Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

TL;DR

This paper investigates the structure and properties of unramified Iwasawa modules over multiple zp-extensions of CM-fields, providing formulas for Galois coinvariants and criteria for cyclicity in special cases.

Contribution

It offers new descriptions of Galois coinvariants and conditions for cyclicity of Iwasawa modules in multiple zp-extensions, extending prior understanding of their structure.

Findings

Order of Galois coinvariants expressed via characteristic power series

Necessary and sufficient conditions for cyclicity of Iwasawa modules

Results depend on splitting behavior of primes and Iwasawa invariants

Abstract

For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second one is that if $K$ is an imaginary quadratic field and $p$ does not split in $K$, we give a necessary and sufficient condition for which $X_{\widetilde K}$ is $\mathbb{Z}_p[[{\rm Gal}(\widetilde K/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$, where $\widetilde K$ is the $\mathbb{Z}_p^2$-extension of $K$.

Equations

$$\#{\left(X_{{\tilde{K}}_{\chi }}/(S,T_1,\cdots ,T_{d_{\chi }})\right)}^{\chi }=\#{\mathcal{O}}_{\chi }/f_{\chi }^{\ast },$$

$${\left((X_{{\tilde{K}}^{\prime }}{)}_{\mathrm{Gal}({\tilde{K}}^{\prime }/K_{\infty })}\right)}^{\chi }\simeq {\left((X_{{\tilde{K}}_{\chi }}{)}_{\mathrm{Gal}({\tilde{K}}_{\chi }/K_{\infty })}\right)}^{\chi }.$$

$$\#{\left((X_{\tilde{K}}{)}_{\mathrm{Gal}(\tilde{K}/K)}\right)}^{\chi }=\#{\mathcal{O}}_{\chi }/f_{\chi }^{\ast }.$$

$$\left({\mathrm{char}}_{{\mathbb{Z}}_p[[S,T]]}({\mathfrak{X}}_{\mathfrak{P}}(\tilde{K}))\right){\mathbb{Z}}_p[[S,T]]=\left(\det (S\cdot I_{\lambda -1}-A)\right){\mathbb{Z}}_p[[S,T]],$$

$$\left({\mathrm{char}}_{{\mathbb{Z}}_p[[S,T]]}({\mathfrak{X}}_{\mathfrak{P}}(\tilde{K})){\mid }_{T=0}\right){\mathbb{Z}}_p[[S]]=\left(\frac{{\mathrm{char}}_{{\mathbb{Z}}_p[[S]]}(X_{K_{\infty }})}{S}\right){\mathbb{Z}}_p[[S]]$$

$$\#\left((X_{\tilde{K}}{)}_{\mathrm{Gal}(\tilde{K}/K)}\right)=\#{\mathbb{Z}}_p/\left({\mathrm{char}}_{{\mathbb{Z}}_p[[S,T]]}({\mathfrak{X}}_{\mathfrak{P}}(\tilde{K})){\mid }_{S=T=0}\right).$$

$${\dim }_{{\mathbb{F}}_p}(A_K/p)-1\le {\dim }_{{\mathbb{F}}_p}(X_{\tilde{K}}/(p,S,T))\le {\dim }_{{\mathbb{F}}_p}(A_K/p)+1$$

$${\dim }_{{\mathbb{F}}_p}(X_{\tilde{K}}/(p,S,T))=\{\begin{array}{l}1\text{if LK⊂K},\\ 2\text{otherwise}.\end{array}$$

$$\begin{array}{ccccccccc}0& \to & L& \to & M& \to & N& \to & 0\\ & & & & & & & & \\ \alpha \downarrow & & \beta \downarrow & & \gamma \downarrow & & & & \\ & & & & & & & & \\ \to & L& \to & M& \to & N& \to & & \end{array}$$

$$\#\frac{M}{M[\beta ]+\beta M}=\#\frac{L}{L[\alpha ]+\alpha L}\cdot \#\frac{M[\beta ]\cap \beta M}{L[\alpha ]\cap \alpha L}$$

$$\begin{array}{ccccccccc}0& \to & L[\alpha ]\oplus \alpha L& \to & M[\beta ]\oplus \beta M& \to & \frac{M[\beta ]}{L[\alpha ]}\oplus \frac{\beta M}{\alpha L}& \to & 0\\ & & & & & & & & \\ l\downarrow & & m\downarrow & & n\downarrow & & & & \\ 0& \to & L& \to & M& \to & N& \to & 0,\end{array}$$

$$\mathrm{Coker}(n)=M/(L+M[\beta ]+\beta M)=M/(L+M[\beta ]).$$

$$0\to \frac{M[\beta ]\cap \beta M}{L[\alpha ]\cap \alpha L}\to \mathrm{Ker}(n)\to \frac{L}{L[\alpha ]+\alpha L}\to \frac{M}{M[\beta ]+\beta M}\to \mathrm{Coker}(n)\to 0.$$

$$\#\frac{M}{M[\beta ]+\beta M}=\#\frac{L}{\alpha L}.$$

$$0\to M_{i+1}\to M_i\to N_i\to 0(i\ge 0).$$

$$\#\frac{M_i}{M_i[\beta ]+\beta M_i}=\#\frac{M_{i+1}}{M_{i+1}[\beta ]+\beta M_{i+1}}\cdot \#\frac{M_i[\beta ]\cap \beta M_i}{M_{i+1}[\beta ]\cap \beta M_{i+1}}.$$

$$0\to \frac{M[\beta ]\cap \beta M}{L[\alpha ]\cap \alpha L}\to \frac{L[\alpha ]}{L[\alpha ]\cap \alpha L}\to \frac{M[\beta ]}{M[\beta ]\cap \beta M}$$

$$\#\frac{M}{M[\beta ]+\beta M}=\#\frac{L}{L[\alpha ]+\alpha L}\cdot \#\frac{M[\beta ]\cap \beta M}{L[\alpha ]\cap \alpha L}=\#\frac{L}{\alpha L}$$

$$0\to X\to E\to C\to 0$$

$$\#(X/X[S]+SX)=\#\mathcal{O}/f^{\ast }.$$

$$E[S]=0,E/E[S]+SE\simeq \mathcal{O}/(f(0{)}^n).$$

$$SE=0,E/E[S]+SE=0.$$

$$\mathrm{Gal}({\tilde{K}}_{\chi }/K_{\infty })\simeq {\mathcal{O}}_{\chi }$$

$$1\to X_{{\tilde{K}}_{\chi }}\to \mathrm{Gal}(L_{{\tilde{K}}_{\chi }}/K_{\infty })\to {\mathcal{O}}_{\chi }\to 1$$

$$1\to X_{{\tilde{K}}^{\prime }}\to \mathrm{Gal}(L_{{\tilde{K}}^{\prime }}/K_{\infty })\to \mathrm{Gal}({\tilde{K}}^{\prime }/K_{\infty })\to 1.$$

$$0\to Y^{\chi }\to X_{K_{\infty }}^{\chi }\to {\mathcal{O}}_{\chi }\to 0.$$

$$H_2({\mathcal{O}}_{\chi },{\mathbb{Z}}_p)\to (X_{{\tilde{K}}_{\chi }}{)}_{\mathrm{Gal}({\tilde{K}}_{\chi }/K_{\infty })}\to X_{K_{\infty }}\to {\mathcal{O}}_{\chi }\to 0.$$

$$H_2(\mathrm{Gal}({\tilde{K}}^{\prime }/K_{\infty }),{\mathbb{Z}}_p)\to (X_{{\tilde{K}}^{\prime }}{)}_{\mathrm{Gal}({\tilde{K}}^{\prime }/K_{\infty })}\to X_{K_{\infty }}\to \mathrm{Gal}({\tilde{K}}^{\prime }/K_{\infty })\to 0$$

$$\#(X_{{\tilde{K}}_{\chi }}/(S,T_1,\dots ,T_d){)}^{\chi }$$

$$\#(X_{{\tilde{K}}_{\chi }}/(S,T_1,\dots ,T_d){)}^{\chi }=\#{\mathcal{O}}_{\chi }/f_{\chi }^{\ast }.$$

$$[\mathrm{Gal}(\tilde{K}/K):\mathfrak{D}]=\#({\mathbb{Z}}_p/f^{\ast }),$$

Full text

1 Introduction

1.1 The unramified Iwasawa modules

Let $p$ be an arbitrary prime number, $K$ a finite extension of the rational number field $\mathbb{Q}$ and $K_{\infty}/K$ the cyclotomic $\mathbb{Z}_{p}$-extension. For an arbitrary algebraic number field $F$, we denote by $X_{F}$ the Galois group of the maximal unramified abelian $p$-extension over $F$. The module $X_{K_{\infty}}$ is called the unramified Iwasawa module of $K_{\infty}$. In the case where $K$ is totally real, little is known about the structure of $X_{K_{\infty}}$, although the Iwasawa main conjecture gives us highly nontrivial information about the minus part of $X_{K_{\infty}}$ in the case where $K$ is a CM-field. Greenberg conjectured in [9] that $X_{K_{\infty}}$ would be finite if $K$ is totally real, which is called Greenberg’s conjecture. A lot of efforts by a number of mathematicians have revealed that this conjecture holds true in many cases, but it still remains unsolved (in general).

