Fitting ideals in two-variable equivariant Iwasawa theory and an application

Let $\Sigma$ be a finite set of finite places of $K$ which contains $S_{\operatorname{ram}}(L_{\infty}/K)$. Put $\Sigma_{0}=\Sigma\setminus\{\overline{\mathfrak{p}}\}$. As the first object of study, we consider the $\Sigma_{0}$-ramified Iwasawa module $X_{\Sigma_{0}}(L_{\infty})$, which is defined as the Galois group of the maximal abelian $p$-extension of $L_{\infty}$ which is unramified outside $\Sigma_{0}$. It is known that $X_{\Sigma_{0}}(L_{\infty})$ is a finitely generated torsion $\mathcal{R}$-module.

The first goal of this paper is to compute its initial Fitting ideal $\operatorname{Fitt}_{\mathcal{R}}(X_{\Sigma_{0}}(L_{\infty}))$. A significant progress toward this problem was made in a previous paper [9] of the author. Let $\operatorname{Fitt}_{\mathcal{R}}^{[n]}$ denote the $n$-th shift of $\operatorname{Fitt}_{\mathcal{R}}$, which was introduced in [9]; see Theorem 2.1 for the definition. Then it is shown in [9, Theorem 5.16] that the ideals $\operatorname{Fitt}_{\mathcal{R}}(X_{\Sigma_{0}}(L_{\infty}))$ and $\operatorname{Fitt}_{\mathcal{R}}^{[2]}(\mathbb{Z}_{p})$ differ only by an invertible ideal. Moreover, the author claimed that the invertible ideal should be described as a certain $p$-adic $L$-function. Note here that invertible ideals of $\mathcal{R}$ are principal since $\mathcal{R}$ is a product of local rings.

Now we state the first main result in this paper, which asserts that the prediction is true. In Definition 2.14, we will introduce an invertible ideal $\mathfrak{L}_{\Sigma_{0}}(L_{\infty}/K)$ of $\mathcal{R}$ which is related to a Katz-type $p$-adic $L$-function.

The proof of Theorem 1.1 will be outlined just after Theorem 1.3 below.

More generally than $X_{\Sigma_{0}}(L_{\infty})$, we consider the $S$-ramified Iwasawa module $X_{S}(L_{\infty})$ for any finite set $S$ of finite places of $K$ such that $\mathfrak{p}\in S$ and $\overline{\mathfrak{p}}\not\in S$. In particular, we are concerned with the minimal case $S=\{\mathfrak{p}\}$. Although $X_{\{\mathfrak{p}\}}(L_{\infty})$ looks more fundamental than $X_{\Sigma_{0}}(L_{\infty})$, the structure of $X_{\{\mathfrak{p}\}}(L_{\infty})$ is more complicated from our perspective.

In our two-variable setting, we obtain the following result.

The definitions of $Z_{\Sigma\setminus S}^{0}(L_{\infty}/K)$ and $\mathfrak{L}_{\Sigma,S}(L_{\infty}/K)$ will be given in Section 2. See also Remark 2.11 for the computation of $\operatorname{Fitt}_{\mathcal{R}}^{[1]}(Z_{\Sigma\setminus S}^{0}(L_{\infty}/K))$.

By Lemma 2.9(1) below and $\mathfrak{L}_{\Sigma_{0}}=\mathfrak{L}_{\Sigma,\Sigma_{0}}$ by Definition 2.14, it follows that Theorem 1.3 for $S=\Sigma_{0}$ is equivalent to Theorem 1.1. For that reason, in this paper we give only a proof of Theorem 1.3.

Now we outline the proof of Theorem 1.3. The main ingredient is the result by Johnson-Leung–Kings [8], which is mentioned above. However, we have to develop algebraic theory to connect the result [8, Theorem 5.7], which uses determinant modules, and Theorem 1.3. Indeed, in this paper we will show that the notions of determinant modules of complexes and Fitting ideals of modules are essentially equivalent. Then we reduce the proof of Theorem 1.3 to Theorem 5.4 on the determinant of a certain complex $C_{\Sigma,S}$, and prove Theorem 5.4 via the result of [8]. Here we also need the work by de Shalit [1], since the main result of [8] uses elliptic units instead of $p$-adic $L$-functions.

It is known that the Iwasawa modules over imaginary quadratic fields are related to the Selmer groups of elliptic curves with complex multiplication. That fact is used to prove results of Birch–Swinnerton-Dyer type (for example, Rubin [14, §11, §12], de Shalit [1, Chapter IV]). Motivated by those works, we apply the results of the preceding sections to elliptic curves with complex multiplication.

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has complex multiplication by the ring of integers $\mathcal{O}_{K}$ of $K$. Suppose that $E$ has good reduction at $p$; since $p$ splits in $K$, the elliptic curve $E$ must have ordinary reduction at $p$. Similarly as above, let $L/K$ be a finite abelian extension and $S$ a finite set of finite places of $K$ such that $\mathfrak{p}\in S$ and $\overline{\mathfrak{p}}\not\in S$. We shall study the $\mathfrak{p}$-torsion part of the $S$-imprimitive Selmer group $\operatorname{Sel}_{S}(E/L)[\mathfrak{p}^{\infty}]$ (introduced in Definition 6.5).

