1. Introduction
Let
$K$
be a number field and
$p$
an odd prime. Two elliptic curves
$E_1$
and
$E_2$
defined over
$K$
are said to be
$p$
-congruent if
$E_1[p] \cong E_2[p]$
as
$\textrm {Gal}(\bar {K}/K)$
-modules, where
$\bar {K}$
denotes any fixed algebraic closure of
$K$
. More generally, we will study in this article pairs of elliptic curves
$E_1$
and
$E_2$
such that
$E_1[p^i] \cong E_2[p^i]$
for some positive integer
$i$
, which we will simply call congruent elliptic curves in this paper. Let
$K_{\infty }/K$
be a
$\mathbb{Z}_p$
-extension and let
$\textrm {Sel}(E_i/K_{\infty })$
be the
$p$
-primary classical Selmer group of
$E_i$
over
$K_{\infty }$
. We denote the Pontryagin dual of
$\textrm {Sel}(E_i/K_{\infty })$
by
$X(E_i/K_{\infty })$
. Now assume for
$i=1,2$
that
$E_i$
has good ordinary reduction at all primes of
$K$
dividing
$p$
and that
$X(E_i/K_{\infty })$
is a
$\Lambda \,:\!=\,{\mathbb{Z}}_p[[T]] \cong {\mathbb{Z}_p}[[\textrm {Gal}(K_\infty /K)]]$
-torsion module. When
$K={\mathbb{Q}}$
(so
$K_{\infty }={\mathbb{Q}}_{cyc}$
, the cyclotomic
$\mathbb{Z}_p$
-extension of
$\mathbb{Q}$
) the Greenberg-Vatsal paper [Reference Greenberg and Vatsal20] studies the relations between the lambda invariants
$\lambda (X(E_1/K_\infty ))$
and
$\lambda (X(E_2/K_\infty ))$
under the assumption that
$\mu (X(E_i/K_{\infty }))=0$
for
$i=1$
or
$2$
. These results were later generalized by Kidwell [Reference Kidwell28] to arbitrary number fields. The paper of Kundu and Ray [Reference Kundu and Ray37] refines the results of Greenberg-Vatsal and Kidwell, and the authors use their results to construct elliptic curves
$E$
over an imaginary quadratic field
$K$
such that
$X(E/K_{ac})$
has a large
$\lambda$
-invariant where
$K_{ac}$
is the anticyclotomic
$\mathbb{Z}_p$
-extension of
$K$
.
In a similar fashion to the above results, the aim of this paper is to study relationships between generalized Iwasawa invariants of Pontryagin duals of Selmer groups for congruent elliptic curves over
${\mathbb{Z}}_p^d$
-extensions. The generalized Iwasawa invariants were introduced by Cuoco and Monsky in [Reference Cuoco and Monsky11]. These are defined as follows: Let
$G$
be a profinite group isomorphic to
${\mathbb{Z}}_p^d$
for some
$d \geq 1$
. The Iwasawa algebra
$\Lambda _d = {\mathbb{Z}}_p[[G]]$
attached to
$G$
may be identified (by choosing a set of topological generators of
$G$
) with
${\mathbb{Z}}_p[[T_1,T_2,\ldots , T_d]]$
. Now let
$M$
be a finitely generated torsion
$\Lambda _d$
-module that is not pseudo-null. Then there exists a pseudo-null module
$A$
and an exact sequence
\begin{align} 0 \longrightarrow \bigoplus _{i=1}^s \Lambda _d/p^{m_i} \oplus \bigoplus _{j=1}^t \Lambda _d/h_j^{n_j} \longrightarrow M \longrightarrow A \longrightarrow 0. \end{align}
In the above, the exponents are natural numbers and the
$h_j$
are irreducible elements of
$\Lambda _d$
which are prime to
$p$
. The module
$\bigoplus _{i=1}^s \Lambda _d/p^{m_i} \oplus \bigoplus _{j=1}^t \Lambda _d/h_j^{n_j}$
is called an elementary
$\Lambda _d$
-module attached to
$M$
. The element
$f_M=\prod _{i=1}^s p^{m_i} \cdot \prod _{j = 1}^t h_j^{n_j}$
is the characteristic power series of
$M$
. One then defines the generalized Iwasawa invariants of
$M$
as follows. First,
$m_0(M) = \sum _{i=1}^s m_i$
. The definition of the second invariant is more involved. We write
$f_M = p^{m_0(M)} \cdot g_M$
, and we consider the (nonzero) image
$\overline {g_M}$
of
$g_M$
in the quotient ring
$\Omega _d = \Lambda _d/p$
. We let
$l_0(M)$
be the sum of the valuations
$v_{\mathfrak{p}}(\overline {g_M})$
, where
$\mathfrak{p}$
runs over the primes of
$\Omega _d$
of the form
$\overline {\gamma - 1}$
, with
$\gamma \in G$
not a
$p$
-th power (since
$\Lambda _d/p$
is a unique factorization domain, this sum is in fact finite). Finally, for a pseudo-null
$\Lambda _d$
-module
$M$
, we define
$f_M = 0$
and
$m_0(M) = l_0(M) = 0$
.
In [Reference Kleine and Matar35], we introduced a modified
$l_0$
-invariant which is defined as follows. Let
$d \gt 1$
. Recall that
$\Lambda _d \cong {\mathbb{Z}}_p[[G]]$
, where
$G$
is topologically isomorphic to
${\mathbb{Z}}_p^d$
. As above, let
$M$
be any finitely generated torsion
$\Lambda _d$
-module that is not pseudo-null, and let
$f_M$
be the characteristic power series of
$M$
. Write
$f_M = p^{m_0(M)} \cdot g_M$
with
$p \nmid g_M$
, as usual, and consider the image
$\overline {g_M}$
of
$g_M$
in
$\Omega _d$
.
Definition 1.1.
Let
$\mathcal{T}$
be the set of tuples
$(\gamma _1, \ldots , \gamma _{d-1})$
of elements of
$G$
which topologically generate subgroups isomorphic to
${\mathbb{Z}}_p^{d-1}$
. We let
\begin{equation*} \widehat {l_0}(M) = \widehat {l_0}(\overline {g_M}) = \sum _{\mathfrak{p}} v_{\mathfrak{p}}(\overline {g_M}), \end{equation*}
where
$\mathfrak{p}$
runs over the minimal prime ideals of
$(\overline {g_M}) \subseteq \Omega _d$
which are contained in some prime ideal
$\mathcal{P}_t$
of the form
$(\overline {\gamma _1 - 1}, \ldots , \overline {\gamma _{d-1}-1})$
, for any tuple
$t = (\gamma _1, \ldots , \gamma _{d-1})$
of
$\mathcal{T}$
.
Note that
$\widehat {l_0}(M) \ge l_0(M)$
by the definitions, and that
$\widehat {l_0}(M) = l_0(M)$
if
$d = 2$
.
If
$d = 1$
, that is, for
$\Lambda$
-modules, we can define Iwasawa invariants even for non-torsion modules. To be more precise, let
$M$
be a finitely generated (non-necessarily torsion)
$\Lambda$
-module. Then there exist finite
$\Lambda$
-modules
$A$
and
$B$
and an exact sequence
\begin{align} 0 \longrightarrow A \longrightarrow M \longrightarrow \Lambda ^r \oplus \bigoplus _{i=1}^s \Lambda /p^{m_i} \oplus \bigoplus _{j=1}^t \Lambda /h_j^{n_j} \longrightarrow B \longrightarrow 0. \end{align}
In the above, the exponents are again natural numbers and the
$h_j$
are now irreducible polynomials in
${\mathbb{Z}}_p[T]$
which are prime to
$p$
. The module
$\Lambda ^r \oplus \bigoplus _{i=1}^s \Lambda /p^{m_i} \oplus \bigoplus _{j=1}^t \Lambda /h_j^{n_j}$
is again called an elementary
$\Lambda _d$
-module attached to
$M$
,
$r$
is called the rank of
$M$
, and
$f_M=\prod _{i=1}^s p^{m_i} \cdot \prod _{j = 1}^t h_j^{n_j}$
is the characteristic power series of
$M$
, which is now a polynomial in
${\mathbb{Z}}_p[T]$
. In fact,
$\prod _{j = 1}^t h_j^{n_j}$
is a so-called distinguished polynomial, that is, it is monic and congruent to some power of
$T$
modulo
$p$
in the ring
$\Lambda$
. We define the Iwasawa invariants of
$M$
as follows:
$\mu (M) = \sum _i m_i$
(
$= m_0(M)$
for a torsion
$\Lambda$
-module
$M$
) and we let
$\lambda (M)$
be the degree of
$f_M$
(i.e.
$\prod _{j = 1}^t h_j^{n_j} \equiv T^{\lambda (M)} \pmod {p}$
). One can show that
$\lambda (M) = l_0(M)$
for a torsion
$\Lambda$
-module
$M$
.
Let
$\mathbb{L}_{\infty }$
be a fixed
$\mathbb{Z}_p^d$
-extension of
$K$
. If
$\sigma _1, \sigma _2,\ldots ,\sigma _d$
are topological generators of
$\textrm {Gal}(\mathbb{L}_{\infty }/K)$
, we let
$T_i=\sigma _i-1$
for each
$i$
. Suppose that
$E_1$
and
$E_2$
are two elliptic curves over
$K$
. We let
$\Lambda _d={\mathbb{Z}_p}[[\textrm {Gal}(\mathbb{L}_{\infty }/K)]]$
be the Iwasawa algebra attached to
$\mathbb{L}_{\infty }/K$
, identified with the power series ring
${\mathbb{Z}_p}[[T_1,T_2,\ldots ,T_d]]$
. We assume that neither
$X(E_1/\mathbb{L}_{\infty })$
nor
$X(E_2/\mathbb{L}_{\infty })$
is a pseudo-null
$\Lambda _2$
-module, that is, their characteristic power series are nonzero. If
$X(E_j/\mathbb{L}_{\infty })$
is a torsion
$\Lambda _d$
-module, then let
$m_{0,j}=m_0(X(E_j/\mathbb{L}_{\infty }))$
,
$\widehat {l_{0,j}}=\widehat {l_0}(X(E_j/\mathbb{L}_{\infty }))$
, and
$l_{0,j} = l_0(X(E_j/\mathbb{L}_\infty ))$
and let
$f_j$
be the characteristic power series of
$X(E_j/\mathbb{L}_{\infty })$
. We can write
$f_j=p^{m_{0,j}}g_j$
so that
$p \nmid g_j$
. Let
$S_p$
be the set of primes of
$K$
above
$p$
.
We fix a finite set S of nonarchimedean primes of K containing
$S_p$
and all the primes where either
$E_1$
or
$E_2$
has bad reduction.
Definition 1.2.
Let
$\mathcal{E}$
be the set of
$\mathbb{Z}_p$
-extensions of
$K$
which are contained in our fixed
${\mathbb{Z}}_p^d$
-extension
$\mathbb{L}_\infty$
. Define
$\mathcal{E}_p$
(resp.
$\mathcal{E}_{ns}$
) to be the set of all
$\mathbb{Z}_p$
-extensions
$K_{\infty } \in \mathcal{E}$
in which all primes in
$S_p$
ramify (resp. no prime in
$S$
splits completely).
As examples of
$\mathbb{Z}_p$
-extensions which belong or do not belong to
$E_p$
and
$E_{ns}$
, we mention that the cyclotomic
$\mathbb{Z}_p$
-extension of
$K$
is contained in
$\mathcal{E}_p \cap \mathcal{E}_{ns}$
. If
$K$
is a quadratic imaginary field and the prime of
$\mathbb{Q}$
below some prime in
$S$
is inert in
$K/{\mathbb{Q}}$
, then that prime in
$S$
splits completely in the anticyclotomic
$\mathbb{Z}_p$
-extension of
$K$
and hence this extension is not in
$\mathcal{E}_{ns}$
. Furthermore, for this field
$K$
if
$p$
splits in
$K/{\mathbb{Q}}$
, then
$K$
has two
$\mathbb{Z}_p$
-extensions not contained in
$\mathcal{E}_p$
.
Definition 1.3.
Let
$E$
be an elliptic curve defined over
$K$
. If
$v$
is a prime of
$K$
, then we say that
$E$
satisfies
$({\star})$
at
$v$
if either
$E$
has good reduction at
$v$
or one of the following two conditions is met:
-
(i)
$E$
has additive reduction at
$v$
, and
$p \geq 5$
, or
-
(ii)
$E$
has multiplicative reduction at
$v$
, and
$p \nmid c_v$
and
$p \nmid \#k_v^{\times }$
where
$k_v$
is the residue field of
$K_v$
and
$c_v$
is the Tamagawa number.
Throughout this paper, we assume that
$E_1$
and
$E_2$
have good ordinary reduction at all primes of K above p.
We let
$\mu _p$
denote the group of
$p$
-th roots of unity. Throughout the paper, we make the following assumption on our field
$K$
:
Condition
$(\Delta )$
: For every prime
$\mathfrak{p}$
of
$K$
above
$p$
, we have that the inertia subgroup of the extension
$K_{\mathfrak{p}}(\mu _p)/K_{\mathfrak{p}}$
is nontrivial.
This condition is needed for the proof of the central Lemma 3.5. Note that it is a rather mild assumption (see also Remark 1.6).
The main results of this paper are the following two theorems. Actually, the first theorem was obtained by Lim in a more general setting (see [Reference Lim39, Theorem 4.2.1]). We give a different proof of this result which will, in contrast to Lim’s approach, be strong enough to derive also results on generalized
$\lambda$
-invariants (see Theorem1.5, and cf. also the introduction to Section 6).
Theorem 1.4.
Assume that
$\mathcal{E}_p \neq \emptyset$
and
$\mathcal{E}_{ns} \neq \emptyset$
. If
$X(E_1/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module and
$E_1[p^i] \cong E_2[p^i]$
for some integer
$i \gt m_{0,1}$
, then
$X(E_2/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module with
$m_{0,2}=m_{0,1}$
.
Recall that we denote by
$f_1 = p^{m_{0,1}} g_1$
and
$f_2 = p^{m_{0,2}} g_2$
the characteristic power series of
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
, and that
$f_1, f_2 \ne 1$
since we assume that
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
are not pseudo-null over
$\Lambda _d$
. Let
$\Omega _d = \Lambda _d/p$
be the ‘mod
$p$
’ Iwasawa algebra, and denote by
$\overline {g_1}$
and
$\overline {g_2}$
the cosets of
$g_1$
and
$g_2$
in
$\Omega _d$
.
Theorem 1.5.
Assume that
$\mathcal{E}_p \neq \emptyset$
,
$\mathcal{E}_{ns} \neq \emptyset$
, and that both
$E_1$
and
$E_2$
satisfy condition
$({\star})$
for all
$v \in S$
. Furthermore, we assume
-
(i) Both
$E_1(\mathbb{L}_\infty )[p^\infty ]$
and
$E_2(\mathbb{L}_\infty )[p^\infty ]$
are finite.
-
(ii) If
$d \gt 2$
, the maximal pseudo-null submodules of
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
are finitely generated over
${\mathbb{Z}}_p$
. -
(iii) The decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
are open for each
$v \mid p$
. -
(iv)
$X(E_1/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module and
$E_1[p^i] \cong E_2[p^i]$
for some integer
$i \gt m_{0,1}$
.
Then the following statements hold.
-
(1)
$X(E_2/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module with
$m_{0,2}=m_{0,1}$
. Moreover,
$\widehat {l_{0,1}} \ne 0$
if and only if
$\widehat {l_{0,2}} \ne 0$
. -
(2)
$l_{0,2}=l_{0,1}$
.
Let us comment on the additional hypotheses in Theorem1.5.
-
• Condition (i) is mild as [Reference Kleine, Matar and Sujatha36, Lemma 3.5] shows.
-
• Concerning condition (ii), we need the maximal pseudo-null submodules of the Pontryagin duals of the Selmer groups to be “not too large” in order to be able to apply results from [Reference Kleine and Matar35]. In Section 9, we provide two explicit settings where condition (ii) is met.