We consider the maximal multiple $\mathbb{Z}_{p}$-extension $\widetilde{K}/K$ and its unramified Iwasawa module $X_{\widetilde{K}}$. It is known that $X_{\widetilde{K}}$ is a finitely generated torsion $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}/K)]]$-module (see [8]). There is a conjecture that $X_{\widetilde{K}}$ would be pseudo-null as a $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}/K)]]$-module, which is called Greenberg’s generalized conjecture (“pseudo-null” is defined in §3). Concerning this conjecture and its application, there are many studies (Bleher et al. [1], Fujii [7], Itoh [11], Ozaki [16], and Minardi [14], etc.). However, even if Greenberg’s generalized conjecture is true, it just states that the characteristic ideal of $X_{\widetilde{K}}$ is trivial, so that it seems difficult to consider any analogues of the Iwasawa invariants and the Iwasawa main conjecture for $X_{\widetilde{K}}$. Therefore, it is worthwhile to study not only Greenberg’s generalized conjecture, but also various basic properties of $X_{\widetilde{K}}$, for example, the number of generators as a $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}/K)]]$-module, its Galois (co)invariants, and cohomogical properties.

In the following, we always assume that $p$ is odd. In this paper, we study the Galois coinvariants of the unramified Iwasawa modules of a certain multiple $\mathbb{Z}_{p}$-extension in a relatively general situation. Roughly speaking, this paper consists of two parts “Split case” (§2, 3) and “Non-split case” (§4, 5). In Split case, we consider a certain multiple $\mathbb{Z}_{p}$-extension $\widetilde{K}^{\prime}$ of a CM-field $K$ which satisfies the condition of Gross’s conjecture of rank one. If $K$ is an abelian extension in which $p$ splits completely and the degree of $K$ is coprime to $p$, then $\widetilde{K}^{\prime}$ is coincide with $\widetilde{K}$. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde{K}^{\prime}}$ in terms of the characteristic power series of $X_{K_{\infty}}$ (Theorem 1.1). In addition, for an imaginary quadratic field $K$ in which $p$ splits as $(p)=\mathfrak{P}\overline{\mathfrak{P}}$, we give a theorem which suggests that the characteristic ideal of $\mathfrak{P}$-ramified Iwasawa module of $\widetilde{K}$ relates to the structure of $X_{\widetilde{K}}$ (Theorem 1.4). On the other hand, in Non-split case, we consider $\widetilde{K}$ for an imaginary quadratic field $K$ in which $p$ does not split. Our second main result is to give a necessary and sufficient condition for which $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$ (Theorems 1.5 and 5.11). Such $X_{\widetilde{K}}$ will be useful for studying the Iwasawa theory of multiple $\mathbb{Z}_{p}$-extensions. We remark that our results do not need the assumption that Greenberg’s generalized conjecture holds.

1.2 Notation

Throughout this paper, we use the following notation. Let $p$ be an odd prime number, $k$ a totally real number field, $K$ a CM-field such that $K/k$ is a finite abelian extension of degree coprime to $p$, $K_{\infty}/K$ the cyclotomic $\mathbb{Z}_{p}$-extension, and $\widetilde{K}$ the maximal multiple $\mathbb{Z}_{p}$-extension over $K$. For any (finite or infinite) extension $F$ over $\mathbb{Q}$, we denote by $L_{F}$ and $X_{F}$ the maximal unramified abelian $p$-extension of $F$ and the Galois group of $L_{F}$ over $F$, respectively. If $F$ is a finite extension of $\mathbb{Q}$, we denote by $A_{F}$ the $p$-Sylow subgroup of the ideal class group of $F$. We identify $\mathbb{Z}_{p}[[{\rm Gal}(K_{\infty}/K)]]$ with the ring of formal power series $\mathbb{Z}_{p}[[S]]$ by regarding a fixed generator of ${\rm Gal}(K_{\infty}/K)$ as $1+S$. For a character $\chi\colon{\rm Gal}(K/k)\to\overline{\mathbb{Q}}_{p}^{\times}$, we denote by $\mathcal{O}_{\chi}$ the ring obtained from $\mathbb{Z}_{p}$ by adjoining all values of $\chi$. For any ${\rm Gal}(K/k)$-module $M$, put $M^{\chi}:=M\otimes_{\mathbb{Z}_{p}[{\rm Gal}(K/k)]}\mathcal{O}_{\chi}$. We denote by $\mu_{p}$ the set of all $p$-th roots of unity, and simply by $X/\mathfrak{a}$ a quotient module $X/\mathfrak{a}X$ . Let $\Lambda$ be the ring either $\mathcal{O}[[S]]$ or $\mathcal{O}[[S,T]]$, where $\mathcal{O}$ is $\mathbb{Z}_{p}$ or $\mathcal{O}_{\chi}$. For any finitely generated torsion $\Lambda$-module $X$, we call a generator of the characteristic ideal of $X$ a characteristic power series of $X$, and denote it by ${\rm char}_{\Lambda}(X)\in\Lambda$, which is determined up to $\Lambda^{\times}$.

1.3 Main theorems of “Split case”

Let $\chi\colon{\rm Gal}(K/k)\to\overline{\mathbb{Q}}_{p}^{\times}$ be an odd character. Assume that there is only one prime ideal $\mathfrak{p}$ in $k$ above $p$ which satisfies that $\chi(\mathfrak{p})=1$. Let $\widetilde{K}^{\prime}/K$ be the maximal multiple $\mathbb{Z}_{p}$-extension such that $K_{\infty}\subset\widetilde{K}^{\prime}$ and $\widetilde{K}^{\prime}/K_{\infty}$ is unramified. We know that ${\rm Gal}(\widetilde{K}^{\prime}/K)$ is a ${\rm Gal}(K/k)$-module. Note that if $k=\mathbb{Q}$, then $\widetilde{K}=\widetilde{K}^{\prime}$. By [3] (as a consequence of Lemma 1.5 and Proposition 1.6), we see that ${\rm Gal}(\widetilde{K}^{\prime}/K)^{\chi}\simeq\mathcal{O}_{\chi}$. Let $\widetilde{K}_{\chi}$ be the subextension of $\widetilde{K}/K_{\infty}$ such that ${\rm Gal}(\widetilde{K}_{\chi}/K_{\infty})$ is isomorphic to ${\rm Gal}(\widetilde{K}^{\prime}/K)^{\chi}$. We identify $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}_{\chi}/K)]]=\mathbb{Z}_{p}[[{\rm Gal}(K_{\infty}/K)\times{\rm Gal}(\widetilde{K}_{\chi}/K_{\infty})]]$ with $\mathbb{Z}_{p}[[S,T_{1},\ldots,T_{d_{\chi}}]]$ in a similar way as we did $\mathbb{Z}_{p}[[{\rm Gal}(K_{\infty}/K)]]$ with $\mathbb{Z}_{p}[[S]]$, where $d_{\chi}:=[\mathcal{O}_{\chi}:\mathbb{Z}_{p}]$.

Theorem 1.1**.**

Let $p$, $k$, $K$ be as in §1.2, $\chi\colon{\rm Gal}(K/k)\to\overline{\mathbb{Q}}_{p}^{\times}$ an odd character. Assume that $\mu_{p}\not\subset K$ and that there exists just only one prime ideal $\mathfrak{p}$ in $k$ above $p$ which satisfies that $\chi(\mathfrak{p})=1$. Then ${\rm Gal}(K/k)$-module $(X_{\widetilde{K}_{\chi}})_{{\rm Gal}(\widetilde{K}_{\chi}/K)}=X_{\widetilde{K}_{\chi}}/(S,T_{1},\cdots,T_{d_{\chi}})$ satisfies

[TABLE]

where $f_{\chi}^{*}$ is the first non-vanishing coefficient of a characteristic power series of the $\mathcal{O}_{\chi}[[S]]$-module $X_{K_{\infty}}^{\chi}$. Moreover, assume that Leopoldt’s conjecture holds for the maximal totally real subfield $K^{+}$ and $p$, then there is a canonical isomorphism

[TABLE]

Remark 1.2**.**

Since Gross’s ‘order of vanishing conjecture’ (see Gross [10, Conjecture 1.15] or Dasgupta, Kakde, and Ventullo [4, Conjecture 1], for example) of rank one for $K/k$ holds by the argument in [10, Proposition 2.13], $f_{\chi}^{*}$ turns out to be the coefficient of degree $1$. **

Corollary 1.3**.**

Let $p$ be an odd prime number, $K$ an imaginary abelian finite extension over $\mathbb{Q}$ of degree coprime to $p$ in which $p$ splits completely. Then, for any odd character $\chi$ of ${\rm Gal}(K/k)$, we have

[TABLE]

In §3, we will consider the case where $K$ is an imaginary quadratic field such that $p$ splits completely as $(p)=\mathfrak{P}\overline{\mathfrak{P}}$ and connect the above corollary with the Galois group $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ of the maximal abelian $p$-extension of $\widetilde{K}$ which is unramified outside all primes above $\mathfrak{P}$. It is known that $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ is finitely generated and torsion over $\mathbb{Z}_{p}[[S,T]]=\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}/K)]]$ $(T:=T_{1})$ by [18, Theorem 5.3 (ii)]. Therefore, we can consider the characteristic ideal of $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ as a $\mathbb{Z}_{p}[[S,T]]$-module, which plays an important role in the Iwasawa main conjecture. We denote by $\lambda$ the Iwasawa $\lambda$-invariant of $K_{\infty}/K$. Then it is known that $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ is generated by $\lambda-1$ elements as a $\mathbb{Z}_{p}[[T]]$-module. Moreover, let $I_{\lambda-1}$ be the identity matrix of size $\lambda-1$ and $A$ a matrix associated to multiplication by $S$ on $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ whose entries are in $\mathbb{Z}_{p}[[T]]$. We will show the following theorem which suggests that the characteristic ideal of $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ relates to the structure of $X_{\widetilde{K}}$.

