Higher codimension Iwasawa theory for elliptic curves with supersingular reduction

Let us fix an odd prime number $p$ and a number field $F$. Let $E$ be an elliptic curve over $F$ that has good reduction at all $p$-adic primes of $F$.

Let $K_{\infty}/F$ be an abelian $p$-adic Lie extension. Equivalently, $K_{\infty}/F$ is an abelian extension which is a finite extension of a multiple $\mathbb{Z}_{p}$-extension. We suppose that $K_{\infty}\supset\mu_{p^{\infty}}$, where $\mu_{p^{n}}$ denotes the group of $p^{n}$-th roots of unity and $\mu_{p^{\infty}}=\bigcup_{n}\mu_{p^{n}}$. The associated Iwasawa algebra is denoted by $\mathcal{R}=\mathbb{Z}_{p}[[\operatorname{Gal}(K_{\infty}/F)]]$. In this introduction we always assume Assumptions 2.1 and 2.2 introduced in §2.1.

We write $S_{p}^{\operatorname{ss}}$ for the set of $p$-adic primes of $F$ at which $E$ has supersingular reduction. We call an element of $\prod_{\mathfrak{p}\in S_{p}^{\operatorname{ss}}}\{+,-\}$ a multi-sign. More generally, we call an element of $\prod_{\mathfrak{p}\in S_{p}^{\operatorname{ss}}}\{0,1,+,-,\operatorname{rel}\}$ a multi-index (here, $0$ and $1$ are just symbols and have no relation with the natural numbers).

For each multi-index $\epsilon=(\epsilon_{\mathfrak{p}})_{\mathfrak{p}}$, in §2.2, we will introduce the $\epsilon$-Selmer group $\operatorname{Sel}^{\epsilon}(E/K_{\infty})$. It is defined by imposing $\epsilon_{\mathfrak{p}}$-local condition at each $\mathfrak{p}\in S_{p}^{\operatorname{ss}}$; $0$ denotes the strict condition, $\operatorname{rel}$ denotes the relaxed condition, and the $\pm$-local condition is essentially introduced by Kobayashi [14]. The original work of Kobayashi dealt with the case where $F=\mathbb{Q}$, and B. D. Kim [12] gave an extension to the case where $F$ is an imaginary quadratic field. The definition for general $F$ was given by Lei and Lim [15].

We take a finite set $S$ of non-$p$-adic finite primes of $F$ that satisfies a certain condition that we label as (2.1). Then we also define the $S$-imprimitive $\epsilon$-Selmer group $\operatorname{Sel}^{\epsilon}_{S}(E/K_{\infty})$ by relaxing the local condition at the primes in $S$. We put

where $(-)^{\vee}$ denotes the Pontryagin dual. It is known that $\mathfrak{X}^{\epsilon}$ and $\mathfrak{X}_{S}^{\epsilon}$ are finitely generated $\mathcal{R}$-modules. Moreover, we assume Assumption 2.6, which claims that these modules are torsion if $\epsilon$ is a multi-sign.

For each multi-sign $\epsilon$, in §2.3, we will define an algebraic ($S$-imprimitive) $\epsilon$-$p$-adic $L$-function

by requiring $\operatorname{Fitt}_{\mathcal{R}}(\mathfrak{X}_{S}^{\epsilon})=(\mathcal{L}_{S}^{\epsilon})$, where $\operatorname{Fitt}_{\mathcal{R}}(-)$ denotes the initial Fitting ideal. Note that $\mathcal{L}_{S}^{\epsilon}$ is defined in an algebraic way, not an analytic way. Therefore, a main conjecture is expected to be formulated as an equality between $\mathcal{L}_{S}^{\epsilon}$ and a certain analytic $p$-adic $L$-function, up to unit (see Remark 2.8 for the case $F=\mathbb{Q}$). However, in this paper we do not study such a main conjecture and we only deal with algebraic aspects.