-
• The third hypothesis (iii) in Theorem1.5 ensures that
$\mathbb{L}_\infty$
contains only finitely many primes above
$p$
. In particular, if
$K_\infty \subseteq \mathbb{L}_\infty$
denotes any
${\mathbb{Z}}_p$
-extension of
$K$
, then no prime
$v$
of
$K$
above
$p$
splits completely in
$K_\infty$
(this is needed for Proposition 7.10). An example where (iii) holds is when
$K$
is an imaginary quadratic field (see [Reference Minardi45, Lemma 3.1]). Also, (iii) is easily seen to hold if
$K$
has only one prime above
$p$
.
The above theorems are proven using an up-down approach: to show a relationship between generalized Iwasawa invariants of
$X(E_1/\mathbb{L}_{\infty })$
and
$X(E_2/\mathbb{L}_{\infty })$
, we formulate using results of [Reference Kleine and Matar35] an equivalent statement concerning Iwasawa invariants of
$X(E_1/K_{\infty })$
and
$X(E_2/K_{\infty })$
where
$K_{\infty }/K$
are
$\mathbb{Z}_p$
-extensions inside
$\mathbb{L}_{\infty }$
. We then prove this latter statement using techniques similar to ones used in [Reference Greenberg and Vatsal20], [Reference Kidwell28], and [Reference Kundu and Ray37].
Note that, unlike the former three papers, we do not assume that
$E_j(K)[p]=0$
. A key result of Hachimori and Matsuno [Reference Hachimori and Matsuno23] describing the maximal finite submodule of the Pontryagin dual of
$\textrm {Sel}(E/K_{\infty })$
where
$K_{\infty }/K$
is a
$\mathbb{Z}_p$
-extension will allow us to work in the case
$E_j(K)[p] \neq 0$
.
Remark 1.6.
Assume that the elliptic curves
$E_1$
and
$E_2$
are defined over a subfield
$K'$
of
$K$
and the isomorphism
$E_1[p^i] \cong E_2[p^i]$
in Theorems
1.4
and
1.5
is
$\textrm {Gal}(\bar {K}/K')$
-equivariant. Then the proof of Lemma 3.5 reveals that we can replace the condition
$(\Delta )$
by
Condition
$(\Delta ')$
: For every prime
$\mathfrak{p}$
of
$K'$
above
$p$
, we have that the inertia subgroup of the extension
$K'_{\mathfrak{p}}(\mu _p)/K'_{\mathfrak{p}}$
is nontrivial.
Note that when
$K'={\mathbb{Q}}$
condition
$(\Delta ')$
is true. In fact, condition
$(\Delta ')$
will be true as soon as
$p$
is unramified in
$K'$
or if
$K'$
is a normal extension of
$\mathbb{Q}$
and
$p-1$
does not divide the degree of the extension
$K'/{\mathbb{Q}}$
.
Let us briefly summarize the structure of the paper. The article consists of nine sections, including this introduction. In Section 2, we introduce three kinds of Selmer groups which are used in the literature, and we compare these Selmer groups over (multiple)
${\mathbb{Z}}_p$
-extensions of
$K$
. Although preliminary in nature, we believe that this overview on relations between the different kinds of Selmer groups is of independent interest. Moreover, we bound the ranks of the maximal finite
$\Lambda$
-submodules of Selmer groups over
${\mathbb{Z}}_p$
-extensions by using the results of Hachimori–Matsuno mentioned above. In Section 3, we explore the property
$({\star})$
from Definition 1.3, and we prove several control theorems.
The Iwasawa
$\mu$
- and
$\lambda$
-invariants of Selmer groups over
${\mathbb{Z}}_p$
-extensions are studied in Section 4. These results form the basis of our up-down proof of Theorem1.4, which is given in Section 5. In Section 6, we describe an alternative proof of Theorem1.4 which is due to Lim (see [Reference Lim39]). Lim’s proof is completely different from ours and relies on asymptotic growth formulas which are due to Cuoco, Monsky, and Perbet. It allows to compare
$\Lambda$
-ranks and
$\mu$
-invariants of Selmer groups of congruent elliptic curves over (multiple)
${\mathbb{Z}}_p$
-extensions. At the end of Section 6, we show how to derive results on (generalized)
$\lambda$
-invariants from Lim’s approach. It turns out that this approach can deal with
$\lambda$
-invariants only under very restrictive assumptions. Therefore, we return to our up-down approach in Section 7, where we present a proof of Theorem1.5, which works under less restrictive assumptions.
In Section 8, we study the
$\mathfrak{M}_H(G)$
-property (see also [Reference Kleine, Matar and Sujatha36]). This notion goes back to work of Coates et al. [Reference Coates, Fukaya, Kato, Sujatha and Venjakob6, Reference Coates and Sujatha9] on noncommutative Iwasawa theory. The validity of a suitable
$\mathfrak{M}_H(G)$
-property is essential even for the formulation of a main conjecture in noncommutative Iwasawa theory, since it segregates a natural subclass of Iwasawa modules with a sufficiently nice structure theory. Even in the commutative setting of
${\mathbb{Z}}_p^d$
-extensions, there are not many known cases where the property holds, in particular outside of the cyclotomic
${\mathbb{Z}}_p$
-extension. We show that under suitable hypotheses the
$\mathfrak{M}_H(G)$
-property holds for the Selmer groups of an elliptic curve
$E$
if and only if it holds for any elliptic curve
$E_2$
which is congruent to
$E$
. We also prove that the
$\mathfrak{M}_H(G)$
-property holds for a dense subset of
${\mathbb{Z}}_p$
-extensions inside many
${\mathbb{Z}}_p^d$
-extensions
$\mathbb{L}_\infty /K$
; this result generalizes a result from [Reference Kleine, Matar and Sujatha36] from dimension 2 to dimension
$d$
. In Section 9, we study the maximal pseudo-null submodules of Selmer groups over multiple
${\mathbb{Z}}_p$
-extensions of
$K$
. Finally, we end the paper with a section on concrete numerical examples that illustrate Theorems1.4 and 1.5.
2. Comparing Selmer groups
In this section, we define various Selmer groups and prove results that compare these different groups. We recall the definition of the classical Selmer group and then define two other Selmer groups which we call the strict and Greenberg Selmer groups. The comparison theorems in this section will allow us to compare results formulated for these different Selmer groups. This is especially important for Theorems1.4 and 1.5 which are formulated for the classical Selmer group. As it is easier to work with the Greenberg Selmer group rather than the classical Selmer group, we use in the proofs of these theorems our comparison result Corollary 2.4 to change the Selmer group.
In this section, we let
$E$
be any elliptic curve defined over
$K$
and only in this section we let
$S$
be any finite set of nonarchimedean primes of
$K$
containing
$S_p$
and all the primes where
$E$
has bad reduction. Let
$K_S$
be the maximal extension of
$K$
unramified outside
$S$
and the archimedean primes of
$K$
. Suppose now that
$F/K$
is a field extension with
$K \subseteq F \subseteq K_S$
. We let
$G_S(F)=\textrm {Gal}(K_S/F)$
and let
$S_F$
,
$S_{F,p}$
be the sets of primes of
$F$
above those in
$S$
,
$S_p$
, respectively. For every prime
$w \in S_F$
, we define a subgroup
$H^1_f(F_w, E[p^{\infty }])$
of
$H^1(F_w, E[p^{\infty }])$
and let
\begin{equation*}H^1_s(F_w, E[p^{\infty }])=H^1(F_w, E[p^{\infty }])/H^1_f(F_w, E[p^{\infty }]).\end{equation*}
Define
$\mathcal{L}(E/F)=\prod _{w \in S_F} H^1_f(F_w, E[p^{\infty }])$
. The
$p^{\infty }$
-Selmer group of
$E/F$
attached to this data is defined as
\begin{equation*}\displaystyle 0 \longrightarrow \textrm {Sel}(E/F, \mathcal{L}(E/F)) \longrightarrow H^1(G_S(F), E[p^{\infty }]) \longrightarrow \prod _{w \in S_F} H^1_s(F_w, E[p^{\infty }]). \end{equation*}
We now list various local conditions.
-
(1)
$H^1_{kum}(F_w, E[p^{\infty }])=\textrm {img}(E(F_w)\otimes {\mathbb{Q}_p}/{\mathbb{Z}_p} \hookrightarrow H^1(F_w, E[p^{\infty }]))$
Note that because of Mattuck’s theorem, we get that
$H^1_f(F_w, E[p^{\infty }])=0$
for primes
$w$
not dividing
$p$
. -
(2)
Here,
\begin{equation*} H^1_{str}(F_w, E[p^{\infty }])= \begin{cases} \ker (H^1(F_w, E[p^{\infty }]) \longrightarrow H^1(F_w, \tilde {E}[p^{\infty }])) & \text{if } w \mid p,\\ 0 & \text{if } w \nmid p. \end{cases} \end{equation*}
$\tilde {E}$
is the reduced elliptic curve over the residue field.
-
(3)
Here,
\begin{equation*} H^1_{Gr}(F_w, E[p^{\infty }])= \begin{cases} \ker (H^1(F_w, E[p^{\infty }]) \longrightarrow H^1(I_w, \tilde {E}[p^{\infty }])) & \text{if } w \mid p,\\ 0 & \text{if } w \nmid p. \end{cases} \end{equation*}
$I_w$
is the inertia group.
Each of the above local conditions defines a Selmer group:
-
(1) If
$F/K$
is a finite extension, and when
$H^1_f(F_w, E[p^{\infty }])=H^1_{kum}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
then we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}(E/F)$
. This is the classical Selmer group of
$E/F$
. If
$F/K$
is an infinite extension, then when
$H^1_f(F_w, E[p^{\infty }])=H^1_{kum}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}_{\infty }(E/F)$
. For any
$F/K$
(finite or infinite), we write
$H^1_{kum,s}(F_w, E[p^{\infty }])$
for
$H^1_s(F_w, E[p^{\infty }])$
. -
(2) If
$F/K$
is a finite extension, then when
$H^1_f(F_w, E[p^{\infty }])=H^1_{str}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}^{str}(E/F)$
. We call this the strict Selmer group. If
$F/K$
is an infinite extension, then when
$H^1_f(F_w, E[p^{\infty }])=H^1_{str}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}^{str}_{\infty }(E/F)$
. For any
$F/K$
(finite or infinite), we write
$H^1_{str,s}(F_w, E[p^{\infty }])$
for
$H^1_s(F_w, E[p^{\infty }])$
. -
(3) If
$F/K$
is a finite extension, then when
$H^1_f(F_w, E[p^{\infty }])=H^1_{Gr}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}^{Gr}(E/F)$
. We call this the Greenberg Selmer group (see [Reference Greenberg15]). If
$F/K$
is an infinite extension, then when
$H^1_f(F_w, E[p^{\infty }])=H^1_{Gr}(F_w, E[p^{\infty }])$
for all
$w \in S_F$
we denote
$\textrm {Sel}(E/F, \mathcal{L}(E/F))$
by
$\textrm {Sel}^{Gr}_{\infty }(E/F)$
. For any
$F/K$
(finite or infinite), we write
$H^1_{Gr,s}(F_w, E[p^{\infty }])$
for
$H^1_s(F_w, E[p^{\infty }])$
.
Now let
$L/K$
be a
$p$
-adic Lie extension with
$K \subseteq L \subset K_S$
. We define
\begin{equation*}\textrm {Sel}(E/L)=\mathop {\varinjlim }\limits \textrm {Sel}(E/F), \quad \textrm {Sel}^{str}(E/L)=\mathop {\varinjlim }\limits \textrm {Sel}^{str}(E/F)\end{equation*}
and
\begin{equation*}\textrm {Sel}^{Gr}(E/L)=\mathop {\varinjlim }\limits \textrm {Sel}^{Gr}(E/F)\end{equation*}
where the direct limits run over all finite extensions
$F$
of
$K$
contained in
$L$
with respect to restriction. We call these, respectively, the classical, strict, and Greenberg Selmer groups.
We denote the Pontryagin duals of
$\textrm {Sel}(E/L)$
,
$\textrm {Sel}^{str}(E/L)$
, and
$\textrm {Sel}^{Gr}(E/L)$
by
$X(E/L)$
,
$X^{str}(E/L)$
, and
$X^{Gr}(E/L)$
, respectively.
Suppose that
$F/K$
is a field extension with
$K \subseteq F \subseteq K_S$
. Sometimes,
$H^1_f(F_w, E[p^{\infty }])$
for primes
$w \nmid p$
is defined as:
\begin{equation*}H^1_{ur}(F_w, E[p^{\infty }])=\ker (H^1(F_w, E[p^{\infty }]) \longrightarrow H^1(I_w, E[p^{\infty }])).\end{equation*}
This is equal to
$H^1(\textrm {Gal}(\overline {F_w}/F_w)/I_w, E[p^{\infty }]^{I_w})$
. If
$F/K$
is a
${\mathbb{Z}}_p^d$
-extension and a prime
$v$
of
$K$
not dividing
$p$
does not split completely in
$F/K$
, then for any prime
$w$
of
$F$
above
$v$
the pro-
$p$
part of
$\textrm {Gal}(\overline {F_w}/F_w)/I_w$
is trivial. Therefore, in this case, we see that we get the same definition of the Selmer group of
$F/K$
if we choose
$H^1_f(F_w, E[p^{\infty }])$
to be either zero or
$H^1_{ur}(F_w, E[p^{\infty }])$
.
For the rest of this section, we define
$\Omega$
to be the set of finite extensions
$F$
of
$K$
inside
$\mathbb{L}_{\infty }$
for a
${\mathbb{Z}}_p^d$
-extension
$\mathbb{L}_\infty$
of
$K$
. We will compare the three kinds of Selmer groups over
$\mathbb{L}_{\infty }$
. First, we have the following important observation.
Lemma 2.1. We have equalities
-
(a)
$\textrm {Sel}(E/\mathbb{L}_{\infty })=\textrm {Sel}_{\infty }(E/\mathbb{L}_{\infty })$
, -
(b)
$\textrm {Sel}^{str}(E/\mathbb{L}_{\infty })=\textrm {Sel}^{str}_{\infty }(E/\mathbb{L}_{\infty })$
, -
(c)
$\textrm {Sel}^{Gr}(E/\mathbb{L}_{\infty })=\textrm {Sel}^{Gr}_{\infty }(E/\mathbb{L}_{\infty })$
.
Moreover, each of these Selmer groups is independent of the set
$S$
.
Proof.
We prove (a). The proof of (b) and (c) will be similar. To simplify the notation, we denote
$\mathbb{L}_{\infty }$
by
$L$
. From the definitions, we have exact sequences
\begin{equation*}\displaystyle 0 \longrightarrow \textrm {Sel}_{\infty }(E/L) \longrightarrow H^1(G_S(L), E[p^{\infty }]) \longrightarrow \prod _{w \in S_L} H^1(L_w, E)[p^{\infty }], \end{equation*}
\begin{equation*}\displaystyle 0 \longrightarrow \textrm {Sel}(E/L) \longrightarrow H^1(G_S(L), E[p^{\infty }]) \longrightarrow \bigoplus _{v \in S} \left (\mathop {\varinjlim }\limits _{F \in \Omega } \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }]\right ), \end{equation*}
where the direct limit in the last term in the second exact sequence runs over all finite extensions
$F/K$
inside
$L$
.
Define
\begin{equation*}X\,:\!=\,\bigoplus _{v \in S} \left (\mathop {\varinjlim }\limits _{F \in \Omega } \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }]\right ), \end{equation*}
\begin{equation*}Y\,:\!=\,\prod _{w \in S_L} H^1(L_w, E)[p^{\infty }]. \end{equation*}
To prove the lemma, it will be enough to exhibit an injection
$\Xi : X \hookrightarrow Y$
. To obtain such an injection, we need to show that for any
$v \in S$
, we have an injection
$\Xi _v\,:\, X_v \hookrightarrow Y_v$
, where
${X_v\,:\!=\,\mathop {\varinjlim }\limits _{F \in \Omega } \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }]}$
and
$Y_v\,:\!=\,\prod _{w | v} H^1(L_w, E)[p^{\infty }]$
.
For any
$F \in \Omega$
, the restriction maps induce a map
$\theta _F\,:\, \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }] \longrightarrow \prod _{w | v} H^1(L_w, E)[p^{\infty }]$
. The maps
$\{\theta _F\}_{F \in \Omega }$
are compatible under restriction and hence from the universal property of the direct limit we have a homomorphism
$\Xi _v\,:\, X_v \longrightarrow Y_v$
. We now show that
$\Xi _v$
is an injection.