In order to state the result, put $L^{\mathfrak{p}}=L(E[\mathfrak{p}^{\infty}])$, which is a one-variable extension of $K$. Put $\mathcal{R}^{\mathfrak{p}}=\mathbb{Z}_{p}[[\operatorname{Gal}(L^{\mathfrak{p}}/K)]]$ and let $\pi_{L^{\mathfrak{p}}/L}:\mathcal{R}^{\mathfrak{p}}\to\mathbb{Z}_{p}[\operatorname{Gal}(L/K)]$ be the projection map. We denote by $\chi_{E,\mathfrak{p}}$ the character defined by the action of $E[\mathfrak{p}^{\infty}]$:

Then $\chi_{E,\mathfrak{p}}$ induces a twisting algebra isomorphism $\mathcal{R}^{\mathfrak{p}}\to\mathcal{R}^{\mathfrak{p}}$, which we denote by $\widetilde{\chi_{E,\mathfrak{p}}}$. We denote by $(-)^{\vee}$ the Pontryagin dual.

The definitions of $Z_{\Sigma\setminus S}^{0}(L^{\mathfrak{p}}/K)$ and $\mathfrak{L}_{\Sigma,S}(L^{\mathfrak{p}}/K)$ will be given in Section 2. Note that the key ingredient of the proof is Theorem 5.4 rather than Theorem 1.3.

The rest of this paper is organized as follows. In Section 2, in order to complete the statements of the main theorems, we give precise definitions of several objects like $\operatorname{Fitt}^{[n]}$. In Section 3, we show the general relation between determinant modules and Fitting ideals. In Section 4, we review properties of various arithmetic complexes in derived categories. In Section 5, we prove Theorem 1.3 in the way we already outlined. In Section 6, we deduce Theorem 1.4 from Theorem 1.3.

In this subsection, we review the results of [9] on shifts of Fitting invariants.

Though we can deal with more general commutative rings, for the sake of simplicity, we only consider the following situation. Let $G$ be a profinite abelian group and consider the completed group ring $R=\mathbb{Z}_{p}[[G]]$. Suppose that $G$ has an open subgroup $G^{\prime}$ which is isomorphic to $\mathbb{Z}_{p}^{d}$ for some $d\geq 0$. Put $R^{\prime}=\mathbb{Z}_{p}[[G^{\prime}]]$, which is isomorphic to the ring of formal power series of $d$ variables over $\mathbb{Z}_{p}$. Note that the choice of $G^{\prime}$ is auxiliary and the following argument does not depend on $G^{\prime}$.

An $R$-module is said to be torsion (resp. pseudo-null) if it is torsion (resp. pseudo-null) as an $R^{\prime}$-module. For an $R$-module $X$, we denote by $\operatorname{pd}_{R}(X)$ the projective dimension of $X$ over $R$. As already mentioned, any invertible ideal of $R$ is principal.

We also use the following variant.

The role of $\operatorname{Fitt}_{\mathcal{R}}^{\langle n\rangle}$ in this paper is auxiliary compared to $\operatorname{Fitt}^{[n]}_{\mathcal{R}}$. An advantage is that $\operatorname{Fitt}_{\mathcal{R}}^{\langle n\rangle}$ behaves better with respect to exact sequences. A disadvantage is that we do not have $\operatorname{Fitt}_{\mathcal{R}}^{\langle 0\rangle}(X)=\operatorname{Fitt}_{\mathcal{R}}(X)$ in general, while we always have $\operatorname{Fitt}^{[0]}_{\mathcal{R}}(X)=\operatorname{Fitt}_{\mathcal{R}}(X)$. We have a sufficient condition for $\operatorname{Fitt}_{\mathcal{R}}^{\langle n\rangle}$ and $\operatorname{Fitt}^{[n]}_{\mathcal{R}}$ to coincide:

We record a simple proposition.

Let us return to the setting in Section 1. We define various $\mathcal{R}$-modules in the same way as in [4, Subsection 1.1].

Next we show a lemma on the $\operatorname{Fitt}^{[1]}$ in the right hand side of Theorem 1.3. The first assertion enables us to deduce Theorem 1.1 from Theorem 1.3.

As we will recall later (see the proof of Theorem 5.10), the results of de Shalit [1] involve a base change from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{p}^{\operatorname{ur}}$. Here, $\mathbb{Z}_{p}^{\operatorname{ur}}$ is the ring of integers of the completion $\mathbb{Q}_{p}^{\operatorname{ur}}$ of the maximal unramified extension of $\mathbb{Q}_{p}$. We will freely use the following lemma that fractional ideals of Iwasawa algebras are characterized by their base changes to $\mathbb{Z}_{p}^{\operatorname{ur}}$.

Now we introduce $p$-adic $L$-functions. See [1, Theorem II.4.14] for more information.