Let us take distinct multi-signs $\epsilon_{1},\dots,\epsilon_{n}\in\prod_{\mathfrak{p}\in S_{p}^{\operatorname{ss}}}\{+,-\}$ with $n\geq 2$ (this forces $S_{p}^{\operatorname{ss}}\neq\emptyset$). We then define multi-indices $\overline{\epsilon}$ and $\underline{\epsilon}$ by

for each $\mathfrak{p}\in S_{p}^{\operatorname{ss}}$. Put

For each finite prime $v$ of $F$, we put $K_{\infty,v}=K_{\infty}\otimes_{F}F_{v}$. For $1\leq i\leq n$, we define

where $E^{\pm}(K_{\infty,\mathfrak{p}})$ denotes the $\pm$-norm subgroup of $E(K_{\infty,\mathfrak{p}})$ (see §2.1) and we use the Kummer map to embed $E^{\pm}(K_{\infty,\mathfrak{p}})\otimes(\mathbb{Q}_{p}/\mathbb{Z}_{p})$ into $H^{1}(K_{\infty,\mathfrak{p}},E[p^{\infty}])$. By comparing the local conditions between $\mathfrak{X}_{S}^{\underline{\epsilon}}$ and $\mathfrak{X}_{S}^{\overline{\epsilon}}$, we obtain a natural exact sequence of $\mathcal{R}$-modules

As we will see in §7.1, the generic ranks of both $\mathfrak{D}_{i}$ and $\mathfrak{X}_{S}^{\overline{\epsilon}}$ as $\mathcal{R}$-modules are $l$. It is of critical importance that $\mathfrak{D}_{i}$ is free as an $\mathcal{R}$-module. To prove this fact, in §4 we study the structure of $E^{\pm}(K_{\infty,\mathfrak{p}})$ closely.

Then (1.1) implies that the first map to $\mathfrak{X}_{S}^{\overline{\epsilon}}$ has information about $\mathfrak{X}_{S}^{\underline{\epsilon}}$. Motivated by the work [3], we take the exterior powers and obtain a map

The module in Theorem 1.1 below involving exterior powers denotes the cokernel of this map after taking the quotient of the target module by its torsion part.

In general, for a finitely generated $\mathcal{R}$-module $M$ and $i\geq 0$, we put $E^{i}(M)=\operatorname{Ext}_{\mathcal{R}}^{i}(M,\mathcal{R})$. We define $M_{\operatorname{tor}}$ (resp. $M_{\operatorname{PN}}$) as the maximal torsion (resp. pseudo-null) submodule of $M$ and we put $M_{/\operatorname{tor}}=M/M_{\operatorname{tor}}$ (resp. $M_{/\operatorname{PN}}=M/M_{\operatorname{PN}}$).

The proof of Theorem 1.1 will be given in §7.1. The structure of $E^{1}(\mathfrak{X}_{S}^{\overline{\epsilon}})$ will be the theme of Theorem 1.3 below. Actually, by combining Theorems 1.1 and 1.3, we obtain an analogue of [8, Theorem 5.3] for classical Iwasawa modules for CM-fields. (The statement of the results in this paper is much simpler than that of [8]; this is essentially because we have $H^{0}(K_{\infty,\mathfrak{p}},E[p^{\infty}])=0$ for any $p$-adic prime $\mathfrak{p}$ of $F$.)

We also have a refined version of Theorem 1.1 for the $l=1$ case. Note that $l=1$ is equivalent to that $n=2$ and the two multi-signs $\epsilon_{1}$ and $\epsilon_{2}$ differ at a single component. By assuming $l=1$, thanks to (1.1), we can immediately reformulate Theorem 1.1 as claim (1) of the following corollary; claim (2) will be proved in §7.2.

Note that the pseudo-nullity of $E^{1}(\mathfrak{X}_{S}^{\overline{\epsilon}})$ is closely related to that of $\mathfrak{X}^{\eta}$ in Theorem 1.3 below (as long as we assume condition ($\star$) there). Taking [4, Conjecture B] of Coates and Sujatha into account, we may expect that $\mathfrak{X}^{\eta}$ is often pseudo-null.

The following is the second main result of this paper (the proof will be given in §8). It gives an alternative description of $E^{1}(\mathfrak{X}_{S}^{\overline{\epsilon}})$ in Theorem 1.1; we set $\epsilon$ as $\overline{\epsilon}$ in Theorem 1.1.