Assume that
$x \in \ker \Xi _v$
. We have that for some
$F \in \Omega$
,
$x=\alpha _F(x_F)$
where
$x_F \in \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }]$
and
$\alpha _F \,:\, \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }] \longrightarrow X_v$
is the map into the direct limit. Since
$\Xi _v \circ \alpha _F = \theta _F$
, we have that
$\theta _F(x_F)=0$
.
Write
$x_F=(z_w)$
where the component
$z_w$
is contained in
$H^1(F_w,E)[p^{\infty }]$
. Let
$w$
be a prime of
$F$
above
$v$
. Choose a prime
$w'$
of
$L$
above
$w$
. Since
$\theta _F(x_F)=0$
, it follows that
$\textrm {res}_{L_{w'}}(z_w)=0$
, where
\begin{align*}{\textrm {res}_{L_{w'}}\,:\, H^1(F_w,E)[p^{\infty }] \longrightarrow H^1(L_{w'}, E)[p^{\infty }]}\end{align*}
is the restriction map. It is well known that we have
$H^1(L_{w'}, E) = \mathop {\varinjlim }\limits _{\tilde {F} \in \Omega } H^1(\tilde {F}_{w'}, E)$
, where for
$\tilde {F} \in \Omega$
,
$\tilde {F}_{w'}$
means the completion of
$\tilde {F}$
at the prime of
$\tilde {F}$
below
$w'$
. Hence,
$H^1(L_{w'}, E)[p^{\infty }] = \mathop {\varinjlim }\limits _{\tilde {F} \in \Omega } H^1(\tilde {F}_{w'}, E)[p^{\infty }]$
. Therefore from this and the fact that
$\textrm {res}_{L_{w'}}(z_w)=0$
, we see that there exists
$F^{(w)} \in \Omega$
that is Galois over
$F$
with
$\textrm {res}_{(F^{(w)})_{w'}}(z_w)=0$
, where
$\textrm {res}_{(F^{(w)})_{w'}}\,:\, H^1(F_w,E)[p^{\infty }] \longrightarrow H^1((F^{(w)})_{w'}, E)[p^{\infty }]$
is the restriction map.
Since
$F^{(w)}$
is Galois over
$F$
, if
$\tilde {w}$
is any prime of
$F^{(w)}$
above
$w$
, then the completion
$(F^{(w)})_{\tilde {w}}$
is equal to
$(F^{(w)})_{w'}$
. Therefore, for any prime
$\tilde {w}$
of
$F^{(w)}$
above
$w$
we have
$\textrm {res}_{(F^{(w)})_{\tilde {w}}}(z_w)=0$
.
Now let
$F'$
be the composite of the fields
$F^{(w)}$
where
$w$
runs over all primes of
$F$
above
$v$
. Then we see that
$\beta _{F,F'}(x_F)=0$
, where
$\beta _{F,F'}\,:\, \bigoplus _{w|v}H^1(F_w,E)[p^{\infty }] \longrightarrow \bigoplus _{w|v}H^1(F'_w,E)[p^{\infty }]$
is the map coming from restriction in each component. Therefore,
$x=\alpha _F(x_F)=0$
and hence
$\ker \Xi _v=0$
.
Finally, to prove that each of the Selmer groups in the statement of the lemma is independent of the set
$S$
, we only need to show that this result is true if
$\mathbb{L}_{\infty }$
is replaced with
$F \in \Omega$
. To this end, let
$F \in \Omega$
and let
$S$
be any finite set of nonarchimedean primes of
$K$
containing
$S_p$
and all the primes where
$E$
has bad reduction. By [Reference Milne44, Proposition I-6.5], we have an exact sequence
\begin{equation} 0 \longrightarrow H^1(G_S(F), E[p^{\infty }]) \longrightarrow H^1(F, E[p^{\infty }]) \longrightarrow \prod _{w \notin S_F} H^1(F_w, E)[p^{\infty }]. \end{equation}
We should note that loc. cit. assumes that
$S_F$
contains all archimedean primes of
$F$
. For any archimedean prime
$w$
of
$F$
, we have
$H^1(F_w, E[p^{\infty }])=0$
because
$p$
is odd. Also recall from the beginning of this section that we defined
$K_S$
to be the maximal extension of
$K$
unramified outside
$S$
and the archimedean primes of
$K$
. Thus, we do in fact have an exact sequence (3).
Let
$\bullet \in \{kum, str, Gr\}$
. We now show that
$\textrm {Sel}^{\bullet }(E/F)$
is independent of the set
$S$
, where
$\textrm {Sel}^{kum}(E/F)=\textrm {Sel}(E/F)$
. For
$\bullet =kum$
, this is proven in [Reference Milne44, Corollary I-6.6]. We can easily now adapt the proof to also work for the strict and Greenberg Selmer groups. Consider the exact sequence
\begin{equation} H^1(F, E[p^{\infty }]) \xrightarrow {\,\,\theta\,\, } \prod _{\text{all} \, w} H^1_{\bullet ,s}(F_w, E[p^{\infty }]) \xrightarrow {\,\,\pi\,\, } \prod _{w \notin S_F} H^1_{\bullet ,s}(F_w, E[p^{\infty }]). \end{equation}
In the above sequence,
$\theta$
is the localization map and
$\pi$
is just the projection. This sequence of maps induces an exact sequence
\begin{equation} 0 \longrightarrow \ker \theta \longrightarrow \ker \pi \circ \theta \longrightarrow \ker \pi . \end{equation}
If
$w \notin S_F$
, then in particular
$w$
does not divide
$p$
. By Mattuck’s theorem
$E(F_w) \cong \mathbb{Z}_l^d \times T$
, where
$l$
is the residue characteristic of
$F_w$
,
$d=[F_w\,:\,{\mathbb{Q}_l}]$
and
$T$
is a finite group. Since
$l \ne p$
, we have
$E(F_w)\otimes {\mathbb{Q}_p}/{\mathbb{Z}_p}=0$
and so
\begin{equation*}H^1_{\bullet ,s}(F_w, E[p^{\infty }])=H^1(F_w, E[p^{\infty }])=H^1(F_w, E)[p^{\infty }].\end{equation*}
Taking this into account, we get from (3) that
$\ker \pi \circ \theta = H^1(G_S(F), E[p^{\infty }])$
. Therefore, we get from (5) an exact sequence
\begin{equation*}0 \longrightarrow \ker \theta \longrightarrow H^1(G_S(F), E[p^{\infty }]) \longrightarrow \prod _{w \in S_F} H^1_{\bullet ,s}(F_w, E[p^{\infty }]).\end{equation*}
So
$\ker \theta = \textrm {Sel}^{\bullet }(E/F)$
. Since
$\ker \theta$
does not depend on the set
$S$
, the Selmer group
$\textrm {Sel}^{\bullet }(E/F)$
is independent of the set
$S$
.
We will use the equalities in the previous lemma throughout the paper without reference. In what follows, let
$\mathbb{L}_\infty /K$
be an arbitrary, but fixed
${\mathbb{Z}}_p^d$
-extension.
Proposition 2.2.
We have an equality
$\textrm {Sel}^{Gr}(E/\mathbb{L}_{\infty }) = \textrm {Sel}^{str}(E/\mathbb{L}_{\infty })$
.
Proof.
To prove the proposition, it will clearly suffice to show that for any
$F \in \Omega$
we have
${\textrm {Sel}_p}^{Gr}(E/F) = {\textrm {Sel}_p}^{str}(E/F)$
. Let
$F \in \Omega$
and consider the commutative diagram
Applying the snake lemma to the above diagram, we get an exact sequence
\begin{equation*}0 \longrightarrow \textrm {Sel}^{str}(E/F) \xrightarrow {\,\,\iota\,\,} \textrm {Sel}^{Gr}(E/F) \longrightarrow \textrm {img} \lambda _F \cap \ker \alpha \longrightarrow 0. \end{equation*}
Therefore to prove the result, it will suffice to show that
$\ker \alpha =0$
. It is easy to see that we have an injection
$\ker \alpha \hookrightarrow \bigoplus _{w \in S_{F,p}} H^1(\mathfrak{h}_{F,w}, \tilde {E}[p^{\infty }])$
, where
$\mathfrak{h}_{F,w}=\textrm {Gal}(\overline {F_w}/F_w)/I_w$
and
$I_w$
is the inertia subgroup.
As the residue field of
$F_w$
is finite,
$\mathfrak{h}_{F,w}$
acts nontrivially on
$\tilde {E}[p^{\infty }]$
. Now,
\begin{align*}H^1(\mathfrak{h}_{F,w}, \tilde {E}[p^{\infty }])=\tilde {E}[p^{\infty }]/(g-1)\tilde {E}[p^{\infty }]\end{align*}
where
$g$
is a topological generator of
$\mathfrak{h}_{F,w}$
. Since
$\tilde {E}[p^{\infty }]$
is isomorphic to
${\mathbb{Q}_p}/{\mathbb{Z}_p}$
and
$\mathfrak{h}_{F,w}$
acts nontrivially on it, it follows that
$\tilde {E}[p^{\infty }]/(g-1)\tilde {E}[p^{\infty }]=0$
. So,
$H^1(\mathfrak{h}_{F,w}, \tilde {E}[p^{\infty }])=0$
as desired.
Proposition 2.3. The following assertions hold.
-
(a) If
$F/K$
is a finite subextension of
$\mathbb{L}_{\infty }/K$
and
$w \in S_{F,p}$
, we let
$k_{F_w}$
denote the residue field of
$F_w$
. We have an inclusion map
$\iota \,:\, \textrm {Sel}(E/\mathbb{L}_{\infty }) \longrightarrow \textrm {Sel}^{str}(E/\mathbb{L}_{\infty })$
and an embedding
where the inverse limit is taken with respect to corestriction.
\begin{align*}\theta \,:\, \textrm {coker} \iota \hookrightarrow \Big(\mathop {\varprojlim }\limits _{F \in \Omega } \bigoplus _{w \in S_{F,p}} \tilde {E}(k_{F_w})[p^{\infty }]\Big )^{\vee },\end{align*}
-
(b) If
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion and
$E(\mathbb{L}_{\infty })[p^{\infty }]$
is finite, then
$\theta$
is an isomorphism.
-
(c) If every prime in
$S_p$
ramifies in
$\mathbb{L}_{\infty }/K$
, then
$\textrm {Sel}(E/\mathbb{L}_{\infty })= \textrm {Sel}^{str}(E/\mathbb{L}_{\infty })$
. -
(d)
$X(E/\mathbb{L}_{\infty })$
is a finitely generated
$\Lambda _d$
-torsion module if and only if
$X^{str}(E/\mathbb{L}_{\infty })$
is a finitely generated
$\Lambda _d$
-torsion module.
Proof.
By the discussion in [Reference Coates and Greenberg7, pg. 152–153], we have for any finite extension
$F/K$
inside
$\mathbb{L}_{\infty }$
and
$w \in S_{F,p}$
that
$H^1_{kum}(F_w, E[p^{\infty }]) \subseteq H^1_{str}(F_w, E[p^{\infty }])$
. Therefore, we have an inclusion
\begin{equation*}\iota \,:\, \textrm {Sel}(E/\mathbb{L}_{\infty }) \longrightarrow \textrm {Sel}^{str}(E/\mathbb{L}_{\infty }).\end{equation*}
Let
$S_{\infty }$
be the set of primes of
$\mathbb{L}_{\infty }$
above those in
$S$
. We have the following commutative diagram
Applying the snake lemma to the above diagram, we get an exact sequence
\begin{equation*}0 \longrightarrow \textrm {Sel}(E/\mathbb{L}_{\infty }) \stackrel {\iota }{\longrightarrow } \textrm {Sel}^{str}(E/\mathbb{L}_{\infty }) \longrightarrow \textrm {img} \lambda _{\infty } \cap \ker \alpha \longrightarrow 0.\end{equation*}
We have an isomorphism
\begin{equation*}\ker \alpha \cong \mathop {\varinjlim }\limits _{F \in \Omega } \bigoplus _{w \in S_F} H^1_{str}(F_w, E[p^{\infty }])/H^1_{kum}(F_w, E[p^{\infty }]). \end{equation*}
Let
$F \in \Omega$
and let
$w \in S_{F,p}$
. Let
$C_w=\mathcal{F}(\bar {\mathfrak{m}})[p^{\infty }]$
where
$\bar {\mathfrak{m}}$
is the maximal ideal of
$\overline {F_w}$
and
$\mathcal{F}$
is the formal group of
$E$
. First, we note that Tate local duality [Reference Neukirch, Schmidt and Wingberg47, Theorem 7.2.6] together with the Weil pairing yields a non-degenerate pairing
\begin{equation*}\langle \; , \;\rangle \,:\, H^2(F_w, T_p(C_w)) \times H^0(F_w, \tilde {E}[p^{\infty }]) \longrightarrow {\mathbb{Q}_p}/{\mathbb{Z}_p}\end{equation*}
where
$T_p(C_w)$
is the
$p$
-adic Tate module of
$C_w$
. If
$F'_w/F_w$
is a finite extension, let
\begin{align*}\textrm {res}\,:\, H^2(F_w, T_p(C_w)) \longrightarrow H^2(F'_w, T_p(C_w))\end{align*}
be the restriction map and let
$\textrm {cor}\,:\, H^0(F'_w, \tilde {E}[p^{\infty }]) \longrightarrow H^0(F_w, \tilde {E}[p^{\infty }])$
be the corestriction (norm) map. For
$a \in H^2(F_w, T_p(C_w))$
and
$b \in H^0(F'_w, \tilde {E}[p^{\infty }])$
, a property of Tate local duality gives
$\langle \textrm {res} a, b \rangle =\langle a, \textrm {cor} b \rangle$
. We have maps
\begin{equation*}\kappa _{F_w}\,:\, E(F_w)\otimes {\mathbb{Q}_p}/{\mathbb{Z}_p} \hookrightarrow H^1(F_w, E[p^{\infty }]), \end{equation*}
\begin{equation*}\lambda _{F_w}\,:\, H^1(F_w, C_w) \longrightarrow H^1(F_w, E[p^{\infty }]). \end{equation*}
Taking into account [Reference Coates and Greenberg7, Proposition4.5], the proof of [Reference Coates and Greenberg7, Proposition 4.6] shows that we have an isomorphism
$\theta _{F_w}\,:\, \textrm {img} \lambda _{F_w}/\textrm {img} \kappa _{F_w} \stackrel {\sim }{\longrightarrow } (\tilde {E}(k_{F_w})[p^{\infty }])^{\vee }$
where
$k_{F_w}$
is the residue field of
$F_w$
. Consider the exact sequence
\begin{equation*}0 \longrightarrow C_w \longrightarrow E[p^{\infty }] \longrightarrow \tilde {E}[p^{\infty }] \longrightarrow 0. \end{equation*}
From this exact sequence, we see that
$\textrm {img} \lambda _{F_w}=H^1_{str}(F_w, E[p^{\infty }])$
. Therefore, taking into account the property of Tate local duality above and the description of the map
$\theta _{F_w}$
, we have an isomorphism
\begin{equation*}\mathop {\varinjlim }\limits _{F \in \Omega } \bigoplus _{w \in S_{F,p}} \textrm {img} \lambda _{F_w}/\textrm {img} \kappa _{F_w} \cong \bigg(\mathop {\varprojlim }\limits _{F \in \Omega }\bigoplus _{w \in S_{F,p}} \tilde {E}(k_{F_w})[p^{\infty }]\bigg)^{\vee }. \end{equation*}
The direct limits are taken with respect to restriction and inverse limits are taken with respect to corestriction. This proves (a).
If
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion and
$E(\mathbb{L}_{\infty })[p^{\infty }])$
is finite, then by [Reference Hachimori and Venjakob25, Theorem 7.2] the map
$\lambda _{\infty }$
in the commutative diagram above is surjective. Therefore by the proof of part (a), we see that
$\theta$
is an isomorphism. This proves (b).