Theorem 1.4**.**

Let $p$ be an odd prime number, $K$ an imaginary quadratic field such that $p$ splits completely as $(p)=\mathfrak{P}\overline{\mathfrak{P}}$ and $\widetilde{K}$ the unique $\mathbb{Z}_{p}^{2}$-extension of $K$. Assume that $L_{K}\subset\widetilde{K}$ or that the characteristic ideal of $X_{K_{\infty}}$ does not have any square factor. Then the characteristic power series ${\rm char}_{\mathbb{Z}_{p}[[S,T]]}(\mathfrak{X}_{\mathfrak{P}}(\widetilde{K}))$ of $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ satisfies

[TABLE]

as ideals. In particular, combining these with Corollary 1.3, we have

[TABLE]

1.4 Main theorems of “Non-split case”

In the latter half of this paper, we consider the case where $K$ is an imaginary quadratic field such that $p$ does not split. Then it is well known that the number of generators of $X_{K_{\infty}}$ as a $\mathbb{Z}_{p}[[S]]$-module is equal to $\dim_{\mathbb{F}_{p}}(A_{K}/p)$. Our target is the number of generators of $X_{\widetilde{K}}$ as a $\mathbb{Z}_{p}[[S,T]]$-module, which is equal to $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))$ by Nakayama’s lemma. It is easy to see that

[TABLE]

(see §4.1). In §4.1, we describe a system of generators of $X_{\widetilde{K}}$ and show the following theorem which classifies $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))$ in the case where $A_{K}$ is a cyclic abelian group.

Theorem 1.5**.**

Let $p$ be an odd prime number and $K$ an imaginary quadratic field such that $p$ does not split.

• (i)

(trivial case)* Assume that $L_{K}\cap\widetilde{K}=K$, then $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=\dim_{\mathbb{F}_{p}}(A_{K}/p).$*

• (ii)

Suppose that $L_{K}\cap\widetilde{K}\neq K$, and that $\dim_{\mathbb{F}_{p}}(A_{K}/p)=1$.

• (ii-a)

If $\lambda=1$, then $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=1.$

• (ii-b)

If $\lambda\geq 2$, then

[TABLE]

From the point of view of applications, we are interested in the condition for $X_{\widetilde{K}}$ to be $\mathbb{Z}_{p}[[S,T]]$-cyclic. In §5, we classify when $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic in a certain case with $\dim_{\mathbb{F}_{p}}(A_{K}/p)=1$ and $\lambda=2$ (Theorem 5.11). We are going to give some numerical examples about it and introduce a method of calculating these examples in the forth coming paper.

2 Proof of Theorem 1.1

2.1 Preliminary from general module theory

For a module $M$ and a morphism $\varphi\colon M\to M$, we define $M[\varphi]:={\rm Ker}(\varphi)$ and $\varphi M:={\rm Im}(M)$.

Proposition 2.1**.**

Let

[TABLE]

be an exact commutative diagram of modules. Regard $L$ as a submodule in $M$. If $\gamma N=0$ and $\#(M/M[\beta]+\beta M)<\infty$, then the following equation

[TABLE]

holds.

Proof.

We consider an exact commutative diagram

[TABLE]

where the vertical maps are defined by the difference between the two components, for example, $l\colon(x,y)\mapsto x-y$. We will show ${\rm Ker}(n)\simeq{\rm Coker}(n)$. Since $\gamma N=0$, the image of $\beta M/\alpha L$ by $n$ is zero. Also the natural map $M[\beta]/L[\alpha]\to N[\gamma]=N$ is injective by the snake lemma. Hence we obtain ${\rm Ker}(n)=\beta M/\alpha L.$ On the other hand, we see

[TABLE]

Hence an exact sequence $0\to L+M[\beta]\to M\stackrel{{\scriptstyle\beta}}{{\to}}\beta M/\alpha L\to 0$ induces an isomorphism $\beta M/\alpha L\simeq{\rm Coker}(n)$. Therefore we obtain ${\rm Ker}(n)\simeq{\rm Coker}(n)$. Applying the snake lemma to the first exact commutative diagram, we have an exact sequence

[TABLE]

From this and the assumption that $\#M/M[\beta]+\beta M<\infty$, we have the claim. ∎

Corollary 2.2**.**

For the exact commutative diagram (2.1) in Proposition 2.1, suppose that $\#(M/M[\beta]+\beta M)<\infty$.

• (i)

Assume that there exists some integer $n>0$ such that $\gamma^{n}N=0$. Then the equation (2.2) holds. In particular, if all modules in (2.1) are $\mathbb{Z}_{p}[[S]]$-modules, the maps $\alpha$, $\beta$, $\gamma$ are multiplication by $S$, and $N$ is finite, then the equation (2.2) holds.

• (ii)

Assume that $\gamma N=0$, $M[\beta]+\beta M/\beta M$ is torsion-free, and $L/\alpha L$ is finite. Then

[TABLE]

Proof.

(i) Put $M_{0}:=M$, $M_{i}:=\beta^{i}M+L$ ($i\geq 0$). Note that $M_{n}=L$, since $0=\gamma^{n}N\simeq\beta^{n}M+L/L$. Define $N_{i}$ by the exact sequence

[TABLE]

Then we have $\beta N_{i}=0$. Therefore we can easily check that $\#(M_{i}/M_{i}[\beta]+\beta M_{i})<\infty$ for $i=0,\ldots,n$, using the same method as in the proof of Proposition 2.1 and the induction on $i$. Therefore we can apply Proposition 2.1 to the above exact sequence, so that

[TABLE]

Taking the products from $i=0$ to $i=n-1$, we have the claim since $M_{n}=L$.

(ii) Note that $\beta M\subset L$ and $L[\alpha]/L[\alpha]\cap\alpha L$ is finite by the assumption. Hence the exact sequence

[TABLE]

implies that $({M[\beta]\cap\beta M})/({L[\alpha]\cap\alpha L})\simeq{L[\alpha]+\alpha L}/{\alpha L}$, since ${M[\beta]+\beta M}/{\beta M}$ is torsion-free. Therefore,

[TABLE]

by (i). This completes the proof. ∎

Let $\mathcal{O}$ be the ring either $\mathbb{Z}_{p}$ or $\mathcal{O}_{\chi}$. Applying the corollary to $\mathcal{O}[[S]]$-modules, we obtain the following corollary. Let $X$ be a finitely generated torsion $\mathcal{O}[[S]]$-module which has no nontrivial finite $\mathcal{O}[[S]]$-submodules. Then, by the structure theorem of finitely generated torsion $\mathcal{O}[[S]]$-modules (see [19, Theorem 13.12]), there is an exact sequence of $\mathcal{O}[[S]]$-modules

[TABLE]

such that $E$ is described as $E=\bigoplus_{j}\mathcal{O}[[S]]/(f_{j}(S)^{n_{j}})$ and $C$ is finite. Here, each $f_{j}(S)\in\mathcal{O}[S]$ is an irreducible distinguished polynomial or a uniformizer in $\mathcal{O}$.

Corollary 2.3**.**

With the notation as above, we denote the first non-vanishing coefficient of $\prod_{j}f_{j}(S)^{n_{j}}$ by $f^{*}$. If $S^{2}$ does not divide $f_{j}(S)^{n_{j}}$ for any $j$, then

[TABLE]

Proof.

In the case where $E=\mathcal{O}[[S]]/(f(S)^{n})$ with $S\nmid f(S)$, we have

[TABLE]

And also, in the case where $E=\mathcal{O}[[S]]/(S)$, we see

[TABLE]

Hence, in both cases, $E[S]\cap SE=0$. Applying Corollary 2.2 (i) to the exact sequence (2.3), we have $\#(X/X[S]+SX)=\#(E/E[S]+SE)=\#\mathcal{O}/f^{*}.$ This completes the proof. ∎

2.2 Galois coinvariants

We use the notation in §1.2. Let $\chi\colon{\rm Gal}(K/k)\to\overline{\mathbb{Q}}_{p}^{\times}$ be an odd character. Suppose that the assumption in Theorem 1.1. In other words, we assume that $\mu_{p}\not\subset K$ and there is only one prime ideal $\mathfrak{p}$ lying above $p$ in $k$ which satisfies $\chi(\mathfrak{p})=1$. Recall that $\widetilde{K}^{\prime}$ is the maximal multiple $\mathbb{Z}_{p}$-extension such that $K_{\infty}\subset\widetilde{K}^{\prime}$ and $\widetilde{K}^{\prime}/K_{\infty}$ is unramified. Then there is a unique abelian extension $\widetilde{K}_{\chi}/K$ contained in $\widetilde{K}^{\prime}$ such that

[TABLE]

as ${\rm Gal}(K/k)$-modules by [3, Lemma 1.5]. Also, recall that we identify $\mathbb{Z}_{p}[[{\rm Gal}(\widetilde{K}_{\chi}/K)]]$ with $\mathbb{Z}_{p}[[S,T_{1},\ldots,T_{d_{\chi}}]]$, where $d_{\chi}:=[\mathcal{O}_{\chi}:\mathbb{Z}_{p}]$. Since $L_{\widetilde{K}_{\chi}}/k$ is a Galois extension because of the maximality of $L_{\widetilde{K}_{\chi}}$, ${\rm Gal}(\widetilde{K}_{\chi}/k)$ acts on $X_{\widetilde{K}_{\chi}}$ by the inner product: $\alpha(x):=\widetilde{\alpha}x\widetilde{\alpha}^{-1}$ for $\alpha\in{\rm Gal}(\widetilde{K}_{\chi}/k)$ and $x\in X_{\widetilde{K}_{\chi}}$, where $\widetilde{\alpha}\in{\rm Gal}(L_{\widetilde{K}_{\chi}}/k)$ is a fixed extension of $\alpha$. Note that the action is independent of the choice of extensions $\widetilde{\alpha}$. By this action, $X_{\widetilde{K}_{\chi}}$ becomes a ${\rm Gal}(\widetilde{K}_{\chi}/k)$-module. We have two exact sequences

[TABLE]

and

[TABLE]

Put $Y:=X_{\widetilde{K}_{\chi}}/(T_{1},\ldots,T_{d_{\chi}})$. The first part of the following lemma is a partial generalization of Ozaki [16, Lemma 1].