Here, $\bigoplus_{v\not\in S_{p}(F)\cup S}$ denotes the direct sum for the finite primes $v$ of $F$ with $v\not\in S_{p}(F)\cup S$ (see Remark 1.4 below). We write $\iota$ for the involution on $\mathcal{R}$ that inverts every group element. For an $\mathcal{R}$-module $M$, we write $M^{\iota}$ for the $\mathcal{R}$-module whose additive structure is the same as $M$ and the action of $\mathcal{R}$ is twisted by $\iota$.

Note that, in condition $(\star)$, the residue degree is not finite in general, and in that case we regard it as a supernatural number (cf. Definition 4.1). Without the assumption $(\star)$, the description of $E^{1}(\mathfrak{X}_{S}^{\epsilon})$ seems to get harder.

In §2, we give the definitions of Selmer groups and algebraic $p$-adic $L$-functions. In §3, we illustrate the main results of this paper in special cases; in particular, we explain how to recover a main result of [16]. In §4, we study the structures of $\pm$-norm subgroups. In §§5 and 6, we review facts on perfect complexes and then introduce arithmetic complexes whose cohomology groups know the Selmer groups concerned. In §§7 and 8, we prove the first and the second main results, respectively.

As already remarked, the key idea to define the signed Selmer groups is given by Kobayashi [14], and there are a number of subsequent studies to generalize the idea. The definition below basically follows [15, Definition 4.7].

As usual, for each prime $\mathfrak{p}\in S_{p}(F)$, we put $a_{\mathfrak{p}}(E)=(1+\#\mathbb{F}_{\mathfrak{p}})-\#\widetilde{E}(\mathbb{F}_{\mathfrak{p}})$, where $\mathbb{F}_{\mathfrak{p}}$ denotes the residue field of $F$ at $\mathfrak{p}$ and $\widetilde{E}$ denotes the reduction of $E$ at $\mathfrak{p}$. We also put $\deg(\mathfrak{p})=[F_{\mathfrak{p}}:\mathbb{Q}_{p}]$.

In order to use the $\pm$-theory, we need to assume the following.

We also assume the non-anomalous condition at ordinary primes:

Let $\mathfrak{p}\in S_{p}^{\operatorname{ss}}$. We define $M^{\mathfrak{p}}$ as the inertia field in the given extension $K_{\infty}/F$ at $\mathfrak{p}$. Thanks to Assumption 2.1 and $K_{\infty}\supset\mu_{p^{\infty}}$, it is easy to see $K_{\infty}=M^{\mathfrak{p}}(\mu_{p^{\infty}})$ and $\operatorname{Gal}(K_{\infty}/M^{\mathfrak{p}})\simeq\mathbb{Z}_{p}^{\times}$. For each integer $n\geq-1$, we put $M^{\mathfrak{p}}_{n}=M^{\mathfrak{p}}(\mu_{p^{n+1}})$.

As in §1.1, we write $\mathfrak{X}^{\epsilon}$ and $\mathfrak{X}_{S}^{\epsilon}$ for the Pontryagin duals of $\operatorname{Sel}^{\epsilon}(E/K_{\infty})$ and $\operatorname{Sel}^{\epsilon}_{S}(E/K_{\infty})$, respectively. As stated in [15, Conjecture 4.11], it is natural to conjecture the following.

When $F=\mathbb{Q}$, Assumption 2.6 is known to be true, thanks to the celebrated work [10] of Kato (see [14, Theorem 1.2] or [6, Proposition 2.3]). See the last paragraph of [16, §6.2] for progress on the case where $F$ is an imaginary quadratic field.

It is convenient to introduce an order on the 5-element set $\{0,1,+,-,\operatorname{rel}\}$ defined by

(there is no order between $+$ and $-$). We extend this order to the set $\prod_{\mathfrak{p}\in S_{p}^{\operatorname{ss}}}\{0,1,+,-,\operatorname{rel}\}$ by defining $\epsilon^{\prime}\leq\epsilon$ if and only if $\epsilon^{\prime}_{\mathfrak{p}}\leq\epsilon_{\mathfrak{p}}$ for every $\mathfrak{p}\in S_{p}^{\operatorname{ss}}$. Then we have $\operatorname{Sel}^{\epsilon^{\prime}}_{S}(E/K_{\infty})\subset\operatorname{Sel}^{\epsilon}_{S}(E/K_{\infty})$ if $\epsilon^{\prime}\leq\epsilon$ since we have the corresponding inclusions concerning the local conditions. Note also that the definition of $\overline{\epsilon},\underline{\epsilon}$ in §1.2 can be rephrased as

Next we define the algebraic $p$-adic $L$-functions.