For part (c), we note that if
$L'_w/L_w$
is a finite totally ramified extension, then the norm map from
$\tilde {E}(k_{L'_w})[p^{\infty }]$
to
$\tilde {E}(k_{L_w})[p^{\infty }]$
is multiplication by
$[L'_w\,:\,L_w]$
. Therefore, if every prime in
$S_p$
ramifies in
$\mathbb{L}_{\infty }/K$
, we see from this that
$(\!\mathop {\varprojlim }\limits _{F \in \Omega }\bigoplus _{w \in S_{F,p}} \tilde {E}(k_{F_w})[p^{\infty }])^{\vee }=0$
and so
$\textrm {Sel}(E/\mathbb{L}_{\infty })= \textrm {Sel}^{str}(E/\mathbb{L}_{\infty })$
. Alternatively, (c) follows from [Reference Coates and Greenberg7, Prop. 4.3].
From (a), we see that to prove (d), we only have to show that the Pontryagin dual of
${A\,:\!=\,\Big(\mathop {\varprojlim }\limits _{F \in \Omega }\bigoplus _{w \in S_{F,p}} \tilde {E}(k_{F_w})[p^{\infty }]\Big)^{\vee }}$
is a finitely generated
$\Lambda _d$
-torsion module. For
$v \in S_p$
, we let
${B_v\,:\!=\, \Big(\mathop {\varprojlim }\limits _{F \in \Omega }\bigoplus _{w|v} \tilde {E}(k_{F_w})[p^{\infty }]\Big)^{\vee }}$
where the direct sum ranges over all primes
$w$
of
$F \in \Omega$
dividing
$v$
. To show that
$A^{\vee }$
is a finitely generated torsion
$\Lambda _d$
-module, it suffices to show that for any
$v \in S_p$
the
$\Lambda _d$
-module
$(B_v)^{\vee }$
is finitely generated torsion. Let
$v \in S_p$
and let
$G=\textrm {Gal}(\mathbb{L}_{\infty }/K)$
. By Shapiro’s lemma, we have
$(B_v)^G \cong \Big(\Big(\mathop {\varprojlim }\limits _{F \in \Omega } \tilde {E}(k_{F_w})[p^{\infty }]\Big)^{\vee }\Big)^{G_w}$
where
$w$
is some prime of
$\mathbb{L}_{\infty }$
above
$v$
and
$G_w$
is the decomposition group. As
$\mathop {\varprojlim }\limits \tilde {E}(k_{F_w})[p^{\infty }]$
is a pro-
$p$
procyclic group, it is a quotient of
$\mathbb{Z}_p$
(see [Reference Ribes and Zalesskii51, Proposition 2.7.1]). Therefore,
$(B_v)^G$
is cofinitely generated over
$\mathbb{Z}_p$
, and so
$(B_v)^\vee$
is a finitely generated
$\Lambda _d$
-module.
To prove that
$(B_v)^{\vee }$
is a torsion
$\Lambda _d$
-module, we consider two cases. First assume that
$v$
splits completely in
$\mathbb{L}_{\infty }/K$
. Let
$w$
be a prime of
$\mathbb{L}_{\infty }$
above
$v$
. Then
$\mathop {\varprojlim }\limits \tilde {E}(k_{F_w})[p^{\infty }]=\tilde {E}(k_{K_v})[p^{\infty }]$
is finite. Therefore, some power of
$p$
annihilates
$B_v$
whence
$(B_v)^{\vee }$
is a torsion
$\Lambda _d$
-module.
Now assume that
$v$
does not split completely in
$\mathbb{L}_{\infty }/K$
. Choose some
$\mathbb{Z}_p$
-extension
$K_{\infty }/K$
inside
$\mathbb{L}_{\infty }$
where
$v$
does not split completely and let
$H=\textrm {Gal}(\mathbb{L}_{\infty }/K_\infty )$
. Let
$T$
be the set of primes of
$K_{\infty }$
above
$v$
. Then by Shapiro’s lemma it follows that
$(B_v)^H \cong \Big(\Big(\mathop {\varprojlim }\limits _{F \in \Omega } \bigoplus _{w \in T} \tilde {E}(k_{F_w})[p^{\infty }]\Big)^{\vee }\Big)^{H_w}$
where for each
$w \in T$
we have also written
$w$
for a fixed prime of
$\mathbb{L}_{\infty }$
and
$H_w$
is the decomposition group. Since the set
$T$
is finite, we see that
$(B_v)^H$
is cofinitely generated over
$\mathbb{Z}_p$
. Therefore,
$(B_v)^{\vee }$
is finitely generated over
${\mathbb{Z}_p}[[H]]$
whence it is a torsion
$\Lambda _d$
-module.
Recall the notion of
$\mathcal{E}_p$
from Definition 1.2. From Propositions 2.2 and 2.3, we get
Corollary 2.4.
Assume that
$\mathcal{E}_p \neq \emptyset$
. Then
\begin{equation*}\textrm {Sel}(E/\mathbb{L}_{\infty })=\textrm {Sel}^{str}(E/\mathbb{L}_{\infty })=\textrm {Sel}^{Gr}(E/\mathbb{L}_{\infty }).\end{equation*}
Definition 2.5.
For an abelian group
$G$
, we define its
$p$
-rank as
\begin{equation*}r_p(G)=\dim _{{\mathbb{F}_p}}(G[p]).\end{equation*}
Under certain conditions, we will need to bound the
$p$
-rank of the maximal finite
$\Lambda$
-submodule of
$X^{Gr}(E/K_{\infty })$
as
$K_{\infty }$
runs over the set
$\mathcal{E}$
. To achieve this, we use the result of Hachimori–Matsuno [Reference Hachimori and Matsuno23] to bound the order of the maximal finite submodule of
$X(E/K_{\infty })$
. First, we need
Lemma 2.6.
Let
$K_{\infty } \in \mathcal{E}$
. We write
$\Gamma = \textrm {Gal}(K_\infty /K)$
, and we let
$\Gamma _n = \textrm {Gal}(K_\infty /K_n)$
for each
$n \in {\mathbb{N}}$
. Let
$s_n\,:\, \textrm {Sel}(E/K_n) \longrightarrow \textrm {Sel}(E/K_{\infty })^{\Gamma _n}$
be the map induced by restriction. Then
$\ker s_n$
is finite with
${r_p(\!\ker s_n) \le 2}$
, and the order of
$\ker s_n$
is bounded as
$n$
varies.
Proof.
For any
$n$
, consider the restriction map
\begin{equation*} g_n\,:\, H^1(G_S(K_n), E[p^{\infty }]) \longrightarrow H^1(G_S(K_{\infty }), E[p^{\infty }])^{\Gamma _n}. \end{equation*}
Since
$\ker s_n$
injects into
$\ker g_n$
, it will suffice to prove the desired result for
$\ker g_n$
instead. We have
$\ker g_n=H^1(\Gamma _n, E(K_{\infty })[p^{\infty }])$
. Let
$W=E(K_{\infty })[p^{\infty }]$
. If
$\gamma$
is a topological generator of
$\textrm {Gal}(K_{\infty }/K)=\Gamma$
, then
$\ker g_n=W/(\gamma ^{p^n}-1)W$
. The kernel of
$\gamma ^{p^n}-1$
acting on
$W$
is
$E(K_n)[p^{\infty }]$
which is finite.
Let
$W_{\text{div}}$
be the maximal divisible subgroup of
$W$
. Since
$W_{\text{div}}$
has finite
$\mathbb{Z}_p$
-corank and the kernel of
$\gamma ^{p^n}-1$
acting on
$W$
is finite, it follows that
$(\gamma ^{p^n}-1)W_{\text{div}}=W_{\text{div}}$
. Therefore, we see that we have a surjection
$W/W_{\text{div}} \twoheadrightarrow \ker g_n$
. This shows that
$|\ker g_n|$
is bounded by
$[W\,:\, W_{\text{div}}]$
and also that
\begin{align*}r_p(\ker g_n) \leq r_p(W/W_{\text{div}}) \le 2.\end{align*}
Definition 2.7.
If
$K_{\infty } \in \mathcal{E}$
, we denote the maximal finite
$\Lambda$
-submodule of
$X(E/K_{\infty })$
(resp.
$X^{Gr}(E/K_{\infty })$
) by
$\mathfrak{X}(E/K_{\infty })$
(resp.
$\mathfrak{X}^{Gr}(E/K_{\infty })$
).
Now, we have the following important theorem
Theorem 2.8 (Hachimori–Matsuno [Reference Hachimori and Matsuno23]). Suppose that
$K_{\infty } \in \mathcal{E}$
and that
$X(E/K_{\infty })$
is
$\Lambda$
-torsion. Then
$\mathfrak{X}(E/K_{\infty })$
is isomorphic to
$\mathop {\varprojlim }\limits \ker s_n$
where
$s_n\,:\, \textrm {Sel}(E/K_n) \longrightarrow \textrm {Sel}(E/K_{\infty })^{\Gamma _n}$
and the inverse limit is taken with respect to the corestriction maps.
From Theorem2.8 and Lemma 2.6, we have
Proposition 2.9.
For all
$K_{\infty } \in \mathcal{E}$
with
$X(E/K_{\infty })$
$\Lambda$
-torsion, we have
$r_p(\mathfrak{X}(E/K_{\infty })) \le 2$
.
Proof.
Let
$K_{\infty } \in \mathcal{E}$
with
$X(E/K_{\infty })$
$\Lambda$
-torsion. Let
\begin{equation*} s_n\,:\, \textrm {Sel}(E/K_n) \longrightarrow \textrm {Sel}(E/K_{\infty })^{\Gamma _n}\end{equation*}
and let
$A_n\,:\!=\,\ker s_n$
. From Lemma 2.6, we know that
$A_n$
is finite with bounded order as
$n$
varies. For any
$n$
, consider the exact sequence
$A_n \xrightarrow {\times p} A_n \longrightarrow A_n/p \longrightarrow 0$
. Since
$A_n/p$
is finite, it follows by taking inverse limits with respect to corestriction that we have an exact sequence (see [Reference Ribes and Zalesskii51, Lemma 1.1.5])
\begin{equation*}\mathop {\varprojlim }\limits A_n \xrightarrow {\times p} \mathop {\varprojlim }\limits A_n \longrightarrow \mathop {\varprojlim }\limits (A_n/p) \longrightarrow 0. \end{equation*}
Taking Theorem2.8 into account, the above exact sequence implies
\begin{equation*}\mathfrak{X}(E/K_{\infty })/p\cong (\mathop {\varprojlim }\limits A_n)/p \cong \mathop {\varprojlim }\limits (A_n/p). \end{equation*}
By Lemma 2.6, for any
$n$
, we have
$\dim _{{\mathbb{F}_p}}(A_n/p) \le 2$
. From this, it follows that the inverse system coming from the groups
$A_n/p$
is Mittag-Leffler. For any
$i \in {\mathbb{N}}$
let
$B_i \subseteq A_i/p$
be the stable image of
$\overline {\text{cor}}(A_j/p)$
for
$j \geq i$
where
$\overline {\text{cor}}$
is the map induced by corestriction. Then
$\mathop {\varprojlim }\limits (A_n/p) = \mathop {\varprojlim }\limits B_n$
. By the choice of the groups
$B_i$
, the inverse system coming from these groups has surjective maps. As each
$B_i$
is finite, therefore for some
$N \geq 0$
we have for
$j \geq i \geq N$
that the map
$\overline {\text{cor}}\,:\, B_j \longrightarrow B_i$
is an isomorphism. Then
$\mathop {\varprojlim }\limits B_n \cong B_N$
. Whence we have
\begin{equation*} \dim _{{\mathbb{F}_p}}(\!\mathop {\varprojlim }\limits B_n)=\dim _{{\mathbb{F}_p}}(B_N) \leq \dim _{{\mathbb{F}_p}}(A_N/p) \leq 2.\end{equation*}
This completes the proof.
Proposition 2.10.
Assume that
$E(\mathbb{L}_{\infty })[p^{\infty }]$
is finite and that the decomposition subgroups
${D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)}$
are open for each
$v \mid p$
. Then there exists a constant
$C^{Gr} \geq 0$
such that for all
$K_{\infty } \in \mathcal{E}$
with
$X^{Gr}(E/K_{\infty })$
$\Lambda$
-torsion we have
$r_p(\mathfrak{X}^{Gr}(E/K_{\infty })) \le C^{Gr}$
.
Proof.
Let
$t$
be the number of primes of
$\mathbb{L}_{\infty }$
above
$p$
. We claim that
$C^{Gr}=t+2$
has the required property. Let
$K_{\infty } \in \mathcal{E}$
be such that
$X^{Gr}(E/K_{\infty })$
is
$\Lambda$
-torsion. Propositions 2.2 and 2.3 hold for any
${\mathbb{Z}}_p^d$
-extension
$\mathbb{L}_{\infty }/K$
. Applying them to the
$\mathbb{Z}_p$
-extension
$K_{\infty }/K$
, we see that
$X(E/K_{\infty })$
is
$\Lambda$
-torsion. Therefore, by Proposition 2.9
$r_p(\mathfrak{X}(E/K_{\infty })) \le 2$
.
By Propositions 2.2 and 2.3, we have a surjection
\begin{equation*}\Psi \,:\, X^{Gr}(E/K_{\infty }) \twoheadrightarrow X(E/K_{\infty }). \end{equation*}
Therefore, we have an exact sequence
\begin{equation*}0 \longrightarrow \mathfrak{X}^{Gr}(E/K_{\infty }) \cap \ker \Psi \longrightarrow \mathfrak{X}^{Gr}(E/K_{\infty }) \longrightarrow \mathfrak{X}(E/K_{\infty }) \longrightarrow 0.\end{equation*}
From this exact sequence and the fact that
$r_p(\mathfrak{X}(E/K_{\infty })) \le 2$
, we see that to show that
${r_p(\mathfrak{X}^{Gr}(E/K_{\infty })) \le t+2}$
it will suffice to show that
$r_p(\ker \Psi ) \leq t$
.
With the notation of Proposition 2.3 let
\begin{equation*}A\,:\!=\, \left (\mathop {\varprojlim }\limits _{F \in \Omega } \bigoplus _{w \in S_{F,p}} \tilde {E}(k_{F_w})[p^{\infty }]\right )^{\vee }. \end{equation*}
It is easy to see that
$\dim _{{\mathbb{F}_p}}(A/p) \leq t$
. This fact together with Propositions 2.2 and 2.3 implies that
$r_p(\ker \Psi ) \leq t$
. This completes the proof.
Remark 2.11. The main tool needed to prove Theorem 2.8 of Hachimori–Matsuno is the Cassels–Tate pairing for the classical Selmer group:
\begin{equation*}\textrm {Sel}(E/K_n) \times \textrm {Sel}(E/K_n) \longrightarrow {\mathbb{Q}_p}/{\mathbb{Z}_p}. \end{equation*}
Here
$K_n$
is any tower field of our
$\mathbb{Z}_p$
-extension
$K_{\infty }/K$
. In Li Guo’s paper [Reference Guo21], a slightly modified version of
$\textrm {Sel}^{str}(E/K_n)$
is defined which we denote by
$\textrm {Sel}^{str*}(E/K_n)$
. loc. cit. defines a pairing:
\begin{equation*}\textrm {Sel}^{str*}(E/K_n) \times \textrm {Sel}^{str*}(E/K_n) \longrightarrow {\mathbb{Q}_p}/{\mathbb{Z}_p}. \end{equation*}
With this pairing, one can prove the analog of the Hachimori–Matsuno result for
$\mathop {\varinjlim }\limits \textrm {Sel}^{str*}(E/K_n)$
. We need to impose some conditions in order to get
$\mathop {\varinjlim }\limits \textrm {Sel}^{str*}(E/K_n)=\textrm {Sel}^{str}(E/K_{\infty })$
. Therefore to avoid additional hypotheses, we have proven Proposition 2.10 using Propositions 2.9
,
2.2 and 2.3
.
3. Results on
$\textbf{Sel}^{\textbf{Gr}}({\textbf{E/K}}_{\boldsymbol{\infty} })$
for
$K_{\infty } \in \mathcal{E}$
In this section, we prove some important results on
${\textrm {Sel}_p}^{Gr}(E/K_{\infty })$
for
$K_{\infty } \in \mathcal{E}$
. Let
$E$
be either
$E_1$
or
$E_2$
. Recall from the introduction that
$S$
is a finite set of nonarchimedean primes of
$K$
containing
$S_p$
and all the primes where either
$E_1$
or
$E_2$
has bad reduction. Suppose now that
$F$
is a field with
$K \subseteq F \subseteq K_S$
.