Lemma 2.4**.**

• (i)

There is an exact sequence of $\mathbb{Z}_{p}[[S]]$-modules

[TABLE]

• (ii)

Assume that Leopoldt’s conjecture holds for $K^{+}$ and $p$, then the latter half of Theorem 1.1 holds, in other words, there is a canonical isomorphism $\left((X_{\widetilde{K}^{\prime}})_{{\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})}\right)^{\chi}\simeq Y^{\chi}$.

Proof.

(i) Taking the Hochschild-Serre spectral sequence of (2.4), we obtain an exact sequence of ${\rm Gal}(K_{\infty}/k)$-modules

[TABLE]

Since ${\rm Gal}(K/K^{+})$ acts trivially on $H_{2}(\mathcal{O}_{\chi},\mathbb{Z}_{p})\simeq\mathcal{O}_{\chi}\wedge_{\mathbb{Z}_{p}}\mathcal{O}_{\chi}$ (here, ${\rm Gal}(K/K^{+})$ acts on the right hand side diagonally), its $\chi$-component is trivial. Hence we obtain the claim.

(ii) In the same way, we have

[TABLE]

from (2.5). Combining this with (i), we have only to show that $\left({\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})\wedge_{\mathbb{Z}_{p}}{\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})\right)^{-}=0$, since ${\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})^{\chi}={\rm Gal}(\widetilde{K}_{\chi}/K_{\infty})$. For this, it is enough to show that ${\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})^{+}=0$. Assume that it does not hold. Then ${\rm Gal}(\widetilde{K}^{\prime}/K_{\infty})^{+}$ is nontrivial and torsion-free, which implies that there exists a $\mathbb{Z}_{p}$-extension of $K^{+}$ different from the cyclotomic $\mathbb{Z}_{p}$-extension. This contradicts our assumption that Leopoldt’s conjecture holds. ∎

It is known that $(X_{K_{\infty}}/X_{K_{\infty}}[S]+SX_{K_{\infty}})^{\chi}$ is finite and $(X_{K_{\infty}}[S]+SX_{K_{\infty}}/SX_{K_{\infty}})^{\chi}$ is torsion-free (see Kurihara [13], for example), since Gross’s order of vanishing conjecture holds. Applying Corollary 2.2 (ii) to the exact sequence (2.6), we have

[TABLE]

since the actions of ${\rm Gal}(K/k)$ and ${\rm Gal}(K_{\infty}/K)$ are commutative. Again, since the Gross’s conjecture holds, we can apply Corollary 2.3 to get

[TABLE]

This completes the proof of Theorem 1.1.

Remark 2.5**.**

Under several assumptions, we can calculate the value $f^{*}$ in Corollary 1.3. More precisely, let $K$ be an imaginary quadratic field and $p$ an odd prime number. We suppose that $L_{K}\subset\widetilde{K}$. We also suppose that $p\geq 5$ if $p$ does not split in $K$. Then we have

[TABLE]

where $\mathfrak{D}$ is the decomposition group in $\mathrm{Gal}(\widetilde{K}/K)$ of a prime lying above $p$ (Murakami [15, Proposition 3.4]). **

3 Proof of Theorem 1.4

We consider the case where $K$ is an imaginary quadratic field such that $p$ splits completely as $(p)=\mathfrak{P}\overline{\mathfrak{P}}$. Then $\widetilde{K}^{\prime}=\widetilde{K}$. Let $\Lambda$ be the ring either $\mathbb{Z}_{p}[[S]]$ or $\mathbb{Z}_{p}[[S,T]]$. Recall that, for any finitely generated torsion $\Lambda$-module $X$, we chose a characteristic power series ${\rm char}_{\Lambda}(X)\in\Lambda$ of $X$. A $\Lambda$-module $X$ is called pseudo-null if $X$ has two relatively prime annihilators in $\Lambda$.

Lemma 3.1**.**

Let $0\to X\to E\to C\to 0$ be an exact sequence of finitely generated torsion $\Lambda$-modules. For a $\Lambda$-module $M$, denote by ${\rm Ann}_{\Lambda}(M)$ the annihilator ideal of $M$.

• (i)

If $E$ has no non-trivial pseudo-null $\Lambda$-submodules and $C$ is a pseudo-null $\Lambda$-module, then we have ${\rm Ann}_{\Lambda}(X)={\rm Ann}_{\Lambda}(E)$.

• (ii)

Furthermore, suppose that $E=\bigoplus^{s}_{i=1}\Lambda/\mathfrak{p}_{i}^{n_{i}}$ where each $\mathfrak{p}_{i}$ is a height one prime ideal with $\mathfrak{p}_{i}\neq\mathfrak{p}_{j}$ for $i\neq j$. Then we have ${\rm Ann}_{\Lambda}(X)=({\rm char}_{\Lambda}(X))$.

Proof.

(i) The inclusion ${\rm Ann}_{\Lambda}(X)\supset{\rm Ann}_{\Lambda}(E)$ is obvious. We will show the other inclusion. Let $f\in\Lambda$. Then we have an exact sequence

[TABLE]

Suppose that $f\in{\rm Ann}_{\Lambda}(X)$. Then we have $X/f=X$, hence we have an commutative diagram

[TABLE]

By snake lemma, we have $0\to fE\to fC\to 0$, hence $fE$ is a pseudo-null $\Lambda$-module. Since $E$ has no non-trivial pseudo-null $\Lambda$-submodules, we have $fE=0$. Thus we have $f\in{\rm Ann}_{\Lambda}(E)$, hence we have proved ${\rm Ann}_{\Lambda}(X)\subset{\rm Ann}_{\Lambda}(E)$.

(ii) This follows easily since ${\rm Ann}_{\Lambda}(E)=({\rm char}_{\Lambda}(X))$ in this case. ∎

Let $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ be the Galois group of the maximal $p$-extension of $\widetilde{K}$ unramified outside all primes above $\mathfrak{P}$. Put

[TABLE]

for simplicity. We may assume that $F(S)$ is a distinguished polynomial with degree $\lambda$, where $\lambda$ is the Iwasawa $\lambda$-invariant of $K_{\infty}/K$. From Lemma 2.4 (i), we have the exact sequence

[TABLE]

Therefore $F(S)$ is written as $F(S)=SF^{*}(S)$ for some distinguished polynomial $F^{*}(S)$ which is coprime to $S$ with degree $\lambda-1$, and $X_{\widetilde{K}}/T$ has no nontrivial finite $\mathbb{Z}_{p}[[S]]$-submodules. On the other hand, by Fujii [6, Lemma 3], $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ has no nontrivial pseudo-null submodules and

[TABLE]

since $K$ is an imaginary quadratic field. Therefore $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})/T$ is $\mathbb{Z}_{p}[[S]]$-torsion. This yields that $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})[T]$ is a pseudo-null $\mathbb{Z}_{p}[[S,T]]$-module by Perrin-Riou [17, Chapitre I Lemme 4.2 in page 12], so that $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})[T]=0$. Hence we have

[TABLE]

as ideals in $\mathbb{Z}_{p}[[S]]$, again by the same lemma in [17]. By (3.2), we can take $\lambda-1$ generators $\xi_{1},\ldots,\xi_{\lambda-1}$ of $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ as a $\mathbb{Z}_{p}[[T]]$-module. Define a matrix $A$ and $f(S,T)\in\mathbb{Z}_{p}[[T]][S]$ by

[TABLE]

and $f(S,T)=\det(S\cdot I_{\lambda-1}-A)$, respectively. Then $f(S,T)$ annihilates $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$, and also $X_{\widetilde{K}}$. Hence, $f(S,0)$ annihilates $X_{\widetilde{K}}/T$. Note that the degree of $f(S,T)$ with respect to $S$ is $\lambda-1$. Now we show Theorem 1.4.

Theorem 3.2**.**

(Theorem 1.4)* Suppose that $L_{K}\subset\widetilde{K}$ or $F(S)$ does not have any square factor. Then the following equalities*

[TABLE]

as ideals hold.

Proof.

First, suppose that $F(S)$ does not have any square root. Combining the equation $(F^{*}(S))=(\mathcal{F}_{\mathfrak{P}}(S,0))$ with this assumption, we see that $\mathcal{F}_{\mathfrak{P}}(S,T)$ does not have any square factor. By Lemma 3.1 (ii),

[TABLE]

for some $g(S,T)\in\mathbb{Z}_{p}[[S,T]]$. Since $\deg F^{*}(S)=\lambda-1=\deg f(S,0)$ and $f(S,0)$ is monic, we obtain that $F^{*}(S)$ is equal to $f(S,0)=\mathcal{F}_{\mathfrak{P}}(S,0)g(S,0)$ up to multiplication by unit. Hence, $g(S,T)\in\mathbb{Z}_{p}[[S,T]]^{\times}$, which yields the claim.