We suppose $F=\mathbb{Q}$, so we consider an abelian extension $K_{\infty}/\mathbb{Q}$ which is a finite extension of $\mathbb{Q}(\mu_{p^{\infty}})$. Let $E/\mathbb{Q}$ be an elliptic curve which has good supersingular reduction at $p$. We moreover suppose $a_{p}(E)=0$ (this is automatically true if $p\geq 5$ by the Hasse bound). Since $S_{p}^{\operatorname{ss}}$ is a singleton, a multi-index can be simply denoted by an element of $\{0,1,+,-,\operatorname{rel}\}$. Note that Assumptions 2.1, 2.2, and 2.6 hold automatically.

The unique choice (up to permutation) of distinct multi-signs is $\epsilon_{1}=+$ and $\epsilon_{2}=-$. Then we have $l=1$ and $\overline{\epsilon}=\operatorname{rel}$, $\underline{\epsilon}=1$. As a consequence of Corollary 1.2 and Theorem 1.3 (note that the condition $(\star)$ trivially holds), we obtain the following.

We shall deduce a main result of [16] from Corollary 1.2 and Theorem 1.3. Let us suppose that $F$ is an imaginary quadratic field in which $p$ splits into $\mathfrak{p}$ and $\overline{\mathfrak{p}}$. We also suppose that the dimension of $\operatorname{Gal}(K_{\infty}/F)$ is two. Equivalently, $K_{\infty}/F$ is an abelian extension which is a finite extension of $\widetilde{F}(\mu_{p})$, where $\widetilde{F}$ denotes the unique $\mathbb{Z}_{p}^{2}$-extension of $F$.

Let $E/F$ be an elliptic curve which has good supersingular reduction at both $\mathfrak{p}$ and $\overline{\mathfrak{p}}$, and we suppose that $a_{\mathfrak{p}}(E)=a_{\overline{\mathfrak{p}}}(E)=0$ so that Assumptions 2.1 and 2.2 hold. Fixing the order $\mathfrak{p},\overline{\mathfrak{p}}$, we express a multi-index $\epsilon$ simply by writing $(\epsilon_{\mathfrak{p}},\epsilon_{\overline{\mathfrak{p}}})$.

Then there are, up to permutation, exactly 4 choices of $\epsilon_{1},\epsilon_{2}$ such that $l=1$, namely

Moreover, we define $\eta$ as in Theorem 1.3 for $\overline{\epsilon}$; in other words, $\eta$ equals $\underline{\epsilon}$ with $1$ replaced by $0$. For instance, when $\epsilon_{1}=(+,+)$ and $\epsilon_{2}=(+,-)$, we have $\overline{\epsilon}=(+,\operatorname{rel})$, $\underline{\epsilon}=(+,1)$, and $\eta=(+,0)$.

As a consequence of Corollary 1.2 and Theorem 1.3, we obtain a completely analogous theorem to Theorem 3.1. We omit to state it. Note that we have to assume the validity of Assumption 2.6, and in addition condition ($\star$) to apply Theorem 1.3.

Our theorem recovers [16, Theorem 1] (except for the cases involving $\theta^{\textrm{Gr}}_{4,2}$) in the following way. We consider $K_{\infty}=\widetilde{F}(\mu_{p})$. Then $S_{\operatorname{ram},p}=\emptyset$, so $S=\emptyset$ satisfies (2.1). We write $\mathcal{L}^{\epsilon}=\mathcal{L}_{\emptyset}^{\epsilon}$ for each multi-sign $\epsilon$. Note that $\mathcal{R}$ is a finite product of regular local rings. For a pseudo-null $\mathcal{R}$-module $M$, as in [2, §1.1], we define the second Chern class $c_{2}(M)$ by a formal sum

where $\mathfrak{q}$ runs over the prime ideals of $\mathcal{R}$ of height $2$.