For a prime
$v \in S \setminus S_p$
, we define
\begin{equation*}\mathcal{H}_v(E/F)=\prod _{w|v}H^1(F_w,E[p^{\infty }])\end{equation*}
where the sum is over all primes
$w$
dividing
$v$
.
Proposition 3.1.
Let
$K_{\infty }/K$
be a
$\mathbb{Z}_p$
-extension. Assume that
$v \notin S_p$
does not split completely in
$K_{\infty }/K$
. Then
$\mathcal{H}_v(E/K_{\infty })^{\vee }$
is a finitely generated free
$\mathbb{Z}_p$
-module with
$\textrm {rank}_{{\mathbb{Z}_p}}(\mathcal{H}_v(E/K_{\infty })^{\vee })\le 2n$
where
$n$
is the number of primes of
$K_{\infty }$
above
$v$
.
Proof.
Let
$w$
be a prime of
$K_{\infty }$
above
$v$
. First, we show that
$H^1(K_{\infty ,w}, E[p^{\infty }])$
is divisible. Since
$v$
does not split completely in
$K_{\infty }/K$
, it follows from [Reference Neukirch, Schmidt and Wingberg47, Theorem 7.1.8(i)] that
$cd_p(K_{\infty ,w}) \le 1$
. The exact sequence
$0 \longrightarrow E[p] \longrightarrow E[p^{\infty }] \xrightarrow {\times p} E[p^{\infty }] \longrightarrow 0$
induces an exact sequence
\begin{equation*}H^1(K_{\infty ,w}, E[p^{\infty }]) \xrightarrow {\times p} H^1(K_{\infty ,w}, E[p^{\infty }]) \longrightarrow H^2(K_{\infty ,w}, E[p]). \end{equation*}
The last term is zero because
$cd_p(K_{\infty ,w}) \le 1$
whence
$H^1(K_{\infty ,w}, E[p^{\infty }])$
is divisible.
Let
$B=\textrm {Hom}_{{\mathbb{Z}_p}}(T_p(E[p^{\infty }]), \mu _{p^{\infty }})$
where
$T_p(E[p^{\infty }])$
is the
$p$
-adic Tate module of
$E[p^{\infty }]$
and
$\mu _{p^{\infty }}$
is the group of
$p$
-power roots of unity. By [Reference Greenberg15, Proposition 2],
$H^1(K_{\infty ,w}, E[p^{\infty }])$
is cofinitely generated over
$\mathbb{Z}_p$
with
$\textrm {corank}_{{\mathbb{Z}_p}}(H^1(K_{\infty ,w}, E[p^{\infty }]))=\textrm {corank}_{{\mathbb{Z}_p}}(H^0(K_{\infty ,w}, B))$
. By the Weil pairing
$B \cong E[p^{\infty }]$
and therefore
$\textrm {corank}_{{\mathbb{Z}_p}}(H^1(K_{\infty ,w}, E[p^{\infty }]))=\textrm {corank}_{{\mathbb{Z}_p}}(E(K_{\infty ,w})[p^{\infty }])$
. This completes the proof.
Proposition 3.2.
Let
$K_{\infty }/K$
be a
$\mathbb{Z}_p$
-extension. Let
$v \notin S_p$
. Then the following two assertions hold true.
-
(i) If
$E$
has additive reduction at
$v$
and
$p \geq 5$
, then
$\mathcal{H}_v(E/K_{\infty })=0$
. -
(ii) If
$E$
has multiplicative reduction at
$v$
,
$p \nmid c_v$
and
$p \nmid \#k_v^{\times }$
where
$k_v$
is the residue field of
$K_v$
, then
$\mathcal{H}_v(E/K_{\infty })=0$
.
Proof.
Let
$w$
be a prime of
$K_{\infty }$
above
$v$
. If
$v$
does not split completely in
$K_{\infty }/K$
, then
$K_{\infty ,w}/K_v$
is a
$\mathbb{Z}_p$
-extension. Let
$\Gamma _w=\textrm {Gal}(K_{\infty ,w}/K_v)$
. Then
$cd_p(\Gamma _w)=1$
and so we get a surjection
\begin{align*}H^1(K_v, E[p^{\infty }]) \twoheadrightarrow H^1(K_{\infty ,w}, E[p^{\infty }])^{\Gamma _w}.\end{align*}
Since
$\Gamma _w$
is pro-
$p$
, we see that to show that
$H^1(K_{\infty ,w}, E[p^{\infty }])=0$
, it will suffice to show that
$H^1(K_v, E[p^{\infty }])=0$
. If
$v$
splits completely in
$K_{\infty }/K$
, then
$H^1(K_{\infty ,w}, E[p^{\infty }])=H^1(K_v, E[p^{\infty }])$
. Therefore, we see that in either case that we must show that
$H^1(K_v, E[p^{\infty }])=0$
.
By Mattuck’s Theorem, we have that
$E(K_v) \cong \mathbb{Z}_l^d \times T$
, where
$l$
is the residue characteristic of
$K_v$
,
$d=[K_v\,:\,{\mathbb{Q}_l}]$
and
$T$
is a finite group. Since
$l \ne p$
, we have
$E(K_v)\otimes {\mathbb{Q}_p}/{\mathbb{Z}_p}=0$
and so
$H^1(K_v, E[p^{\infty }])=H^1(K_v, E)[p^{\infty }]$
. By Tate duality for abelian varieties over local fields [Reference Milne44, Corollary I-3.4], we have an isomorphism
$H^1(K_v, E)^{\vee } \cong E(K_v)$
. Therefore,
$(H^1(K_v, E)[p^{\infty }])^{\vee } \cong \mathop {\varprojlim }\limits E(K_v)/p^nE(K_v)$
. Since
${E(K_v) \cong \mathbb{Z}_l^d \times T}$
and
$l \ne p$
, we have that
$\mathop {\varprojlim }\limits E(K_v)/p^nE(K_v)$
is just the (finite)
$p$
-primary subgroup of
$E(K_v)$
. It follows from the previous facts that
$H^1(K_v, E[p^{\infty }])$
is isomorphic to
$E(K_v)[p^{\infty }]$
and so we only need to show that
$E(K_v)[p^{\infty }]=0$
.
Assume that
$E$
has additive reduction at
$v$
and
$p$
and
$p \geq 5$
. Let
$E_0(K_v)$
be the group of points of
$E(K_v)$
of nonsingular reduction. Since
$p \geq 5$
, by [Reference Silverman57, pg. 448] the cardinality of
$E(K_v)/E_0(K_v)$
is prime to
$p$
. Therefore,
$E(K_v)[p^{\infty }]=E_0(K_v)[p^{\infty }]$
. Let
$\hat {E}$
be the formal group of
$E/K_v$
and
$\mathfrak{m}_v$
the maximal ideal of
$O_v$
(the ring of integers of
$K_v$
). Since
$v \nmid p$
,
$\hat {E}(\mathfrak{m}_v)$
has no elements of order
$p$
and so
$E(K_v)[p^{\infty }]=E_0(K_v)[p^{\infty }]$
injects into the set
$\tilde {E}_{ns}(k_v)$
of nonsingular points of the reduction of
$E$
modulo the residue field
$k_v$
. But as
$E$
has additive reduction at
$v$
we have
$\tilde {E}_{ns}(k_v)\cong k_v$
. Since
$k_v$
has no points of order
$p$
(because
$v \nmid p$
), we see that
$E(K_v)[p^{\infty }]=0$
. This proves (i).
Now assume that
$E$
has multiplicative reduction at
$v$
,
$p \nmid c_v$
and
$p \nmid \#k_v^{\times }$
. Since
$p \nmid c_v$
,
$\#E(K_v)/E_0(K_v)$
is prime to
$p$
and so
$E(K_v)[p^{\infty }]=E_0(K_v)[p^{\infty }]$
. As above we have that
$E(K_v)[p^{\infty }]=E_0(K_v)[p^{\infty }]$
injects into
$\tilde {E}_{ns}(k_v)$
. As
$E$
has multiplicative reduction at
$v$
,
$\tilde {E}_{ns}(k_v)\cong k_v^{\times }$
. Since
$p \nmid \#k_v^{\times }$
by assumption, we can conclude that
$E(K_v)[p^{\infty }]=0$
. This proves (ii).
Let
$K_{\infty }/K$
be a
$\mathbb{Z}_p$
-extension. Let
$S_{K_{\infty }}$
(resp.
$S_{K_{\infty ,p}}$
) be the sets of primes of
$K_{\infty }$
above those in
$S$
(resp.
$S_p$
). We now define the “non-primitive” Greenberg Selmer group as
\begin{align} \displaystyle 0 \longrightarrow \textrm {Sel}^{Gr(p)}(E/K_{\infty }) \longrightarrow H^1(G_S(K_{\infty }), E[p^{\infty }]) \longrightarrow \prod _{w \in S_{K_{\infty ,p}}} H^1_{Gr,s}(K_{\infty ,w}, E[p^{\infty }]). \end{align}
We will denote the Pontryagin dual of
$\textrm {Sel}^{Gr(p)}(E/K_{\infty })$
by
$X^{Gr(p)}(E/K_{\infty })$
. We now define the reduced Greenberg Selmer groups and reduced non-primitive Greenberg Selmer groups. To define these, we first define the reduced local conditions. For
$i \in {\mathbb{N}}$
, we define
\begin{equation*} H^1_{Gr}(K_{\infty ,w}, E[p^i])= \begin{cases} \ker (H^1(K_{\infty ,w}, E[p^i]) \longrightarrow H^1(I_w, \tilde {E}[p^i])) & \text{if } w \mid p,\\ 0 & \text{if } w \nmid p. \end{cases} \end{equation*}
For any prime
$w$
of
$K_{\infty }$
, we define
\begin{equation*}H^1_{Gr,s}(K_{\infty ,w}, E[p^i])=H^1(K_{\infty ,w}, E[p^i])/H^1_{Gr}(K_{\infty ,w}, E[p^i]). \end{equation*}
We define
\begin{equation*}\displaystyle 0 \longrightarrow \textrm {Sel}^{Gr}(E[p^i]/K_{\infty }) \longrightarrow H^1(G_S(K_{\infty }), E[p^i]) \longrightarrow \prod _{w \in S_{K_{\infty }}} H^1_{Gr,s}(K_{\infty ,w}, E[p^i]) , \end{equation*}
\begin{equation*} \displaystyle 0 \longrightarrow \textrm {Sel}^{Gr(p)}(E[p^i]/K_{\infty }) \longrightarrow H^1(G_S(K_{\infty }), E[p^i]) \longrightarrow \prod _{w \in S_{K_{\infty ,p}}} H^1_{Gr,s}(K_{\infty ,w}, E[p^i]). \end{equation*}
Recall from Definition 2.5 that for an abelian group
$G$
, we define its
$p$
-rank as
$r_p(G)=\dim _{{\mathbb{F}_p}}(G[p])$
. We have the following result.
Proposition 3.3.
For any
$K_{\infty } \in \mathcal{E}_p$
and each
$i \in {\mathbb{N}}$
, the natural map
\begin{equation*} s_i\,:\, \textrm {Sel}^{Gr(p)}(E[p^i]/K_{\infty }) \longrightarrow \textrm {Sel}^{Gr(p)}(E/K_{\infty })[p^i]\end{equation*}
is surjective with finite kernel satisfying
$r_p(\ker s_i) \leq 2$
and
$|\ker (s_i)| \le p^{2i}$
.
Proof. Consider the commutative diagram
The map
$g_i$
is surjective with
$\ker g_i = E(K_{\infty })[p^{\infty }]/p^i$
, whence
$r_p(\ker g_i) \leq 2$
. It follows from this and the snake lemma that the desired result will follow if we can show that the map
$h_i$
is injective. This in turn will follow if we can show that the map
$h_{w,i}\,:\, H^1(I_w, \tilde {E}[p^i]) \longrightarrow H^1(I_w, \tilde {E}[p^{\infty }])[p^i]$
is injective where
$w$
is a prime of
$K_{\infty }$
above
$p$
. But
$I_w$
acts trivially on
$\tilde {E}$
, so
\begin{equation*} \ker h_{w,i} = H^0(I_w, \tilde {E}[p^{\infty }])/p^i=\tilde {E}[p^{\infty }]/p^i =0.\end{equation*}
This completes the proof.
Remark 3.4.
The proof of the above lemma shows that
$s_i$
is an isomorphism if
$E(K_\infty )[p^\infty ] = 0$
.
Lemma 3.5.
For any
$K_{\infty } \in \mathcal{E}_p$
and every
$i \in {\mathbb{N}}$
,
$i \gt 0$
, the isomorphism
$E_1[p^i] \cong E_2[p^i]$
of Galois modules induces an isomorphism
\begin{equation*}\textrm {Sel}^{Gr(p)}(E_1[p^i]/K_{\infty }) \cong \textrm {Sel}^{Gr(p)}(E_2[p^i]/K_{\infty }). \end{equation*}
Proof.
The isomorphism
$\Phi \,:\, E_1[p^i] \cong E_2[p^i]$
induces an isomorphism
\begin{align*}H^1(G_S(K_{\infty }), E_1[p^i]) \cong H^1(G_S(K_{\infty }), E_2[p^i]).\end{align*}
Let
$v$
be a prime of
$K$
above
$p$
. We want to show that
$\Phi$
induces an isomorphism
$\Phi _v\,:\, {\tilde {E_1}[p^i] \cong \tilde {E_2}[p^i]}$
. We now show that condition
$(\Delta )$
in the introduction implies that such a map
$\Phi _v$
can be induced from
$\Phi$
. Let
$G_{K_v} = \textrm {Gal}(\bar {K}_v/K_v)$
and let
$I_v$
be the inertia subgroup.
Let
$C_v=\mathcal{F}(\bar {\mathfrak{m}})$
where
$\bar {\mathfrak{m}}$
is the maximal ideal of
$\bar {K}_v$
and
$\mathcal{F}$
is the formal group of
$E_1$
. We similarly define
$D_v$
using the formal group of
$E_2$
. Then we have exact sequences
\begin{equation*}0 \longrightarrow C_v[p^n] \longrightarrow E_1[p^n] \longrightarrow \tilde {E}_1[p^n] \longrightarrow 0,\end{equation*}
\begin{equation*}0 \longrightarrow D_v[p^n] \longrightarrow E_2[p^n] \longrightarrow \tilde {E}_2[p^n] \longrightarrow 0.\end{equation*}
From the exact sequences above, we see that to prove that
$\Phi _v$
can be induced from
$\Phi$
we need to show that
$\Phi (C_v[p^n])=D_v[p^n]$
. To show that, we use a similar argument to that of Greenberg and Vatsal, see [Reference Greenberg and Vatsal20, p. 26]. Let
$\mu _{p^n}$
be the group of
$p^n$
-th roots of unity in
$\bar {K}_v$
. It will suffice to prove that
$C_v[p^n]$
is the unique subgroup of
$E_1[p^n]$
that is isomorphic to
$\mu _{p^n}$
as an
$I_v$
-module and similarly for
$E_2[p^n]$
. Note that as
$E_1$
and
$E_2$
have good ordinary reduction at
$v$
; therefore as
${\mathbb{Z}}/p^n{\mathbb{Z}}$
-modules, we have
$\tilde {E}_1[p^n] \cong \tilde {E}_2[p^n] \cong {\mathbb{Z}}/p^n{\mathbb{Z}}$
. As
$I_v$
acts trivially on
$\tilde {E}_1[p^n]$
, we see by the Weil pairing that
$I_v$
acts on
$C_v[p^n]$
by the
$p^n$
-cyclotomic character
$\chi$
, that is,
$C_v[p^n] \cong \mu _{p^n}$
as an
$I_v$
-module. Now we show that
$C_v[p^n]$
is the unique subgroup with this property.
CLAIM:
$I_v$
acts nontrivially on any nontrivial quotient of
$\mu _{p^n}$
. Assume that the claim is not true. Any nontrivial quotient of
$\mu _{p^n}$
is of the form
$\mu _{p^n}/\mu _{p^k}$
where
$k \lt n$
. Let
$\sigma \in I_v$
and let
$\zeta$
be a primitive
$p^n$
-th root of unity. Then
$\sigma (\zeta )\mu _{p^k}=\sigma (\zeta \mu _{p^k})=\zeta \mu _{p^k}$
, that is
$\sigma (\zeta )=\zeta \alpha$
where
$\alpha \in \mu _{p^k}$
. Since
$\alpha =\zeta ^c$
for some integer
$c$
divisible by
$p^{n-k}$
, we have that
$\sigma (\zeta )=\zeta ^{1+c}$
for such an integer
$c$
. Then
$\sigma (\zeta ^{p^{n-1}})=(\zeta ^{p^{n-1}})^{1+c}=\zeta ^{p^{n-1}}$
since
$cp^{n-1}$
is divisible by
$p^n$
. Thus,
$\sigma (\mu _p)=\mu _p$
. This holds for all
$\sigma \in I_v$
. Therefore,
$(\mu _p)^{I_v}=\mu _p$
. This contradicts
$(\Delta )$
and so we see that the claim is true.