Next, suppose that $L_{K}\subset\widetilde{K}$. Then it is known that $\widetilde{K}$ coincides with the maximal abelian $p$-extension of $K$ which is unramified all primes outside above $p$. Hence $X_{K_{\infty}}/S\simeq\mathbb{Z}_{p}$. Applying the snake lemma to (3.1), we obtain an exact sequence $\mathbb{Z}_{p}\to X_{\widetilde{K}}/(S,T)\to X_{K_{\infty}}/S\to\mathbb{Z}_{p}\to 0$. Hence $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic and so $\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ is. This means that there are surjections $\mathbb{Z}_{p}[[S,T]]/f(S,T)\twoheadrightarrow\mathfrak{X}_{\mathfrak{P}}(\widetilde{K})$ and $\mathbb{Z}_{p}[[S]]/f(S,0)\twoheadrightarrow X_{\widetilde{K}}/T$. So, in the same way as above, we obtain the claim. ∎

4 Proof of Theorem 1.5

4.1 A system of generators of $X_{\widetilde{K}}$

First of all, we show a lemma from group theory. Let $p$ be an arbitrary prime number and $G$ a finite abelian $p$-group such that $\dim_{\mathbb{F}_{p}}G/p=g$. For $x\in G$ with order $p^{n}$, $\langle x\rangle_{n}$ denotes the cyclic group generated by $x$ (we also use the notation $\langle x\rangle$ instead of $\langle x\rangle_{n}$).

Lemma 4.1**.**

With the notation as above, let $H$ be a subgroup of $G$ such that $G/H$ is cyclic, then there exists a minimal system of generators $x_{1},x_{2}\ldots,x_{g}\in G$ such that the following holds:

• (i)

$G/H$* is generated by the image of $x_{1}$ under the projection. Moreover, $x_{1}$ has the minimum order among such elements.*

• (ii)

$x_{2},\ldots,x_{g}\in H$, if $g\geq 2$.

Proof.

We denote the image of $x\in G$ in $G/H$ by $\overline{x}$. First, we take a direct sum decomposition

[TABLE]

with $n_{i1}\leq n_{i2}\leq\ldots\leq n_{ig_{i}}$ for each $1\leq i\leq r$ and

[TABLE]

in $G/H$. For simplicity, we put $x_{i}:=x_{i1}$ and $n_{i}:=n_{i1}$ $(1\leq i\leq r)$. For any $i$ and $j$ with $1\leq i\leq r$ and $2\leq j\leq g_{i}$, there exist $h_{ij}\in H$ and $p\nmid a_{ij}$ such that $x_{i}=h_{ij}+a_{ij}x_{ij},$ since $\langle\overline{x_{i}}\rangle_{m_{i}}=\langle\overline{x_{ij}}\rangle_{m_{i}}$. Then, we can easily check that $\langle x_{i}\rangle_{n_{1}}\oplus\langle x_{ij}\rangle_{n_{ij}}=\langle x_{i}\rangle_{n_{1}}\oplus\langle h_{ij}\rangle_{n_{ij}}$. Hence, we may change $\langle x_{ij}\rangle_{n_{ij}}$ in the above decomposition with $\langle h_{ij}\rangle_{n_{ij}}$. Therefore, by changing the names appropriately, we obtain a direct sum decomposition

[TABLE]

for some subset $H^{\prime}$ in $H$. Next, for any $i$ with $2\leq i\leq r$, there exist $h_{i}\in H$ and $p\mid a_{i}$ such that $x_{i}=h_{i}+a_{i}x_{1},$ since $\langle\overline{x_{1}}\rangle_{m_{1}}\supsetneq\langle\overline{x_{i}}\rangle_{m_{i}}$. Then

[TABLE]

and elements in $\bigcup_{i=2}^{r}\{h_{i}\}\cup H^{\prime}$ are linearly independent mod $pG$. Finally, we show that the above $x_{1}$ can be taken such that it has the minimum order among the elements whose images generate $G/H$. Let $x_{1}^{\prime}\in G$ be such an element. Then we have $x_{1}^{\prime}=h^{\prime}+a^{\prime}x_{1}$ for some $h^{\prime}\in H$ and $p\nmid a^{\prime}$. We also obtain $h^{\prime}=b_{1}x_{1}+\sum_{i=2}^{r}b_{i}h_{i}+\sum_{h\in H^{\prime}}b_{h}h$ for some $b_{1},\ldots b_{r},h_{h}\in\mathbb{Z}$. Then $b_{1}x_{1}\in\langle x_{1}\rangle\cap H$, so that $p^{m_{1}}\mid b_{1}$. Since $x_{1}^{\prime}$ has the form $x_{1}^{\prime}=(a^{\prime}+b_{1})x_{1}+\sum_{i=2}^{r}b_{i}h_{i}+\sum_{h\in H^{\prime}}b_{h}h$, we obtain that $G=\langle x_{1}^{\prime}\rangle+\langle h_{2},\ldots,h_{r}\rangle+\bigoplus_{h\in H^{\prime}}\langle h\rangle$. ∎

Let $p$ be an odd prime number, $K$ an imaginary quadratic field such that $p$ does not split. We use the notation in §1.2 for such $p$ and $K$, $\widetilde{K}$, $K_{\infty}$, $X_{\widetilde{K}}$, etc. Put $g:=\dim_{\mathbb{F}_{p}}(A_{K}/p)$. We use Ozaki’s exact sequence in the case where $p$ does not split:

Lemma 4.2**.**

([16, Lemma 1])* There is an exact sequence of $\mathbb{Z}_{p}[[S]]$-modules*

[TABLE]

Lemma 4.3**.**

There is a canonical isomorphism

[TABLE]

Proof.

Let $L^{\prime}$ be the maximal abelian subextension in $L_{K_{\infty}}/K$. We see that $L^{\prime}=K_{\infty}L_{K}$, since $L_{K}$ is the fixed field by the inertia group of a prime lying above $p$ in ${\rm Gal}(L^{\prime}/K)$ and $L^{\prime}/K_{\infty}L_{K}$ is unramified. If we show that

[TABLE]

then we obtain ${\rm Gal}(L_{K_{\infty}}\cap\widetilde{K}/K_{\infty})={\rm Gal}(K_{\infty}(L_{K}\cap\widetilde{K})/K_{\infty})\simeq{\rm Gal}(L_{K}\cap\widetilde{K}/K).$ Let us show (4.1). We have $L_{K}\cap\widetilde{K}\subset L_{K}\subset K_{\infty}L_{K}\subset L_{K_{\infty}}$, so that

[TABLE]

Since $L_{K_{\infty}}\cap\widetilde{K}$ is abelian over $K$ and unramified over $K_{\infty}$, we see that $L_{K_{\infty}}\cap\widetilde{K}\subset L^{\prime}=K_{\infty}L_{K}$. This yields

[TABLE]

The claim follows from (4.2) and (4.3). ∎

Since $X_{K_{\infty}}/(p,S)\simeq A_{K}/p$,

[TABLE]

by Lemmas 4.2 and 4.3. Put

[TABLE]

Now, we show Theorem 1.5 (i). If $m=0$, i.e., $L_{K}\cap\widetilde{K}=K$, then $X_{\widetilde{K}}/T\simeq X_{K_{\infty}}$, so that $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=g.$ This completes the proof of Theorem 1.5 (i). In the following, we assume that $m>0$.

We fix certain generators $x_{1},\ldots,x_{g}$ of $X_{K_{\infty}}$ as a $\mathbb{Z}_{p}[[S]]$-module as follows. Applying Lemma 4.1 to ${\rm Gal}(L_{K}/K)$ and its quotient ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$, we can choose the basis of ${\rm Gal}(L_{K}/K)$ which satisfies the conditions in Lemma 4.1. Moreover, applying Nakayama’s lemma to this basis, then we can choose $x_{1},\ldots,x_{g}$ in $X_{K_{\infty}}$ which satisfy the conditions bellow.

• •

$x_{1},\ldots,x_{g}$ generate $X_{K_{\infty}}$.

• •

The image of $x_{1}$ generates ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$. Also the images of $x_{2},\ldots,x_{g}$ become [math] in ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$.

• •

Moreover, if ${\rm Gal}(L_{K}/K)\simeq{\rm Gal}(L_{K}\cap\widetilde{K}/K)\oplus{\rm Gal}(L_{K}/L_{K}\cap\widetilde{K})$ and the exponent of ${\rm Gal}(L_{K}/L_{K}\cap\widetilde{K})$ is equal to or less than $p^{m}$, then $x_{1}$ can be replaced modulo ($x_{2},\ldots,x_{g})\mathbb{Z}_{p}[[S]]$ (this fact is useful in §5).

Note that $x_{1},\ldots,x_{g}$ are defined modulo $SX_{K_{\infty}}$. In the following, we assign the sum “$\sum_{j=2}^{g}$” to [math] if $g=1$. For any $a_{1},\ldots,a_{m}\in X_{K_{\infty}}$, $(a_{1},\ldots,a_{m})_{X}$ denotes the $\mathbb{Z}_{p}[[S]]$-submodule of $X_{K_{\infty}}$ generated by $a_{1},\ldots,a_{m}$. Put

[TABLE]

By Lemma 4.2, we identify $X_{\widetilde{K}}/T$ with the submodule in $X_{K_{\infty}}$.

Proposition 4.4**.**

We have the following.

• (i)

$X_{\widetilde{K}}/T=M+N$.

• (ii)

$(M+L/L)\cap(N+L/L)=0$, and hence $X_{\widetilde{K}}/(p,S,T)\simeq(M+L/L)\oplus(N+L/L)$.

• (iii)

$N+L/L\simeq\mathbb{F}_{p}^{\oplus(g-1)}$.

Proof.

(i) By Lemmas 4.2 and 4.3, we see $x_{2},\ldots,x_{g}\in X_{\widetilde{K}}/T$, and also

[TABLE]

since ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$ has order $p^{m}$ and the trivial action by ${\rm Gal}(K_{\infty}/K)$. Hence $M+N\subset X_{\widetilde{K}}/T.$ Since $[X_{K_{\infty}}:M+N]\leq p^{m}$, we have the claim.