At first we consider the following specific situation associated to a nonzero integral ideal $\mathfrak{f}$ of $K$ which is prime to $p$. Put $L_{\mathfrak{f}}=K(\mathfrak{f}p)$, the ray class field of $K$ modulo $\mathfrak{f}p$, so $L_{\mathfrak{f},\infty}=K(\mathfrak{f}p^{\infty})$. We take $\Sigma_{\mathfrak{f}}$ as the set of prime divisors of $\mathfrak{f}p$. Put $\mathcal{G}_{\mathfrak{f}}=\operatorname{Gal}(L_{\mathfrak{f},\infty}/K)$, $\mathcal{R}_{\mathfrak{f}}=\mathbb{Z}_{p}[[\mathcal{G}_{\mathfrak{f}}]]$, and $\mathcal{R}^{\operatorname{ur}}_{\mathfrak{f}}=\mathbb{Z}_{p}^{\operatorname{ur}}[[\mathcal{G}_{\mathfrak{f}}]]$.

The crucial property of $\mu_{\mathfrak{f}}$ is that it comes from elliptic units; see Theorem 5.10.

Now, returning to the general situation, we introduce $p$-adic $L$-functions by similar formulas to $\mu_{\mathfrak{f}}$. Let $\mathcal{R}^{\operatorname{ur}}=\mathbb{Z}_{p}^{\operatorname{ur}}[[\mathcal{G}]]$ be the completed group ring of $\mathcal{G}$ over $\mathbb{Z}_{p}^{\operatorname{ur}}$. Note that, as a convention of this paper, $\mathfrak{L}$ denotes invertible ideals of Iwasawa algebras, while $\mathcal{L}$ denotes elements.

Finally we give the definition of one-variable ideals of $p$-adic $L$-functions.

For the well-definedness, see Corollary 6.2. By replacing $L$ by $L(E[\mathfrak{p}])$ in Definition 2.16, the invertible ideal $\mathfrak{L}_{\Sigma,S}(L^{\mathfrak{p}}/K)$ of $\mathcal{R}^{\mathfrak{p}}$, which appear in Theorem 1.4, is defined.

Following the notations in [9], we let $\mathcal{P}_{R}$ (resp. $\mathcal{Q}_{R}$) be the exact category consisting of finitely generated torsion $R$-modules $P$ (resp. $Q$) such that $\operatorname{pd}_{R}(P)\leq 1$ (resp. $\operatorname{pd}_{R}(Q)<\infty$).

For an (essentially small) exact category $\mathcal{C}$ like $\mathcal{P}_{R}$ and $\mathcal{Q}_{R}$, we denote by $K_{0}(\mathcal{C})$ its Grothendieck group. By definition, $K_{0}(\mathcal{C})$ has the following presentation by generators and relations: the generators are $[X]$ for objects $X\in\mathcal{C}$ and the relations are $[X]=[X^{\prime}]+[X^{\prime\prime}]$ for exact sequences $0\to X^{\prime}\to X\to X^{\prime\prime}\to 0$ in $\mathcal{C}$.

Recall that a triangulated category $\mathcal{C}$ is an additive category equipped with a translation functor $\mathcal{C}\to\mathcal{C}$, denoted by $X\mapsto X[1]$, and a notion of (distinguished) triangles

satisfying a couple of axioms.

For an (essentially small) triangulated category $\mathcal{C}$, we denote by $K_{0}(\mathcal{C})$ its Grothendieck group. By definition, $K_{0}(\mathcal{C})$ has the following presentation by generators and relations: the generators are $[X]$ for objects $X\in\mathcal{C}$ and the relations are $[X]=[X^{\prime}]+[X^{\prime\prime}]$ for triangles $X^{\prime}\to X\to X^{\prime\prime}\to$ in $\mathcal{C}$.

We briefly introduce several categories arising from cochain complexes.

Now we introduce determinant modules. We refer to [10] for the detailed construction. One can also refer to [4, Subsection 3.1].

For a finitely generated projective $R$-module $F$, letting $\operatorname{rank}(F)$ denote the (locally constant) rank of $F$, we define the determinant of $F$ by

Moreover, we denote its inverse by

More precisely, we should introduce the grade, but we omit it for simplicity.

For each complex $F^{\bullet}\in Ch^{\operatorname{perf}}(R)$, we define its determinant by

Similarly, we denote by $\operatorname{Det}_{R}^{-1}(F^{\bullet})$ its inverse.

Suppose $F^{\bullet}\in Ch^{\operatorname{perf}}_{\operatorname{tor}}(R)$. Since $\operatorname{Frac}(R)\otimes_{R}F^{\bullet}$ is acyclic, we have a natural isomorphism $\operatorname{Det}_{\operatorname{Frac}(R)}(\operatorname{Frac}(R)\otimes_{R}F^{\bullet})\simeq\operatorname{Frac}(R)$. Therefore, we have a natural injective map

From now on, we identify $\operatorname{Det}_{R}(F^{\bullet})$ with its image in $\operatorname{Frac}(R)$. Therefore, $\operatorname{Det}_{R}(F^{\bullet})$ is an invertible ideal of $R$.

We construct the homomorphism $\varphi$ in the diagram (3.1).