Now suppose that
$U \neq C_v[p^n]$
is a subgroup of
$E_1[p^n]$
that is isomorphic to
$\mu _{p^n}$
as an
$I_v$
-module. Consider the quotient
$U/(U \cap C_v[p^n])$
. Since
$U \neq C_v[p^n]$
, this is a nontrivial quotient of
$U$
and so by the above claim
$I_v$
acts nontrivially on this quotient. But we have a
$({\mathbb{Z}}/p^n{\mathbb{Z}})[I_v]$
-injection
\begin{align*}U/(U \cap C_v[p^n]) \hookrightarrow \tilde {E}_1[p^n].\end{align*}
Since
$I_v$
acts trivially on
$\tilde {E}_1[p^n]$
, this is a contradiction. Therefore, we see that
$U=C_v[p^n]$
, that is,
$C_v[p^n]$
is the unique subgroup of
$E_1[p^n]$
that is isomorphic to
$\mu _{p^n}$
as an
$I_v$
-module. Thus,
$\Phi$
induces an isomorphism
$\Phi _v$
.
It follows from this that for any
$w \in S_p$
, we have an isomorphism
\begin{align*}H^1_{Gr,s}(K_{\infty ,w}, E_1[p^i]) \cong H^1_{Gr,s}(K_{\infty ,w}, E_2[p^i]).\end{align*}
The lemma follows from these observations.
We end this section with the following control theorem (where as usual we assume that
$d \geq 2$
). Let
$E$
be either
$E_1$
or
$E_2$
. Let
$S$
be the set of primes of
$K$
which lie above
$p$
or where
$E$
has bad reduction.
Proposition 3.6.
Let
$K_{\infty } \in \mathcal{E}$
. Consider the dual of the restriction map
\begin{equation*}f^\vee \colon X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })} \longrightarrow X^{Gr}(E/K_{\infty }). \end{equation*}
We have
-
(a)
$\textrm {coker} f^{\vee }$
is a finitely generated
${\mathbb{Z}}_p$
-module with
$\textrm {rank}_{{\mathbb{Z}}_p}(\textrm {coker}\, f^{\vee }) \leq (d-1)$
. -
(b) Let
$m_p=\sum _{v | p} m_v$
where the sum runs over the primes of
$K$
above
$p$
and
$m_v$
is defined to be zero if
$v$
splits completely in
$K_{\infty }$
and equal to the number of primes of
$K_{\infty }$
above
$v$
otherwise. Then
$\ker f^{\vee }$
is a finitely generated torsion
$\Lambda$
-module with
$\lambda (\ker f^{\vee }) \leq (d-1)m_p +(d-1)(d-2)/2$
.
Proof. Taking into account Proposition 2.2, this proposition can be proven in an identical way to [Reference Kleine and Matar35, Lemma 3.11].
Corollary 3.7.
Let
$K_{\infty } \in \mathcal{E}$
, then the following assertions hold.
-
(a)
$\textrm {rank}_{\Lambda }(X^{Gr}(E/K_{\infty })) = \textrm {rank}_{\Lambda }(X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })})$
. -
(b) Assume that
-
(1) no prime in
$S_p$
splits completely in
$K_{\infty }/K$
, -
(2)
$\textrm {rank}_{\Lambda }(X^{Gr}(E/K_{\infty })) = \textrm {rank}_{\Lambda }(X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })})=0$
.
Furthermore, assume either of the following conditions is true
-
(i) No prime in
$S \setminus S_p$
splits completely in
$K_{\infty }/K$
-
(ii)
$E$
satisfies
$({\star})$
from the introduction for all
$v \in S$
Then
$\mu (X^{Gr}(E/K_{\infty })) = \mu (X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })})$
. -
-
(c) If no prime in
$S_p$
splits completely in
$K_{\infty }/K$
, then there exists a neighbourhood
$\mathcal{E}(K_{\infty },n)$
such that
$|\lambda (X^{Gr}(E/K'_{\infty })) - \lambda (X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K'_{\infty })})|$
is bounded as
$K'_\infty$
runs over the elements of
$\mathcal{E}(K_{\infty },n)$
for which
\begin{equation*}\textrm {rank}_{\Lambda }(X^{Gr}(E/K'_{\infty })) = \textrm {rank}_{\Lambda }(X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K'_{\infty })}=0. \end{equation*}
Proof.
Statement (a) follows directly from Proposition 3.6. Now assume that both
$X^{Gr}(E/K_{\infty })$
and
$X^{Gr}(E/\mathbb{L}_{\infty })_{\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })}$
are
$\Lambda$
-torsion. Also assume that no prime in
$S_p$
splits completely in
$K_{\infty }/K$
. By Lemma 2.1, we may assume that
$S$
is precisely the set of primes of
$K$
above
$p$
and where
$E$
has bad reduction. If
$E$
satisfies
$({\star})$
for all
$v \in S$
, then by Proposition 3.2, for any prime of
$K_{\infty }$
lying over a prime of
$S \setminus S_p$
we have
$H^1(K_{\infty ,w}, E[p^{\infty }])=0$
. From this and the proof of Proposition 3.6, we have that under either condition (i) or (ii) both
$\mu (\ker f^{\vee })=0$
and
$\mu (\textrm {coker} f^{\vee })=0$
. This proves (b). We can prove part (c) in an identical way to [Reference Kleine and Matar35, Corollary 3.12(b)].
4. Congruent elliptic curves and
$\mathbb{Z}_p$
-extensions
In this section, we collect some results about relationships between Iwasawa invariants of Pontryagin duals of Selmer groups of congruent elliptic curves over
$\mathbb{Z}_p$
-extensions. Via the up-down approach described in the introduction, we will in the next sections use these results to prove our main theorems.
We need the following auxiliary result, which is due to Lim.
Lemma 4.1 (Lim [Reference Lim39], Lemma 2.4.1). Let
$M$
be a finitely generated
$\Lambda$
-module. Suppose that there is a
$\Lambda$
-module homomorphism
\begin{align} \varphi \colon M[p^\infty ] \longrightarrow \bigoplus _{j = 1}^s \Lambda /(p^{m_j}) \end{align}
with pseudo-null (i.e. finite) kernel and cokernel. Then we have
\begin{equation*} \mu (M/p^i) = i \cdot \text {rank}_\Lambda (M) + \sum _{j = 1}^s \min (i, m_j). \end{equation*}
for every
$i \in {\mathbb{N}}$
,
$i \ge 1$
.
Note that the module
$\bigoplus _{j = 1}^s \Lambda /(p^{m_j})$
is an elementary
$\Lambda$
-module attached to
$M[p^\infty ]$
, as in the exact sequence (2).
Definition 4.2.
Let
$M$
be a finitely generated
$\Lambda$
-module, and consider a pseudo-isomorphism
$\varphi$
as in equation (7). Then we define
$e(M)$
to be the maximum of the exponents
$m_j$
,
$1 \le j \le s$
.
The following theorem illustrates relationships between
$\mu$
-invariants of Pontryagin duals of Selmer groups of congruent elliptic curves. Recall Definitions 1.2 and 1.3 from the introduction.
Theorem 4.3.
Assume that either
$K_{\infty } \in \mathcal{E}_{ns}$
or
$E_1$
and
$E_2$
satisfy condition
$({\star})$
for all
$v \in S$
.
-
(a) Suppose that
$E_1[p^i] \cong E_2[p^i]$
as Galois modules, where
$i$
is larger than both
$e(X^{Gr}(E_1/K_\infty ))$
and
$e(X^{Gr}(E_2/K_\infty ))$
. Then
and
\begin{equation*} \text {rank}_\Lambda (X^{Gr}(E_1/K_\infty )) = \text {rank}_\Lambda (X^{Gr}(E_2/K_\infty ))\end{equation*}
In fact, the two
\begin{equation*} \mu (X^{Gr}(E_1/K_\infty )) = \mu (X^{Gr}(E_2/K_\infty )).\end{equation*}
$\Lambda$
-modules
$X^{Gr}(E_1/K_\infty )[p^\infty ]$
and
$X^{Gr}(E_2/K_\infty )[p^\infty ]$
are pseudo-isomorphic.
-
(b) Now suppose that
$\text {rank}_\Lambda (X^{Gr}(E_1/K_\infty )) \le \text {rank}_\Lambda (X^{Gr}(E_2/K_\infty ))$
and that
$E_1[p^i] \cong E_2[p^i]$
for some
$i \gt e(X^{Gr}(E_1/K_\infty ))$
. Then the
$\Lambda$
-ranks are equal, and the
$p$
-primary torsion submodules of
$X^{Gr}(E_1/K_\infty )$
and
$X^{Gr}(E_2/K_\infty )$
are pseudo-isomorphic as
$\Lambda$
-modules.
-
(c) If
$E_1[p] \cong E_2[p]$
, then
$\textrm {Sel}^{Gr}(E_1/K_{\infty })$
is
$\Lambda$
-cotorsion with
$\mu =0$
if and only if
$\textrm {Sel}^{Gr}(E_2/K_{\infty })$
is
$\Lambda$
-cotorsion with
$\mu =0$
.
Proof. The following proof uses ideas from the proof of [Reference Lim39, Theorem 4.2.1]. But note that Lim’s result was proven only for torsion Iwasawa modules.
For both
$j = 1$
and
$j = 2$
, we have an exact sequence
\begin{equation*}0 \longrightarrow \textrm {Sel}^{Gr}(E_j/K_{\infty }) \longrightarrow \textrm {Sel}^{Gr(p)}(E_j/K_{\infty }) \longrightarrow \prod _{v \in S \setminus S_p} \mathcal{H}_v(E_j/K_{\infty }). \end{equation*}
If
$K_{\infty } \in \mathcal{E}_{ns}$
, then by Proposition 3.1 we see that
$\bigoplus _{v \in S \setminus S_p}\mathcal{H}_v(E_j/K_{\infty })$
is a cotorsion
$\Lambda$
-module with
$\mu =0$
for both
$j = 1,2$
. By Lemma 2.1 in the exact sequence above, we may assume for
$j=1,2$
that
$S$
is precisely the set of primes of
$K$
which lie above
$p$
or where
$E_j$
has bad reduction. If both
$E_1$
and
$E_2$
satisfy condition
$({\star})$
for all
$v \in S$
, then by Proposition 3.2
$\bigoplus _{v \in S \setminus S_p}\mathcal{H}_v(E_j/K_{\infty })=0$
,
$j = 1,2$
. Taking Pontryagin duals in the above exact sequence, we may conclude that in either case we have
\begin{equation*} \text {rank}_\Lambda (X^{Gr}(E_j/K_\infty )) = \text {rank}_\Lambda (X^{Gr(p)}(E_j/K_\infty ))\end{equation*}
and
\begin{equation*} \mu (X^{Gr}(E_j/K_\infty )) = \mu (X^{Gr(p)}(E_j/K_\infty ))\end{equation*}
for both
$j = 1$
and
$j = 2$
. In fact, the
$p$
-torsion submodules are pseudo-isomorphic, that is,
\begin{equation*} e(X^{Gr}(E_j/K_\infty )) = e(X^{Gr(p)}(E_j/K_\infty ))\end{equation*}
for both
$j$
. Moreover, for every
$k \in {\mathbb{N}}$
we have an exact sequence
\begin{equation*} \prod _{v \in S \setminus S_p} \mathcal{H}_v(E_j/K_\infty )^{\vee }/p^k \longrightarrow X^{Gr(p)}/p^k \longrightarrow X^{Gr}/p^k \longrightarrow 0, \end{equation*}
where the first term is finite for each
$k$
.
Suppose first that
$E_1[p^i] \cong E_2[p^i]$
for some
$i \gt 1$
. Then Lemma 3.5 implies
\begin{align*}\textrm {Sel}^{Gr(p)}(E_1[p^i]/K_{\infty }) \cong \textrm {Sel}^{Gr(p)}(E_2[p^i]/K_{\infty }).\end{align*}
From Proposition 3.3, we have for
$j=1,2$
that the natural map
\begin{align*}\textrm {Sel}^{Gr(p)}(E_j[p^i]/K_\infty ) \longrightarrow \textrm {Sel}^{Gr(p)}(E_j/K_\infty )[p^i]\end{align*}
is a surjection with finite kernel. Combining the two previous facts and taking Pontryagin duals, we obtain that we have a pseudo-isomorphism of
$\Lambda$
-modules
\begin{equation*} X^{Gr(p)}(E_1/K_\infty )/p^i \sim X^{Gr(p)}(E_2/K_\infty )/p^i. \end{equation*}
In particular, the
$\mu$
-invariants of these two quotient modules are equal. Note that we also have
${E_1[p^{i-1}] \cong E_2[p^{i-1}]}$
, and therefore we also have an equality of
$\mu$
-invariants
\begin{equation*} \mu (X^{Gr(p)}(E_1/K_\infty )/p^{i-1}) = \mu (X^{Gr(p)}(E_2/K_\infty )/p^{i-1}). \end{equation*}
As we have seen in the first part of the proof, this implies that we have an analogous equation for
$X^{Gr}$
instead of
$X^{Gr(p)}$
. Now we apply Lemma 4.1, both with
$i$
and
$i-1$
, and obtain that
\begin{eqnarray*} i \cdot r_1 + \mu (X^{Gr}(E_1/K_\infty )) & = & i \cdot r_2 + \mu (X^{Gr}(E_2/K_\infty )), \\ (i-1) \cdot r_1 + \mu (X^{Gr}(E_1/K_\infty )) & = & (i-1) \cdot r_2 + \mu (X^{Gr}(E_2/K_\infty )), \end{eqnarray*}
where we let
$r_1 = \text {rank}_\Lambda (X^{Gr}(E_1/K_\infty ))$
and
$r_2 = \text {rank}_\Lambda (X^{Gr}(E_2/K_\infty ))$
for brevity. Here, we use the fact that
$i-1$
is greater than or equal to the maximum of both
$e(X^{Gr(p)}(E_j/K_\infty ))$
,
$j \in \{1,2\}$
. Subtracting the two equations, we obtain that the two
$\Lambda$
-ranks are equal, and then the first equation concludes the proof of the first part of (a). In order to obtain the last part, we just note that our assumptions imply that
${E_1[p^k] \cong E_2[p^k]}$
for each
$k \le i$
. Writing down the corresponding equations from Lemma 4.1 and comparing the coefficients
$\min (k, m_j)$
for the two
$\Lambda$
-modules, starting from
$k = {i-1}$
and going down to
$k =1$
, we can finish the proof of (a) via an induction.
Now suppose that we are the setting of (b). Write down the equation from Lemma 4.1 for each integer
$k \le i$
. Recall that
$i-1 \ge e(X^{Gr}(E_1/K_\infty ))$
. Comparing the equations for
$k = i$
and
$k = i-1$
, we may conclude that
\begin{equation*} \mu (X^{Gr}(E_1/K_\infty )) = \mu (X^{Gr}(E_1/K_\infty )/p^i) = \mu (X^{Gr}(E_1/K_\infty )/p^{i-1})\end{equation*}
equals both
\begin{equation*}i \cdot (r_2 - r_1) + \mu (X^{Gr}(E_2/K_\infty )/p^i)\end{equation*}
and
\begin{equation*} (i-1) \cdot (r_2 - r_1) + \mu (X^{Gr}(E_2/K_\infty )/p^{i-1}). \end{equation*}
Letting
$r \,:\!=\, r_2 - r_1$
for brevity, we obtain that
\begin{equation*} i r + \mu (X^{Gr}(E_2/K_\infty )/p^i) = (i-1) r + \mu (X^{Gr}(E_2/K_\infty )/p^{i-1}). \end{equation*}
Since it is obvious that the first (respectively, the second) summand on the left-hand side of this equation is greater than or equal to the first (respectively, the second) summand on the right-hand side (here we use the assumption that
$r \ge 0$
), we conclude that we must have
\begin{equation*} i r = (i-1) r, \quad \mu (X^{Gr}(E_2/K_\infty /p^i)) = \mu (X^{Gr}(E_2/K_\infty )/p^{i-1}). \end{equation*}
This proves that
$e(X^{Gr}(E_2/K_\infty )) \le i-1$
, and that
$r = 0$
, that is,
$\text {rank}_\Lambda (X^{Gr}(E_1/K_\infty )) = \text {rank}_\Lambda (X^{Gr}(E_2/K_\infty ))$
. Now we proceed iteratively, as in the proof of (a), decreasing the value of
$k$
step by step.