(ii) Any elements in $(M+L/L)\cap(N+L/L)$ are represented by some elements in $X_{K_{\infty}}$ of the form

[TABLE]

The left hand side is congruent to [math] modulo $(p,S)X_{K_{\infty}}$. Since $x_{1},\ldots,x_{g}$ are linearly independent in $A_{K}/p$, we see $b_{2},\ldots,b_{g}=0$. This implies that $(M+L/L)\cap(N+L/L)=0$.

(iii) By an exact sequence $N/(p,S)\to X_{K_{\infty}}/(p,S)\to(X_{K_{\infty}}/N)/(p,S)\to 0$, we have $N/(p,S)\simeq(\mathbb{Z}_{p}/p)^{\oplus(g-1)}$. Therefore we have only to show $N\cap L\subset(p,S)N$, since $[N:N\cap L]\leq p^{g-1}$. Any elements in $N\cap L$ are represented by some elements in $X_{K_{\infty}}$ of the form

[TABLE]

Since $x_{1},\ldots,x_{g}$ are linearly independent modulo $(p,S)X_{K_{\infty}}$, we see $B^{\prime\prime}_{2},\ldots,B^{\prime\prime}_{g}\in(p,S)\mathbb{Z}_{p}[[S]]$. This implies that $N\cap L\subset(p,S)N$. ∎

We denote the image of $x\in X_{K_{\infty}}$ in ${\rm Gal}(L_{K}/K)$ by $\overline{x}$. The following lemma gives criterions in the case where $\langle\overline{x_{1}}\rangle$ is a direct summand of ${\rm Gal}(L_{K}/K)$. We will use the lemma in §4.2 and §5.

Lemma 4.5**.**

Denote the order of $\overline{x_{1}}$ by ${\rm ord}(\overline{x_{1}})$.

• (i)

If $p^{m}={\rm ord}(\overline{x_{1}})$, in other words, ${\rm Gal}(L_{K}/K)\simeq{\rm Gal}(L_{K}\cap\widetilde{K}/K)\oplus{\rm Gal}(L_{K}/L_{K}\cap\widetilde{K})$, then

[TABLE]

• (ii)

Suppose that ${\rm Gal}(L_{K}/K)=\langle\overline{x_{1}}\rangle\oplus\langle\overline{x_{2}},\ldots,\overline{x_{g}}\rangle$ and $0<p^{m}<{\rm ord}(\overline{x_{1}})$, then $p^{m}x_{1}\not\in L$. In particular, $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))\geq g$.

Proof.

(i) By the assumption, we have $p^{m}x_{1}\equiv 0$ mod $SX_{K_{\infty}}$. Therefore, there exist some $A_{1},\ldots,A_{g}\in\mathbb{Z}_{p}[[S]]$ such that

[TABLE]

The second term in the right hand side is in $(p,S)N\subset L$. This implies that $M+L/L$ is generated by $Sx_{1}+L$. Moreover, we see that $p^{m+1}x_{1}$ is written as a linear form of $pSx_{1}$ and elements in $(p,S)N$. We claim that $Sx_{1}\in L$ if and only if $Sx_{1}\in(p,S)N$, which implies the conclusion by Proposition 4.4. Assume that $Sx_{1}\in L$. Then there exist some $A,A^{\prime},B_{j},B^{\prime}_{j}\in\mathbb{Z}_{p}[[S]]$ ($j=2,\ldots,g$) such that

[TABLE]

so that

[TABLE]

This implies that $Sx_{1}\in(p,S)N$ since $1-Ap-A^{\prime}S\in\mathbb{Z}_{p}[[S]]^{\times}$. The converse is trivial.

(ii) Assume that $p^{m}x_{1}\in L$. Then, in a similar way as (i), we see that there exists some $a\in\mathbb{Z}_{p}$ such that

[TABLE]

since ${\rm Gal}(L_{K}/K)=\langle\overline{x_{1}}\rangle\oplus\langle\overline{x_{2}},\ldots,\overline{x_{g}}\rangle$. This implies that $p^{m}x_{1}$ equals [math] in $A_{K}$, since $1-ap\in\mathbb{Z}_{p}^{\times}$, which is a contradiction. ∎

4.2 Classification in the case where $A_{K}$ is cyclic

Now, we show Theorem 1.5 (ii). Suppose that $L_{K}\cap\widetilde{K}\neq K$ and that $\dim_{\mathbb{F}_{p}}(A_{K}/p)=1$, in other words, $g=1$. Then, $N=0$ and there is an isomorphism $X_{K_{\infty}}\simeq\mathbb{Z}_{p}[[S]]/F(S),$ where $F(S)\in\mathbb{Z}_{p}[S]$ is the distinguished polynomial generating the characteristic ideal of $X_{K_{\infty}}$, since $A_{K}$ is cyclic and $X_{K_{\infty}}$ has no nontrivial finite $\mathbb{Z}_{p}[[S]]$-submodules. Note that we have $Sx_{1}\neq 0$, since $S$ dose not divide $F(S)$. In this case, we can apply Lemma 4.5.

Proof of (ii-a). Assume that $\lambda=1$. If $p^{m}=\#A_{K}$, then $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=1$ by Lemma 4.5 (i). On the other hand, assume that $0<p^{m}<\#A_{K}$. Since $F(S)x_{1}=0$ in $X_{K_{\infty}}$ and $F(0)x_{1}\in L$, we obtain $Sx_{1}\in L$, which implies that $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=1$ by Proposition 4.4 (ii), (iii).

Proof of (ii-b). Assume that $\lambda\geq 2$. If $p^{m}=\#A_{K}$, then $\dim_{\mathbb{F}_{p}}(X_{\widetilde{K}}/(p,S,T))=1$. On the other hand, assume that $0<p^{m}<\#A_{K}$. Since we see that $F(S)\equiv F(0)$ mod $(pS,S^{2})$ from $\deg F(S)\geq 2$, we obtain

[TABLE]

This completes the proof of (ii).

5 Conditions of $X_{\widetilde{K}}$ to be cyclic in the case where $\lambda=2$

5.1 Setting and Methods

Let $p$ be an odd prime number, $K$ an imaginary quadratic field such that $p$ does not split. We use the notation in §1.2 for such $p$ and $K$. We consider conditions that $X_{\widetilde{K}}$ becomes $\mathbb{Z}_{p}[[S,T]]$-cyclic. In the case where $\dim_{\mathbb{F}_{p}}(A_{K}/p)=1$, Theorem 1.5 gives the condition, so that we have only to consider the case where $\dim_{\mathbb{F}_{p}}(A_{K}/p)=2$.

In this section, we treat the case where $\dim_{\mathbb{F}_{p}}(A_{K}/p)=2$ and ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$ is a direct summand of ${\rm Gal}(L_{K}/K)\simeq A_{K}$. Moreover, we add more assumptions that $\lambda=2$ and that the roots $\alpha,\beta\in\overline{\mathbb{Q}_{p}}$ of the distinguished polynomial generating the characteristic ideal of $X_{K_{\infty}}$ satisfy $\alpha\neq\beta$. We remark that the latter part of this assumption is expected to be held.

Define $\mathcal{O}:=\mathbb{Z}_{p}[\alpha,\beta]$ and $\Lambda:=\mathcal{O}[[S]]$, and let $\pi$ be a uniformizer in $\mathcal{O}$. Then, by the above assumption, the characteristic ideal of $X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ is described as

[TABLE]

Then, by Koike [12], there exist an integer $k$ with $0\leq k\leq{\rm ord}_{\pi}(\beta-\alpha)$ and an $\mathcal{O}$-basis $\mathbf{e}_{1},\mathbf{e}_{2}$ of $X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ such that the homomorphism of $\Lambda$-modules

[TABLE]

is injective. Note that $k$ depends only on the isomorphism class of $X_{K_{\infty}}$. We regard $X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ as a $\Lambda$-submodule of $\Lambda/(S-\alpha)\oplus\Lambda/(S-\beta)$ by the above injection. We can express the action of $S$ by

[TABLE]

For convenience, we regard $X_{K_{\infty}}\subset X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ by the injection $x\mapsto x\otimes 1$, and put $\gamma:=(\beta-\alpha)\pi^{-k}$ and ${\rm ord}(a):={\rm ord}_{\pi}(a)$ for $a\in\mathcal{O}$. We take such generators $x_{1},x_{2}$ of $X_{K_{\infty}}$ as in §4.1. They are represented as

[TABLE]

for some $\lambda_{ij}\in\mathcal{O}$. We have $A_{K}\otimes_{\mathbb{Z}_{p}}\mathcal{O}=(\mathcal{O}/\pi^{N_{1}})x_{1}\oplus(\mathcal{O}/\pi^{N_{2}})x_{2}$ and ${\rm Gal}(L_{K}\cap\widetilde{K}/K)\otimes_{\mathbb{Z}_{p}}\mathcal{O}=(\mathcal{O}/\pi^{N_{1}})x_{1}$ for some $N_{1},N_{2}\in\mathbb{Z}$. We denote by $\langle a_{1},\ldots,a_{l}\rangle_{\Lambda}$ (resp. $\langle a_{1},\ldots a_{l}\rangle_{\mathcal{O}}$) the $\Lambda$-submodule (resp. $\mathcal{O}$-submodule) in $\Lambda/(S-\alpha)\oplus\Lambda/(S-\beta)$ generated by $a_{1},\ldots,a_{l}$.

Lemma 5.1**.**

We have an isomorphism

[TABLE]

Proof.

We have

[TABLE]

Since $A_{K}\otimes_{\mathbb{Z}_{p}}\mathcal{O}\simeq(X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O})/S$, this yields the claim. ∎

Lemma 5.2**.**

The following conditions hold:

[TABLE]

Proof.