It is convenient to introduce the following formal terminology.

In this section, we basically write $f$ for a positive integer and $g$ for a supernatural number. Note that, for the applications to global settings, we only need to consider supernatural numbers $g$ of the form $g=f$ or $g=fp^{\infty}$ with positive integers $f$.

For each positive integer $f$, let $\mathbb{F}_{p^{f}}$ be the finite field with $p^{f}$ elements (in a fixed algebraic closure $\overline{\mathbb{F}_{p}}$ of $\mathbb{F}_{p}$), and $\mathbb{Q}_{p^{f}}$ the unramified extension of $\mathbb{Q}_{p}$ of degree $f$ (in a fixed maximal unramified extension $\mathbb{Q}_{p}^{\operatorname{ur}}$ of $\mathbb{Q}_{p}$). For each supernatural number $g$, we put

where $f$ runs over the positive integers with $f\mid g$. Then the correspondence $g\mapsto\mathbb{F}_{p^{g}}$ (resp. $g\mapsto\mathbb{Q}_{p^{g}}$) is a bijection between the set of supernatural numbers and the set of intermediate fields of $\overline{\mathbb{F}_{p}}/\mathbb{F}_{p}$ (resp. of $\mathbb{Q}_{p}^{\operatorname{ur}}/\mathbb{Q}_{p}$). We write $\varphi\in\operatorname{Gal}(\mathbb{Q}_{p}^{\operatorname{ur}}/\mathbb{Q}_{p})$ for the arithmetic Frobenius.

Let $g$ be a supernatural number. For an integer $n\geq-1$, we put $\mathbb{Q}_{p^{g},n}=\mathbb{Q}_{p^{g}}(\mu_{p^{n+1}})$ and $R_{g,n}=\mathbb{Z}_{p}[[\operatorname{Gal}(\mathbb{Q}_{p^{g},n}/\mathbb{Q}_{p})]]$. We also put $\mathbb{Q}_{p^{g},\infty}=\mathbb{Q}_{p^{g}}(\mu_{p^{\infty}})$ and $\mathcal{R}_{g}=\mathbb{Z}_{p}[[\operatorname{Gal}(\mathbb{Q}_{p^{g},\infty}/\mathbb{Q}_{p})]]$. Since $\operatorname{Gal}(\mathbb{Q}_{p^{g},0}/\mathbb{Q}_{p^{g},-1})\simeq\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p})/\mathbb{Q}_{p})$ is of order $p-1$, we can decompose the algebra $R_{g,0}$ with respect to the characters of that Galois group. Noting that the trivial character component is isomorphic to $R_{g,-1}$, we write $R_{g,0}^{\operatorname{nt}}$ for the direct product of the non-trivial character components, so we have a natural decomposition as an algebra

Let $E$ be a supersingular elliptic curve over $\mathbb{Q}_{p}$ satisfying $a_{p}(E)=(1+p)-\#\widetilde{E}(\mathbb{F}_{p})=0$.

We first define the $\pm$-norm subgroups in the current local situation.

Note that we have $E^{\pm}(\mathbb{Q}_{p^{g},n})=\bigcup_{f\mid g}E^{\pm}(\mathbb{Q}_{p^{f},n})$, where $f$ runs over all positive integers with $f\mid g$.

For a positive integer $f$, we define $E(\mathbb{Q}_{p^{f},n})_{p}$ as the $p$-adic completion of $E(\mathbb{Q}_{p^{f},n})$, which we regard as a submodule of $E(\mathbb{Q}_{p^{f},n})$ of finite index (the index is prime to $p$). We also define $E^{\pm}(\mathbb{Q}_{p^{f},n})_{p}$ similarly. Then, for each supernatural number $g$, we define $E(\mathbb{Q}_{p^{g},n})_{p}$ and $E^{\pm}(\mathbb{Q}_{p^{g},n})_{p}$ as the union of $E(\mathbb{Q}_{p^{f},n})_{p}$ and $E^{\pm}(\mathbb{Q}_{p^{f},n})_{p}$ for $f\mid g$, respectively.

The goal of this subsection is to prove the following (cf. [6, Theorem 1.2(3)]).