Finally, we prove the last assertion of the theorem. From the structure theorem of finitely generated
$\Lambda$
-modules, it is easy to see that
\begin{align*} &\textrm {Sel}^{Gr(p)}(E_i/K_{\infty }) \text{ is } \Lambda \text{-cotorsion with } \mu =0\\ \Longleftrightarrow\ &\textrm {Sel}^{Gr(p)}(E_i/K_{\infty })[p] \text{ is finite}. \end{align*}
From Proposition 3.3 and Lemma 3.5 it follows that, since
$E_1[p] \cong E_2[p]$
,
$\textrm {Sel}^{Gr(p)}(E_1/K_{\infty })[p]$
is finite if and only if
$\textrm {Sel}^{Gr(p)}(E_2/K_{\infty })[p]$
is finite. Part (c) of the theorem follows by combining these results.
Definition 4.4.
If
$K_{\infty } \in \mathcal{E}$
, we denote the maximal finite
$\Lambda$
-submodule of
$X^{Gr(p)}(E_j/K_{\infty })$
by
$\mathfrak{X}^{Gr(p)}(E/K_{\infty })$
.
Proposition 4.5.
Let
$E$
be either
$E_1$
or
$E_2$
. Assume the following:
-
(1)
$E(\mathbb{L}_{\infty })[p^{\infty }]$
is finite
-
(2) The decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
are open for each
$v \mid p$
-
(3)
$E$
satisfies
$({\star})$
from Definition 1.3 for all
$v \in S$
Then there exists a constant
$C^{Gr} \geq 0$
such that for all
$K_{\infty } \in \mathcal{E}$
with
$X^{Gr}(E/K_{\infty })$
$\Lambda$
-torsion we have
$r_p(\mathfrak{X}^{Gr(p)}(E/K_{\infty })) \le C^{Gr}$
.
Proof.
Let
$K_{\infty } \in \mathcal{E}$
with
$X^{Gr}(E/K_{\infty })$
$\Lambda$
-torsion. We have an exact sequence
\begin{equation*}\prod _{v \in S\setminus S_p} \mathcal{H}_v(E/K_{\infty })^{\vee } \longrightarrow X^{Gr(p)}(E/K_{\infty }) \longrightarrow X^{Gr}(E/K_{\infty }) \longrightarrow 0. \end{equation*}
By Lemma 2.1, we may assume that
$S$
is precisely the set of primes of
$K$
above
$p$
or where
$E$
has bad reduction. By Proposition 3.2, the first term in the above sequence is trivial. Therefore, the desired result follows from Proposition 2.10.
In order to be able to compare
$\lambda$
-invariants, we need
Lemma 4.6.
Assume that both
$E_1$
and
$E_2$
satisfy condition
$({\star})$
for all
$v \in S$
. Then for any
$K_\infty \in \mathcal{E}$
, we have
$\lambda (X^{Gr}(E_j/\tilde {K}_\infty ))= \lambda (X^{Gr(p)}(E_j/\tilde {K}_\infty ))$
.
Proof. We have an exact sequence
\begin{equation*}\prod _{v \in S\setminus S_p} \mathcal{H}_v(E/K_{\infty })^{\vee } \longrightarrow X^{Gr(p)}(E/K_{\infty }) \longrightarrow X^{Gr}(E/K_{\infty }) \longrightarrow 0. \end{equation*}
By Lemma 2.1, we may assume that
$S$
is precisely the set of primes of
$K$
above
$p$
or where
$E$
has bad reduction. By Proposition 3.2, the first term in the above sequence is trivial. The lemma follows.
Greenberg [Reference Greenberg14] introduced the following topology on
$\mathcal{E}$
. For
$L \in \mathcal{E}$
and
$n$
a positive integer, we define
$\mathcal{E}(L,n)\,:\!=\,\{L' \in \mathcal{E} \, | \, [L' \cap L \,:\,K] \geq p^n\}$
. This means that
$\mathcal{E}(L,n)$
consists of all
$\mathbb{Z}_p$
-extensions of
$K$
which coincide with
$L$
at least up to the
$n$
-th layer. Taking
$\mathcal{E}(L,n)$
as a base of neighborhoods of
$L$
gives us a topology on
$\mathcal{E}$
.
Lemma 4.7. Assume the following
-
(1)
$E_1(\mathbb{L}_{\infty })[p^{\infty }]$
and
$E_2(\mathbb{L}_{\infty })[p^{\infty }]$
are finite,
-
(2) the decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
are open for each
$v \mid p$
, -
(3)
$E_1$
and
$E_2$
satisfy
$({\star})$
for all
$v \in S$
.Suppose that
$K_{\infty }$
is a
$\mathbb{Z}_p$
-extension of
$K$
such that
-
(4)
$X^{Gr}(E_1/K_{\infty })$
is
$\Lambda$
-torsion.
Assume that
-
(5)
$E_1[p^i] \cong E_2[p^i]$
for some integer
$i$
such that
$i \gt \mu (X^{Gr}(E_1/K_{\infty }))$
. Then
$X^{Gr}(E_2/K_{\infty })$
is
$\Lambda$
-torsion and
where
\begin{equation*}|\lambda (X^{Gr}(E_1/K_\infty )) - \lambda (X^{Gr}(E_2/K_\infty ))| \le 2 i + C^{Gr}, \end{equation*}
$C^{Gr}$
denotes the constant from Proposition 4.5
.
Proof.
In view of our hypotheses, it follows from Theorem 4.3(b) that
$X^{Gr}(E_2/K_{\infty })$
is
$\Lambda$
-torsion with
$\mu (X^{Gr}(E_2/K_{\infty }))=\mu (X^{Gr}(E_1/K_{\infty }))$
.
Now fix
$K_\infty$
. For both
$j = 1$
and
$j = 2$
, we have a commutative diagram
In the above diagram, the left vertical map is inclusion and the middle vertical map is the identity map. This commutative diagram implies that we have another commutative diagram
By Proposition 3.3, the rows are exact with
$\#B_k \leq p^{2k}$
for both
$k = i$
and
$k = i-1$
. Taking Pontryagin duals and using the snake lemma, the above commutative diagram induces a map
$\theta _j \colon p^{i-1}X^{Gr(p)}(E_j/K_{\infty })/p^i \longrightarrow \ker \widehat {\iota _{j*}}$
where
$\widehat {\iota _{j*}}$
is the dual map to
$\iota _{j*}$
. The map
$\theta _j$
is injective with
$\#\textrm {coker} \theta _j \leq p^{2i}$
.
Now Lemma 3.5 implies that
$\ker \widehat {\iota _{1*}} \cong \ker \widehat {\iota _{2*}}$
. Since
$p^{i-1}$
annihilates the ‘
$\mu$
-parts’ of both
$X^{Gr(p)}(E_j/K_\infty )$
,
$j = 1,2$
, it is easy to see that
$p^{i-1}X^{Gr(p)}(E_j/K_{\infty })/p^i$
are finite for
$j=1,2$
. From these two facts and the above, we have
\begin{align*} \#p^{i-1}X^{Gr(p)}(E_1/K_{\infty })/p^i \cdot \#\textrm {coker} \theta _1 &= \#\ker \widehat {\iota _{1*}}\\ &=\#\ker \widehat {\iota _{2*}}\\ &=\#p^{i-1}X^{Gr(p)}(E_2/K_{\infty })/p^i \cdot \#\textrm {coker} \theta _2. \end{align*}
Therefore, we may conclude that
\begin{equation*}|v_p(|p^{i-1}X^{Gr(p)}(E_1/K_{\infty })/p^i|) - v_p(|p^{i-1}X^{Gr(p)}(E_2/K_{\infty })/p^i|)|\end{equation*}
equals
\begin{equation*}|v_p(|\textrm {coker} \theta _1|) - v_p(|\textrm {coker} \theta _2|)|, \end{equation*}
and therefore it is bounded by
$2i$
.
Recall that
$\mathfrak{X}^{Gr(p)}(E_j/K_\infty ) \subseteq X^{Gr(p)}(E_j/K_\infty )$
denotes the maximal finite
$\Lambda$
-submodule,
$j = 1,2$
. Let
$C^{Gr}$
be the integer in Proposition 4.5. By Proposition 4.5
$r_p(\mathfrak{X}^{Gr(p)}(E_j/K_\infty )) \le C^{Gr}$
for each
$j = 1,2$
.
Using that
$p^{i-1}$
annihilates the ‘
$\mu$
-parts’ of both
$X^{Gr(p)}(E_j/K_\infty )$
,
$j = 1,2$
, we will now show by using [Reference Kleine and Müller33, Lemma 3.8] that the cardinality of
$p^{i-1} X^{Gr(p)}(E_j/K_\infty )/p^{i}$
differs from
$p^{\lambda (X^{Gr(p)}(E_j/K_\infty ))}$
by at most
$p^{r_p(\mathfrak{X}^{Gr(p)}(E_j/K_\infty ))}$
,
$j = 1,2$
. Then it will follow from the above that the difference between the
$\lambda$
-invariants of these two
$\Lambda$
-modules is bounded by
$2i + C^{Gr}$
, and the assertion of the lemma follows from Lemma 4.6.
In order to prove the above claim, we fix
$j$
and abbreviate
$X^{Gr(p)}(E_j/K_\infty )$
to
$X$
. Let
$Y = p^{i-1} X$
. Then
$Y$
is a finitely generated and torsion
$\Lambda$
-module, and it follows from the choice of
$i$
that
$\mu (Y) = 0$
. We let
$\mathfrak{X} \subseteq X$
and
$\mathcal{Y} \subseteq Y$
be the maximal finite
$\Lambda$
-submodules, and we fix an elementary
$\Lambda$
-module
$E_Y$
attached to
$Y$
.
Then we have an exact sequence
\begin{equation*} 0 \longrightarrow \mathcal{Y} \longrightarrow Y \longrightarrow \tilde {E}_Y \longrightarrow 0, \end{equation*}
where
$\tilde {E}_Y \subseteq E_Y$
is a submodule of finite index. For any abelian group
$A$
, we define
$\text {rank}_p(A) = v_p(|A/pA|)$
, provided this number is finite. Since
$\mu (Y) = 0$
, multiplication by
$p$
is injective on
$\tilde {E}_Y \subseteq E_Y$
and we have an equality
\begin{equation*} \text {rank}_p(Y) = \text {rank}_p(\mathcal{Y}) + \text {rank}_p(\tilde {E}_Y) \end{equation*}
of
$p$
-ranks. Since
$p$
does not divide the characteristic power series
$f_Y$
of
$Y$
, it follows from [Reference Kleine31, Proposition 3.4(i) and (iv)] that
$\text {rank}_p(\tilde {E}_Y) = \text {rank}_p(E_Y)$
, and the latter is just
$\lambda (E_Y) = \lambda (Y) = \lambda (X)$
because
$f_Y$
is a distinguished polynomial. On the other hand, since
$\mathcal{Y} \subseteq \mathfrak{X}$
, we have
$\text {rank}_p(\mathcal{Y}) \le \text {rank}_p(\mathfrak{X})$
. This proves the above claim.
5. Proof of the first main theorem
In this section, we prove Theorem1.4. First, we need the following
Lemma 5.1.
Assume that
$\mathcal{E}_{ns} \neq \emptyset$
. For
$j=1,2$
$X^{Gr}(E_j/\mathbb{L}_{\infty })$
is a torsion
$\Lambda _d$
-module with
$m_0(X^{Gr}(E_j/\mathbb{L}_{\infty }))=\alpha$
if and only if there exist a finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
contained in
$\mathbb{L}_\infty$
such that for all
$\tilde {K}_{\infty } \in \mathcal{E}$
that are not contained in these extensions we have that
$X^{Gr}(E_j/\tilde {K}_{\infty })$
is
$\Lambda$
-torsion with
$\mu (X^{Gr}(E_j/\tilde {K}_{\infty }))=\alpha$
.
Proof.
Let
$j \in \{1,2\}$
. For brevity, we let
$X = X^{Gr}(E_j/\mathbb{L}_\infty )$
. We have the following facts:
-
(a) If
$X$
is
$\Lambda _d$
-torsion, then for any
$\tilde {K}_{\infty } \in \mathcal{E}$
and
$\tilde {H} = \textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
it follows from [Reference Monsky46, Theorem 3.2] that
$X_{\tilde {H}}$
is a torsion
$\Lambda$
-module with
$\mu$
-invariant equal to
$m_0(X)$
as long as
$\tilde {K}_\infty$
is not contained in a fixed finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
contained in
$\mathbb{L}_\infty$
. -
(b) Let
$\tilde {K}_{\infty } \in \mathcal{E}$
and
$\tilde {H} = \textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
. From [Reference Lim40, Lemma4.7], it follows that if
$X_{\tilde {H}}$
is
$\Lambda$
-torsion, then
$X$
is
$\Lambda _d$
-torsion. -
(c) Let
$\tilde {K}_{\infty } \in \mathcal{E}$
and
$\tilde {H} = \textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
. Corollary 3.7 implies that-
(i)
$\textrm {rank}_{\Lambda }(X_{\tilde {H}})=\textrm {rank}_{\Lambda }(X^{Gr}(E/\tilde {K}_{\infty }))$
, -
(ii)
$\mu (X_{\tilde {H}}) = \mu (X^{Gr}(E/\tilde {K}_\infty ))$
if there does not exist a prime in
$S$
which splits completely in
$\tilde {K}_{\infty }/K$
.
-
The condition in (c)-(ii) above means that it will suffice if
$\tilde {K}_\infty$
is not contained in the decomposition field of any of the primes
$v \in S$
. Since
$\mathcal{E}_{ns} \neq \emptyset$
, the decomposition subgroup
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
has
$\mathbb{Z}_p$
-rank at least one for each
$v \in S$
. Therefore, condition (c)-(ii) excludes the
${\mathbb{Z}}_p$
-extensions which are contained in a finite set of
${\mathbb{Z}}_p^{d-1}$
-extensions inside
$\mathbb{L}_\infty$
. The statement of the lemma easily follows from this and the facts (a), (b), and (c) above.
We can now prove the first main result, namely Theorem1.4 from the introduction.
Proof of Theorem 1.4. By Propositions 2.2 and 2.3, it suffices to prove the theorem with
$X(E_j/\mathbb{L}_{\infty })$
replaced with
$X^{Gr}(E_j/\mathbb{L}_{\infty })$
for
$j=1,2$
. Assume that
$X^{Gr}(E_1/\mathbb{L}_{\infty })$
is a torsion
$\Lambda _d$
-module with
${E_1[p^i] \cong E_2[p^i]}$
for some integer
$i \gt m_{0,1}$
. By Lemma 5.1 for all
$\tilde {K}_{\infty } \in \mathcal{E}$
that are not contained in a fixed finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
contained in
$\mathbb{L}_\infty$
, we have that
$X^{Gr}(E_1/\tilde {K}_{\infty })$
is
$\Lambda$
-torsion with
$\mu (X^{Gr}(E_1/\tilde {K}_{\infty }))=m_{0,1}$
. Note that
$\mu (X^{Gr}(E_1/K_\infty )) \ge e(X^{Gr}(E_1/K_\infty ))$
for any
$K_\infty \in \mathcal{E}$
. Therefore, we see from Theorem4.3(b) that for all
$\tilde {K}_{\infty } \in \mathcal{E}$
that are not contained in a fixed finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
contained in
$\mathbb{L}_\infty$
we have that
$X^{Gr}(E_2/\tilde {K}_{\infty })$
is
$\Lambda$
-torsion with
$\mu (X^{Gr}(E_2/\tilde {K}_{\infty }))=m_{0,1}$
. Then by Lemma 5.1,
$X^{Gr}(E_2/\mathbb{L}_{\infty })$
is a torsion
$\Lambda _d$
-module with
$m_{0,2}=m_{0,1}$
.