Since $\pi^{N_{1}}x_{1}\in SX_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$, we have $\pi^{N_{1}}(\lambda_{11}\mathbf{e}_{1}+\lambda_{12}\mathbf{e}_{2})\in\langle\alpha\mathbf{e}_{1}+\gamma\mathbf{e}_{2},\ \beta\mathbf{e}_{2}\rangle_{\mathcal{O}}$. Hence, there exist some $s,t\in\mathcal{O}$ such that $\lambda_{11}\pi^{N_{1}}=s\alpha$, $\lambda_{12}\pi^{N_{1}}=s\gamma+t\beta$. They induce the first relations. The rest are verified in the same way. ∎

Recall that we defined $N$ as the $\mathbb{Z}_{p}[[S]]$-submodule of $X_{K_{\infty}}$ generated by $x_{2}$ and that Lemma 4.5 (i) gives a criterion whether $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic or not. In the same way as the proofs of Proposition 4.4 and Lemma 4.5, we can show the following

Lemma 5.3**.**

$X_{\widetilde{K}}$* is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if $Sx_{1}\in(\pi,S)N\otimes_{\mathbb{Z}_{p}}\mathcal{O}$.*

Proof.

Note that there is nothing to show if $\mathcal{O}/\mathbb{Z}_{p}$ is unramified. And also, note that $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if $X_{\widetilde{K}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ is $\mathcal{O}[[S,T]]$-cyclic by Nakayama’s lemma. Put $L^{\prime}:=(\pi,S)(M\otimes_{\mathbb{Z}_{p}}\mathcal{O}+N\otimes_{\mathbb{Z}_{p}}\mathcal{O})$. Then, in the same way as the proofs of Proposition 4.4 (ii) and (iii), we can show that $X_{\widetilde{K}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}\simeq(M\otimes_{\mathbb{Z}_{p}}\mathcal{O}+L^{\prime}/L^{\prime})\oplus(N\otimes_{\mathbb{Z}_{p}}\mathcal{O}+L^{\prime}/L^{\prime})$ and $(N\otimes_{\mathbb{Z}_{p}}\mathcal{O}+L^{\prime})/L^{\prime}\simeq\mathbb{F}_{p}$. Moreover, in the same way as the proof of Lemma 4.5 (i), we can show that $Sx_{1}\in(\pi,S)N\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ if and only if $Sx_{1}\in L^{\prime}$. ∎

Lemma 5.4**.**

$X_{\widetilde{K}}$* is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exist some $f(S),g(S)\in\Lambda$ such that*

[TABLE]

Proof.

We have $Sx_{1}=\begin{bmatrix}\lambda_{11}\alpha\\ \lambda_{11}\beta+\lambda_{12}\beta\pi^{k}\end{bmatrix}$ and

[TABLE]

Then the claim follows immediately by Lemma 5.3. ∎

Now, we consider the case where $N_{1}\geq N_{2}$. Then, as we mentioned in §4.1, $x_{1}$ is defined modulo $(x_{2},SX_{K_{\infty}})\subset X_{K_{\infty}}$, so that we may change $x_{1}$ up to $x_{2}\Lambda$. Furthermore, since $\lambda_{11}\lambda_{22}-\lambda_{12}\lambda_{21}\in\mathcal{O}^{\times}$, at least one (possibly both) of $\lambda_{21}$, $\lambda_{22}$ is in $\mathcal{O}^{\times}$. When $\lambda_{21}\in\mathcal{O}^{\times}$ (resp. $\lambda_{22}\in\mathcal{O}^{\times}$), we may assume that $\lambda_{21}=1$, $\lambda_{11}=0$, and $\lambda_{12}=1$ (resp. $\lambda_{22}=1$, $\lambda_{12}=0$, and $\lambda_{11}=1$). Thus, if $N_{1}\geq N_{2}$, we are reduced to only two cases:

• (I)

$\lambda_{21}=1$, $\lambda_{11}=0$, and $\lambda_{12}=1$. In other words, $x_{1}=\mathbf{e}_{2}$ and $x_{2}=\mathbf{e}_{1}+\lambda_{22}\mathbf{e}_{2}$.

• (II)

$\lambda_{22}=1$, $\lambda_{11}=1$, and $\lambda_{12}=0$. In other words, $x_{1}=\mathbf{e}_{1}$ and $x_{2}=\lambda_{21}\mathbf{e}_{1}+\mathbf{e}_{2}$.

Corollary 5.5**.**

Suppose that $N_{1}\geq N_{2}$. We have the following.

• (i)

In the case of (I), $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exists some $x\in\mathcal{O}$ such that

[TABLE]

• (ii)

In the case of (II), $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exist some $f(S),g(S)\in\Lambda$ such that

[TABLE]

Proof.

There is nothing to show for (ii) by Lemma 5.4. We prove (i). First, assume that $f(S),g(S)\in\Lambda$ satisfy (5.2). Then $f(\alpha)=0$. Therefore, $f(S)$ is divisible $S-\alpha$ by division lemma (see [2, Chapter VII §3.8 Proposition 5]). Put $x:=f(\beta)\pi/(\beta-\alpha)+g(\beta)$. Then we can easily check $x\in\mathcal{O}$ and $\beta=(1+\lambda_{22}\pi^{k})\gamma x$. Conversely, assume that there exist such $x\in\mathcal{O}$. Put $f(S):=0$, $g(S):=x$ Then we can also easily check that they satisfy (5.2). ∎

5.2 The case where $k>0$

Proposition 5.6**.**

If $k>0$ and ${\rm ord}(\gamma)<\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$, then $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic.

Proof.

In this case, $A_{K}\otimes_{\mathbb{Z}_{p}}\mathcal{O}\simeq\mathcal{O}/\gamma\oplus\mathcal{O}/({\alpha\beta}/{\gamma})$ by Lemma 5.1. First, we suppose that $N_{1}={\rm ord}(\gamma)$, $N_{2}={\rm ord}({\alpha\beta}/{\gamma})$. Then $N_{1}<N_{2}$ (note that we cannot apply Corollary 5.5 in this case). Then, $\pi^{N_{1}}/\alpha\notin\mathcal{O}$, so that $\pi\mid\lambda_{11}$ by Lemma 5.2. Hence, $\lambda_{12}\in\mathcal{O}^{\times}$. We may assume that $\lambda_{12}=1$ and also that $\lambda_{21}=1$. By Lemma 5.2, we see that $\lambda_{12}-\lambda_{11}\gamma/\alpha$ must be divided by $\pi$, so that we obtain ${\rm ord}(\lambda_{11})={\rm ord}(\alpha/\gamma)$. Put

[TABLE]

Then $g(S)\in\mathcal{O}$, since $1+\lambda_{22}\pi^{k}\in\mathcal{O}^{\times}$ and ${\rm ord}(\beta-\alpha)-k={\rm ord}(\gamma)<\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$. They satisfy (5.2).

Second, we suppose that $N_{1}={\rm ord}({\alpha\beta}/{\gamma})$, $N_{2}={\rm ord}(\gamma)$. Then $N_{1}>N_{2}$ so that we can apply Corollary 5.5. By Lemma 5.2, we know that $\lambda_{21}\gamma/\alpha\in\mathcal{O}$, and hence ${\rm ord}(\lambda_{21})>0$. This implies that $\lambda_{11},\lambda_{22}\in\mathcal{O}^{\times}$, so we are reduced only to the case of (II). Thus, we may assume that $\lambda_{11}=\lambda_{22}=1$ and $\lambda_{12}=0$. Put $\delta:=(\lambda_{22}-\lambda_{21}\gamma/\alpha)/\beta=(1-\lambda_{21}\gamma/\alpha)/\beta$. Then

[TABLE]

Note that ${\rm ord}(\delta)\geq-{\rm ord}(\gamma)>-\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$ by Lemma 5.2, so that both $1-\alpha\delta$ and $1-\beta\delta$ are units. Now, put

[TABLE]

Here, the sum is convergent since ${\rm ord}(\alpha\delta)>0$ and ${\rm ord}(\beta\delta)>0$. Hence $g(S)\in\mathcal{O}$, and they satisfy the condition in Corollary 5.5 (ii). ∎

The following proposition does not need the assumption that $k>0$.

Proposition 5.7**.**

If ${\rm ord}(\gamma)>\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$, then $X_{\widetilde{K}}$ is not $\mathbb{Z}_{p}[[S,T]]$-cyclic.

Proof.

Note that ${\rm ord}(\alpha)={\rm ord}(\beta)=N_{1}=N_{2}$ in this case, so that we can apply the criterion in Corollary 5.5. Assume that $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic. In the case of (I), the condition that $(1+\lambda_{22}\pi^{k})^{-1}\beta/\gamma\in\mathcal{O}$ contradicts the inequality ${\rm ord}(\gamma)>{\rm ord}(\beta)$. We consider the case of (II). By ${\rm ord}(\gamma)>{\rm ord}(\alpha)$, we have $s:=1-\lambda_{21}\gamma/\alpha\in\mathcal{O}^{\times}.$ By the above assumption and Corollary 5.5 (ii), there are some $f(S),g(S)\in\Lambda$ such that $\alpha=f(\alpha)\lambda_{21}\pi$ and

[TABLE]

Now, $f(\beta)$ has a form $f(\beta)=f(\alpha)+(\beta-\alpha)A(\beta)$ with some $A(S)\in\Lambda$ by division lemma, and $f(\alpha)\pi=\alpha/\lambda_{21}=\gamma(1-s)^{-1}$. Therefore,

[TABLE]

So we have

[TABLE]

The order of the left hand side is equal to or less than $k$, since $s\in\mathcal{O}^{\times}$. On the other hand, the order of $1-{\alpha}s/{\beta}$ is greater than [math]. In fact, $s$ has a form $s=1+\pi t$ with some $t\in\mathcal{O}$, so that

[TABLE]

since ${\rm ord}(\gamma)>{\rm ord}(\beta)$. This is a contradiction. ∎

Proposition 5.8**.**

Suppose that $k>0$ and ${\rm ord}(\gamma)=\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$. Then $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if $\lambda_{21}\in\mathcal{O}^{\times}$.

