in $G(F(L)/\mathbb Q)$ a cyclic subgroup $C$ of order $\ell(\ell-1)$. By the Chebotarev density theorem for any $\mathfrak q'_1$ there are infinitely many prime divisors $\mathfrak Q_1$ of $F(L)$ such that the decomposition subgroup of $\mathfrak Q_1$ coincides with $C$. Let $\varphi_1$ be a Frobenius automorphism in $G(F(L)/\mathbb Q)$ that corresponds to $\mathfrak Q_1$. Without restricting the generality, we can assume that $\varphi_1^{\ell-1}=\varphi$. The divisor $\mathfrak Q_1$ remains prime in the extension $F(L)/F(L)^\varphi$, that is, $\mathfrak Q_1$ is a prime divisor of the field $F(L)^\varphi$. Let $\mathfrak q_1$ be a prime divisor of $L$ under $\mathfrak Q_1$. Then $N(\mathfrak Q_1)=\mathfrak q_1$, where $N$ is the norm in the extension $F(L)^\varphi/L$. So, for the element $\beta_1=\beta_1(\mathfrak q_1)$ such that the principal divisor $(\beta_1)$ is equal to $\mathfrak q_1$, we obtain that the projections $\pi_i(\beta^{\ell-1})$ are equal to $\alpha_i$, that is, $\mathfrak q_1^{\ell-1}$ is a suitable divisor. This proves Theorem 5.2.

Corollary 5.2. For any pair of distinct prime numbers $p_1,p_2\in \mathbb P_0$ there are infinitely many primes $p_3=q\in \mathbb P_0$ such that $[N:L]=\ell^i$ for any even $i$, where $2\leqslant i\leqslant \ell-1$.

Corollary 5.3. For any pair of distinct prime numbers $p_1,p_2\in \mathbb P_0$ there are infinitely many primes $p_3=q\in \mathbb P_0$ such that $[N:L]=\ell^{\ell-1}$, and all the places over $\ell$ split completely in the extension $N/L$, and infinitely many primes $p_3=q\in\mathbb P_0$ such that all the places over $\ell$ have inertia degree $\ell$ in the extension $N/L$.

### § 6. Application to arithmetic of $K$

Proposition 6.1. Assume that $\ell > 3$, $p_3=q$ and $N=N(q)$ has the same meaning as in § 5. Then the following three conditions are equivalent:

(i) $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$;

(ii) $d(\operatorname{Cl}_S(LK)_\ell)\geqslant 1$;

(iii) $d(G(N/L))\geqslant 2$.

Proof. (i) $\Rightarrow$ (ii). Let $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$. This means that there is an unramified extension $N_1/K$ of $K$, with the Galois group $(\mathbb Z/\ell\mathbb Z)^2$ such that all places over $\ell$ split completely in this extension. Then $N_1L/LK$ is an unramified Abelian $\ell$-extension, and all places over $\ell$ split in it, that is, $d(\operatorname{Cl}_S(LK)_\ell)\geqslant 1$.

(ii) $\Rightarrow$ (iii). Let $N_2$ be the maximal Abelian unramified $\ell$-extension of $LK$, in which all places over $\ell$ completely split. Then the Galois group $G(KL/L)\cong \mathbb Z/\ell \mathbb Z$ acts by conjugation on the Abelian $\ell$-group $G(N_2/KL)$. Hence there is a non-trivial factorgroup $C=G(N_2/KL)/B$ with identical action of $G(KL/L)$. Put $N_3=N_2^B$. Then the Galois group $G(N_3/KL)$ contains in the exact sequence

$$$$1\to C \to G(N_3/L)\to G(KL/L)\to 1,$$ \tag{6.1}$$
whence it follows that $G(N_3/L)$ is an Abelian $\ell$-group of order at least $\ell^2$. Only the places $\mathfrak q_1,\dots,\mathfrak q_\ell$ over $q$ may be ramified in the extension $N_3/L$, and $G(N_3/L)$ is generated by the inertia subgroups of these place. Each of these subgroups is of order $\ell$ or $1$. Therefore, $N_3\subseteq N$ and $G(N_3/L)$ is of period $\ell$, so $d(G(N/L))\geqslant d(G(N_3/L))\geqslant 2$.

(iii) $\Rightarrow$ (i). Since $K\subseteq N$ the condition $d(G(N/L))\geqslant 2$ yields $[N:KL]\geqslant \ell$. Thus, $[N:K]\geqslant \ell^2$. Let $N_4$ be the maximal Abelian unramified extension of $KL$, such that all the places over $\ell$ split completely in it. Then, using the same method, that we have used above to construct the field $N_3$, and an analogue of the exact sequence (6.1), which in the present case has the form

$$\begin{equation*} 1\to C'\to G(N_5/K)\to G(KL/K)\to 1, \end{equation*} \notag$$
where $C'$ is the maximal factorgroup of $G(N_4/KL)$ with trivial action of $G(KL/K)$, we obtain an Abelian unramified $\ell$-extension $N_5/K$ of degree $\geqslant\ell^2$ such that all places over $\ell$ split completely in it. By Theorem 4.1 of [1] the $H$-module $G(N_5/K)$ is cyclic. So, by Theorem 3.1 of [1], if the period of $\operatorname{Cl}_S(K)_\ell$ is greater $\ell$, then $d(\operatorname{Cl}_S(K)_\ell)\geqslant \ell-1$. So, the condition $[N_5:K]\geqslant \ell^2$ yields $d({\rm CL}_S(K)_\ell)\geqslant 2$. This proves Proposition 6.1.

The following theorem 6.1 is an immediate consequence of Corollary 5.2 and Proposition 6.1.

Theorem 6.1. Assume that $\ell > 3$ and $K$ is of the form (3.1). In this case we have always $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$.

In the case $\ell=3$ we need the following variant of Proposition 6.1.

Proposition 6.2. Under conditions of Proposition 6.1 assume that $\ell\,{=}\,3$. Let $N_6$ be the maximal Abelian $\ell$-extension of $L$ unramified out of the prime divisors $\mathfrak q_1$, $\mathfrak q_2$, $\mathfrak q_3$, over $(q)=(p_3)$ and such that $G(N_6/L)$ is of period $\ell$. We assume additionally that all the places over $\ell$, must be unramified, but we do not assume that they split completely in $N_6/L$. Then the following three conditions are equivalent:

(i) $d(\operatorname{Cl}(K)_\ell)\geqslant 2$;

(ii) $d(\operatorname{Cl}(KL)_\ell)\geqslant 1$;

(iii) $d(G(N_6/L))\geqslant 2$.

Proof.The proof of this Proposition is completely equivalent to that of Proposition 6.1.

The next theorem is an immediate consequence of Corollary 5.2, Theorem 6.1 and Proposition 6.2.

Theorem 6.2. Let $\ell=3$ and $K$ be of the form (3.1). Then for any pair $p_1,p_2\in \mathbb P_0$ there are infinitely many $p_3\in \mathbb P_0$ such that $d(\operatorname{Cl}_S(K)_\ell)=1$ and infinitely many $p_3$ such that $d(\operatorname{Cl}_S(K)_\ell)>1$.

#### Bibliography

1. L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places”, Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019, 78–99        ; English transl. Proc. Steklov Inst. Math., 307 (2019), 65–84
2. L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. II”, Izv. Ross. Akad. Nauk Ser. Mat., 85:5 (2021), 132–151        ; English transl. Izv. Math., 85:5 (2021), 953–971
3. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ, 1956    ; Russian transl. Inostr. Lit., Moscow, 1960

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