Proof.

Note that ${\rm ord}(\gamma)={\rm ord}(\alpha)={\rm ord}(\beta)=N_{1}=N_{2}$, since $k>0$. Therefore we can apply the criterion in Corollary 5.5. In the case of (I), we see that $(1+\lambda_{22}\pi^{k})^{-1}\beta/\gamma\in\mathcal{O}$, which implies that $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic. We consider the case of (II). In the case where $\lambda_{21}\in\mathcal{O}^{\times}$, we are reduced to the case of (I), so we may assume that $\pi\mid\lambda_{21}$. Then we have only to show that $X_{\widetilde{K}}$ is not $\mathbb{Z}_{p}[[S,T]]$-cyclic. Assume the contrary. Note that $s:=1-\lambda_{21}\gamma/\alpha\in\mathcal{O}^{\times}$. In the same way as Proposition 5.7, we get a contradiction, since $(\beta-\alpha)/\beta\equiv 0$ mod $\pi$ by $k>0$. ∎

5.3 The case where $k=0$

Suppose that $k=0$. Then $X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}\simeq\Lambda/(S-\alpha)\oplus\Lambda/(S-\beta)$. We use the standard basis $\begin{bmatrix}1\\ 0\end{bmatrix}$, $\begin{bmatrix}0\\ 1\end{bmatrix}$, instead of $\begin{bmatrix}1\\ 1\end{bmatrix}$, $\begin{bmatrix}0\\ 1\end{bmatrix}$. If ${\rm ord}(\gamma)>\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$, then $X_{\widetilde{K}}$ is not $\mathbb{Z}_{p}[[S,T]]$-cyclic by Proposition 5.7. So, in the following, we consider the case where ${\rm ord}(\gamma)=\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$. Moreover, we may assume that ${\rm ord}(\alpha)\leq{\rm ord}(\beta)$. Express the generators $x_{1}$, $x_{2}$ as

[TABLE]

for some $\mu_{ij}\in\mathcal{O}$. Then $Sx_{1}\in(\pi,S)N\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ if and only if there exist some $f(S),g(S)\in\Lambda$ such that

[TABLE]

Proposition 5.9**.**

Assume that ${\rm ord}(\beta-\alpha)={\rm ord}(\alpha)\leq{\rm ord}(\beta)$ and that $N_{1}<N_{2}$. Then $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if $\mu_{21}$ is in $\mathcal{O}^{\times}$.

Proof.

In this case, $A_{K}\otimes_{\mathbb{Z}_{p}}\mathcal{O}=(\mathcal{O}/\pi^{N_{1}})x_{1}\oplus(\mathcal{O}/\pi^{N_{2}})x_{2}\simeq\mathcal{O}/\alpha\oplus\mathcal{O}/\beta$. Hence $N_{1}={\rm ord}(\alpha)$, $N_{2}={\rm ord}(\beta)$. Therefore, we have ${\rm ord}(\mu_{12})>0$. In fact, if $\mu_{12}\in\mathcal{O}^{\times}$, then the order of $x_{1}$ in $A_{K}\otimes_{\mathbb{Z}_{p}}\mathcal{O}$ is $\#(\mathcal{O}/\pi^{N_{2}})$, which is a contradiction. Thus, we may assume that $\mu_{11}=\mu_{22}=1$. By (5.3), $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exist some $f(S),g(S)\in\Lambda$ such that

[TABLE]

Assume that ${\rm ord}(\mu_{21})>0$ and that $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic. By division lemma, $f(\alpha)=f(\beta)+(\alpha-\beta)A(\alpha)$ with some $A(S)\in\Lambda$. Then

[TABLE]

This is a contradiction. Hence, if $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic, then $\mu_{21}\in\mathcal{O}^{\times}$. Conversely, let $\mu_{21}$ be in $\mathcal{O}^{\times}$. Put

[TABLE]

Then we can easily check that $\mu_{21}(\pi f(\alpha)+\alpha g(\alpha))=\alpha$ and that $\pi f(\beta)+\beta g(\beta)=\beta\mu_{12}$. Hence $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic. ∎

Proposition 5.10**.**

Assume that ${\rm ord}(\beta-\alpha)={\rm ord}(\alpha)\leq{\rm ord}(\beta)$ and that $N_{1}\geq N_{2}$. Then $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if it holds that $\mu_{21}\in\mathcal{O}^{\times}$ and ${\rm ord}(\mu_{22})={\rm ord}(\beta)-{\rm ord}(\alpha)$.

Proof.

In this case, we may change $x_{1}$ up to modulo $x_{2}\Lambda$ as we mentioned after the proof of Lemma 4.3. As in §5.1, we have only to consider the two cases:

• (I’)

$\mu_{21}=1$, $\mu_{11}=0$, $\mu_{12}=1$.

• (II’)

$\mu_{22}=1$, $\mu_{11}=1$, $\mu_{12}=0$.

First, we deal with the case of (I’). Then ${\rm ord}(\mu_{22})\geq{\rm ord}(\beta)-{\rm ord}(\alpha)$. In fact, if ${\rm ord}(\alpha)={\rm ord}(\beta)$, then the inequality is trivial. And also, if ${\rm ord}(\alpha)<{\rm ord}(\beta)$, then $N_{2}={\rm ord}(\alpha)$ by Lemma 5.1, so that $\pi^{N_{2}}x_{2}\in SX_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}=\left\langle\begin{bmatrix}\alpha\\ 0\end{bmatrix},\begin{bmatrix}0\\ \beta\end{bmatrix}\right\rangle_{\mathcal{O}}$. This implies that $\mu_{22}\alpha/\beta\in\mathcal{O}$. By (5.3), $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exist some $f(S),g(S)\in\Lambda$ such that

[TABLE]

Assume that ${\rm ord}(\mu_{22})>{\rm ord}(\beta)-{\rm ord}(\alpha)$ and $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic. Since $\pi f(\beta)$ has a form $\pi f(\beta)=\pi f(\alpha)+\pi(\beta-\alpha)A(\beta)=-\alpha g(\alpha)+\pi(\beta-\alpha)A(\beta)$ with some $A(S)\in\Lambda$,

[TABLE]

This is a contradiction. Conversely, assume that ${\rm ord}(\mu_{22})={\rm ord}(\beta)-{\rm ord}(\alpha)$. Put

[TABLE]

Then we can easily check $\pi f(\alpha)+\alpha g(\alpha)=0$ and $\mu_{22}(\pi f(\beta)+\beta g(\beta))=\beta$, and hence $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic.

Second, we deal with the case of (II’). By (5.3), $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if there exist some $f(S),g(S)\in\Lambda$ such that

[TABLE]

The similar argument in the case of (I’) shows that ${\rm ord}(\alpha)<{\rm ord}(\beta)$ does not occur. In fact, the assumption ${\rm ord}(\alpha)<{\rm ord}(\beta)$ yields $\mu_{22}\alpha/\beta\in\mathcal{O}$, which is a contradiction since $\mu_{22}=1$. Hence ${\rm ord}(\alpha)={\rm ord}(\beta)$. Using this fact, in the same way as the case of (I’), we can show that if ${\rm ord}(\mu_{21})>0$, then $X_{\widetilde{K}}$ is not $\mathbb{Z}_{p}[[S,T]]$-cyclic. Conversely, if ${\rm ord}(\mu_{21})=0$, then we can prove that $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic by taking

[TABLE]

Then, the proof is completed. ∎

Summarizing all results in this section, we obtain the following.

Theorem 5.11**.**

Let $p$ be an odd prime number, $K$ an imaginary quadratic field such that $p$ does not split. Suppose that

• •

$\dim_{\mathbb{F}_{p}}(A_{K}/p)=2$* and ${\rm Gal}(L_{K}\cap\widetilde{K}/K)$ is a direct summand of ${\rm Gal}(L_{K}/K)$.*

• •

The Iwasawa $\lambda$-invariant of $K_{\infty}/K$ is $2$.

• •

Let $\alpha,\beta\in\overline{\mathbb{Q}_{p}}$ be the roots of the distinguished polynomial generating the characteristic ideal of $X_{K_{\infty}}$. Then $\alpha\neq\beta$.

Put $\mathcal{O}:=\mathbb{Z}_{p}[\alpha,\beta]$ and $l:=\min\{{\rm ord}(\alpha),{\rm ord}(\beta)\}$. Let $x_{2}\in X_{K_{\infty}}$ be a preimage of a generator of ${\rm Gal}(L_{K}/L_{K}\cap\widetilde{K})$. Also, we denote by $\begin{bmatrix}\mu_{21}\\ \mu_{22}\end{bmatrix}$ the image of $x_{2}\otimes 1$ under the injective map $X_{K_{\infty}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}\hookrightarrow\mathcal{O}[[S]]/(S-\alpha)\oplus\mathcal{O}[[S]]/(S-\beta),$ which is defined by (5.1). Then, $X_{\widetilde{K}}$ is $\mathbb{Z}_{p}[[S,T]]$-cyclic if and only if one of the following holds:

[TABLE]

*where each $n_{1}$ and $n_{2}$ is defined by $p^{n_{1}}=\#{\rm Gal}(L_{K}\cap\widetilde{K}/K)$ and $p^{n_{2}}=\#{\rm Gal}(L_{K}/L_{K}\cap\widetilde{K})$, respectively. *