6. Via Lim’s approach
In [Reference Lim39], Lim gave a proof of Theorem1.4 in more generality, using a completely different approach. We will now describe this approach briefly. Then we strengthen Lim’s approach in order to compare also
$l_0$
-invariants of Selmer groups. It will turn out that in order to compare
$l_0$
-invariants via Lim’s approach, a strong additional hypothesis is needed (see Theorem6.15 below). Our up-down approach seems more suitable for handling
$l_0$
-invariants, since it allows us to prove Theorem1.5 which holds under milder assumptions. We will follow that approach in Section 7.
In this section, we will work with
$X^{str}(E/\mathbb{L}_{\infty })$
. By Proposition 2.2, our results apply to
$X^{Gr}(E/\mathbb{L}_{\infty })$
and under suitable assumptions (see Proposition 2.3) our results also apply to
$X(E/\mathbb{L}_{\infty })$
.
Let
$X = X^{str}(E_1/\mathbb{L}_\infty )$
and
$X' = X^{str}(E_2/\mathbb{L}_\infty )$
, and write
$G = \textrm {Gal}(\mathbb{L}_\infty /K) \cong {\mathbb{Z}}_p^d$
. Then
$\mathbb{L}_\infty = \bigcup _n \mathbb{L}_n$
, where each number field
$\mathbb{L}_n = \mathbb{L}_\infty ^{G_n}$
,
$G_n = G^{p^n}$
, satisfies
$\textrm {Gal}(\mathbb{L}_n/K) \cong ({\mathbb{Z}}/p^n{\mathbb{Z}})^d$
. On any finite level
$n$
, we write
$X_n = X^{str}(E_1/\mathbb{L}_n)$
and
$X_n' = X^{str}(E_2/\mathbb{L}_n)$
for brevity.
The strategy is as follows. We encode the information about the
$m_0$
-invariants into asymptotic growth formulas for the orders of the quotients
$X_{G_n}$
and
$X'_{G_n}$
, moding out the submodules of elements which are multiples of
$p^k$
for a sufficiently large
$k$
(it will be important that
$k$
is larger than
$m_0(X)$
). In fact, the
$m_0$
-invariants and the
$\Lambda _d$
-ranks give rise to the leading terms in these asymptotic formulas. Via a control theorem, the quotients
$X_{G_n}/p^k$
and
$X'_{G_n}/p^k$
are related to analogous quotients of the Pontryagin duals of the strict Selmer groups
$X^{str}(E_1/\mathbb{L}_n)$
and
$X^{str}(E_2/\mathbb{L}_n)$
. Via a result similar to Proposition 3.3 (see Lemma 6.4), we can relate these quotients to the Pontryagin duals of residual Selmer groups
$\textrm {Sel}^{str}(E_j[p^k]/\mathbb{L}_n)$
,
$j = 1,2$
, which are defined as the classical strict Selmer groups in Section 2, but with
$E_j[p^\infty ]$
replaced by
$E_j[p^k]$
. If
$E_1[p^k] \cong E_2[p^k]$
as Galois modules, then the cardinalities of the two latter groups are equal (this will follow from a result analogous to Lemma 3.5). All in all we obtain that
$|X_{G_n}/p^k|$
and
$|X'_{G_n}/p^k|$
are equal up to a certain error term which does not affect the leading terms in the above growth formulas, therefore matching up the rank and
$m_0$
-invariant of
$X$
and
$X'$
.
We will now make this more precise. For the first step, we first choose
$k = n$
in the above explanation; this will be eventually larger than the
$m_0$
-invariant of both
$X$
and
$X'$
. We start with three auxiliary lemmas.
Lemma 6.1.
Let
$M$
be a finitely generated
$\Lambda _d$
-module, and let
$r = \text {rank}_{\Lambda _d}(M)$
. Then
\begin{equation*} | v_p(|M_{G_n}/p^n|) - \left ( r \cdot n p^{dn} + m_0(M) \cdot p^{dn}\right )| = O(n p^{(d-1)n}). \end{equation*}
Proof. See [Reference Perbet50, Théorème 2.1 (ii)].
Let
$E_M = \bigoplus _{i=1}^s \Lambda _d/(p^{m_i}) \oplus \bigoplus _{j = 1}^t \Lambda _d/(h_j^{n_j})$
be an elementary
$\Lambda _d$
-module attached to
$M$
. Then
$m_0(M) = \sum _i m_i$
by the definitions. Now let
$k \in {\mathbb{N}}$
be arbitrary, but fixed. Then we define a truncated
$m_0$
-invariant by
\begin{equation*} m_0^{(k)}(M) \,:\!=\, \sum _{i=1}^s \min (k, m_i). \end{equation*}
Note: If
$M$
is
$\Lambda _d$
-torsion, then
$m_0^{(k)}(M) = m_0(M/p^k)$
for each
$k \in {\mathbb{N}}$
.
Corollary 6.2.
Let
$M$
be as in Lemma 6.1
, and let
$k \in {\mathbb{N}}$
be arbitrary but fixed. Suppose that
$M$
is
$\Lambda _d$
-torsion. Then
\begin{equation*} |v_p(|M_{G_n}/p^k|) - m_0^{(k)}(M)p^{dn}| = O(n p^{(d-1)n}). \end{equation*}
Proof.
We apply Lemma 6.1 to the module
$M/p^k$
, noting that
\begin{equation*} M_{G_n}/p^k = (M/p^k)_{G_n}/p^n \end{equation*}
for each
$n \ge k$
.
The next result which we need is a control theorem that was proved by Lim. Recall that
$X = X^{str}(E_1/\mathbb{L}_\infty )$
. For each
$n$
, we let
$X_n = X^{str}(E_1/\mathbb{L}_n)$
for brevity.
Lemma 6.3.
Suppose that no prime in
$S$
is completely split in
$\mathbb{L}_\infty$
, and fix some integer
$k$
. Then
\begin{equation*} |v_p(|X_{G_n}/p^k|) - v_p(|X_n/p^k|)| = O(p^{n(d-1)}) \end{equation*}
and
\begin{equation*} |v_p(|X_{G_n}/p^n|) - v_p(|X_n/p^n|)| = O(n p^{n(d-1)}). \end{equation*}
A similar result holds for
$X' = X(E_2/\mathbb{L}_\infty )$
.
Proof. It has been shown in the proof of [Reference Lim39, Proposition 4.1.3] on page 1174 of loc. cit. that the kernels and cokernels of the maps
\begin{equation*} t_{n,k} \colon X_{G_n}/p^k \longrightarrow X_n/p^k\end{equation*}
are finite abelian
$p$
-groups and that the
$p$
-ranks of these groups are in
$O(p^{n(d-1)})$
where the implicit constant does not depend on
$n$
or
$k$
. Here, we mean by the
$p$
-rank of a finite abelian
$p$
-group
$A$
the dimension of
$A/p$
over the field
${\mathbb{F}}_p$
with
$p$
elements, as in Definition 2.5.
This immediately gives the first claim (for fixed
$k$
). In order to obtain the second claim, we let
$k = n$
(recall that the
$p$
-ranks of the kernels and cokernels are bounded independently in
$k$
).
Let
$X^{str}(E_j[p^k]/\mathbb{L}_n)$
be the Pontryagin dual of
$\textrm {Sel}^{str}(E_j[p^k]/\mathbb{L}_n)$
. We also need the following lemma.
Lemma 6.4.
Suppose that no prime in
$S$
is completely split in the
${\mathbb{Z}}_p^d$
-extension
$\mathbb{L}_\infty /K$
, and fix some integer
$k$
. Then the number of primes above some
$v \in S$
in
$\mathbb{L}_n$
is
$O(p^{n(d-1)})$
,
\begin{equation*} | v_p(|X_n/p^k|) - v_p(|X^{str}(E_1[p^k]/\mathbb{L}_n)|) | = O(p^{(d-1)n}) \end{equation*}
and
\begin{equation*} | v_p(|X_n/p^n|) - v_p(|X^{str}(E_1[p^n]/\mathbb{L}_n)|) | = O(n p^{(d-1)n}). \end{equation*}
A similar result holds for
$X' = X^{str}(E_2/\mathbb{L}_\infty )$
.
Proof. Again, we refer the reader to the proof of [Reference Lim39, Proposition 4.1.3] on page 1174 of loc. cit.
In order to obtain a better error term, we will use the following variant of that lemma.
Lemma 6.5.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension, and suppose that the following hypotheses hold.
-
(i)
$E_1$
satisfies condition
$({\star})$
from the introduction for all
$v \in S$
. -
(ii) The decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
have
${\mathbb{Z}}_p$
-ranks at least two for each
$v \in S_p$
.
Then the natural map
\begin{equation*} s_n \colon X_n/p^n \longrightarrow X^{str}(E_1[p^n]/\mathbb{L}_n)\end{equation*}
has finite kernel and cokernel, and the
$p$
-valuation of the order of the kernel and cokernel of
$s_n$
is in
$O(n p^{(d-2)n})$
.
A similar result holds for
$X' = X^{str}(E_2/\mathbb{L}_\infty )$
.
Proof.
By Lemma 2.1, we may assume that
$S$
is precisely the set of primes of
$K$
which lie above
$p$
or where
$E$
has bad reduction. Let
$S_n$
be the set of primes of
$\mathbb{L}_n$
above those in
$S$
and let
$S_{p,n}$
be the primes of
$\mathbb{L}_n$
above
$p$
. Consider the maps
\begin{equation*}\theta _n\,:\, \bigoplus _{w \in S_{p,n}} H^1(\mathbb{L}_{n,w}, \tilde {E}[p^n]) \longrightarrow \bigoplus _{w \in S_{p,n}} H^1(\mathbb{L}_{n,w}, \tilde {E}[p^{\infty }])[p^n], \end{equation*}
\begin{equation*}\psi _n\,:\, \bigoplus _{w \in S_n \setminus S_{p,n}} H^1(\mathbb{L}_{n,w}, E[p^n]) \longrightarrow \bigoplus _{w \in S_n \setminus S_{p,n}} H^1(\mathbb{L}_{n,w}, E[p^{\infty }])[p^n]. \end{equation*}
As in the proof of [Reference Lim39, Proposition 4.1.3], we see that to prove the lemma we must show that
$v_p(|\ker \theta _n|)=O(np^{(d-2)n})$
and
$v_p(|\ker \psi _n|)=O(np^{(d-2)n})$
. Condition (ii) implies that the number of primes of
$\mathbb{L}_n$
above any prime in
$S_p$
is
$O(p^{(d-2)n})$
. Then as is the proof of loc. cit., this immediately proves that
$v_p(|\ker \theta _n|)=O(np^{(d-2)n})$
.
We now turn to
$\ker \psi _n$
. We now show that condition (i) of the lemma implies that
$H^1(\mathbb{L}_{n,w}, E[p^n])=0$
for all
$n \geq 0$
and
$w \in S_n \setminus S_{p,n}$
. This implies that
$\ker \psi _n=0$
. Let
$n \geq 0$
and
$w \in S_n \setminus S_{p,n}$
. Then we have an exact sequence
\begin{equation*}0 \longrightarrow E(\mathbb{L}_{n,w})/p^n \longrightarrow H^1(\mathbb{L}_{n,w}, E[p^n]) \longrightarrow H^1(\mathbb{L}_{n,w}, E)[p^n] \longrightarrow 0.\end{equation*}
By Mattuck’s Theorem, we have that
$E(\mathbb{L}_{n,w}) \cong \mathbb{Z}_l^d \times T$
, where
$l$
is the residue characteristic of
$\mathbb{L}_{n,w}$
,
$d=[\mathbb{L}_{n,w}\,:\,{\mathbb{Q}_l}]$
and
$T$
is a finite group. Since
$l \ne p$
, we have
$E(\mathbb{L}_{n,w})/p^n \cong E(\mathbb{L}_{n,w})[p^n]$
. By Tate duality for abelian varieties over local fields (see [Reference Milne44, Corollary I-3.4]), we have an isomorphism
$H^1(\mathbb{L}_{n,w}, E)^{\vee } \cong E(\mathbb{L}_{n,w})$
. Therefore,
$(H^1(\mathbb{L}_{n,w}, E)[p^n])^{\vee } \cong E(\mathbb{L}_{n,w})/p^n$
. So as above we have
$H^1(\mathbb{L}_{n,w}, E)[p^n] \cong (H^1(\mathbb{L}_{n,w}, E)[p^n])^{\vee } \cong E(\mathbb{L}_{n,w})[p^n]$
. Combining the two previous facts with the exact sequence above, we see that
$|H^1(\mathbb{L}_{n,w}, E[p^n])|=|E(\mathbb{L}_{n,w})[p^n]|^2$
.
The proof of Proposition 3.2 shows that condition (i) implies that
$E(K_v)[p^{\infty }]=0$
where
$v$
is the prime of
$K$
below
$w$
. So
$(E(\mathbb{L}_{n,w})[p^{\infty }])^{\textrm {Gal}(\mathbb{L}_{n,w}/K_v)}=E(K_v)[p^{\infty }]=0$
. Therefore,
$E(\mathbb{L}_{n,w})[p^{\infty }]=0$
whence by the result above
$H^1(\mathbb{L}_{n,w}, E[p^n])=0$
. This completes the proof.
Now suppose that
$E_1[p^n] \cong E_2[p^n]$
for all
$n \in {\mathbb{N}}$
(we will relax drastically this assumption below). Then it follows from Lemmas 6.1, 6.3, and 6.4 that
\begin{equation*} |\left (r n p^{dn} + m_0(X) p^{dn} \right )- \left ( r' n p^{dn} + m_0(X') p^{dn} \right )| = O(n p^{n(d-1)}). \end{equation*}
Here, we let
$r = \text {rank}_{\Lambda _d}(X)$
and
$r' = \text {rank}_{\Lambda _d}(X')$
. This immediately implies that
$r = r'$
, and that
$m_0(X) = m_0(X')$
.
Now we improve on the above approach. First, we want to get rid of the assumption that
${E_1[p^n] \cong E_2[p^n]}$
for all
$n$
. Second, we would like to say something about
$l_0$
-invariants.
To the first aim, we suppose from now on that
$X$
and
$X'$
are both torsion
$\Lambda _d$
-modules. Instead of choosing
$k = n$
in the above quotients, we let
$k$
vary from 1 to
$m_0(X) + 1$
(it suffices to replace
$m_0(X)$
by the exponent
$e(X)$
of
$X$
, see below). Using Corollary 6.2 and the first formulas in Lemmas 6.3 and 6.4, we obtain that for each such
$k$
we have
\begin{equation*} \left |\left (m_0^{(k)}(X) p^{dn} \right )- \left (m_0^{(k)}(X') p^{dn} \right )\right | = O(p^{n(d-1)}). \end{equation*}
Now suppose that
$E_1[p^i] \cong E_2[p^i]$
for some
$i \gt e(X)$
(here
$e(X)$
means the maximum of the exponents
$m_i$
in the elementary
$\Lambda _d$
-module attached to
$M$
, as in Definition 4.2). Then
$m_0^{(k)}(X) = m_0^{(k)}(X')$
for each
$k \le e(X) + 1$
. In particular,
\begin{equation*} m_0^{(e(X) + 1)}(X') = m_0^{(e(X) + 1)}(X) = m_0^{(e(X))}(X) = m_0^{(e(X))}(X'). \end{equation*}
This shows that
$m_0(X') = m_0(X)$
and that in fact the
$p$
-torsion submodules of the elementary
$\Lambda _d$
-modules attached to
$X$
and
$X'$
are isomorphic, that is,
$X[p^\infty ]$
is pseudo-isomorphic to
$X'[p^\infty ]$
. We summarize this result as
Theorem 6.6 (Lim). Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension such that no prime in
$S$
is totally split in
$\mathbb{L}_\infty$
. Suppose that
$X = X^{str}(E_1/\mathbb{L}_\infty )$
and
$X' = X^{str}(E_2/\mathbb{L}_\infty )$
are both
$\Lambda _d$
-torsion modules, and that
$E_1[p^i] \cong E_2[p^i]$
for some
$i \gt e(X)$
.
Then there exists a pseudo-isomorphism
\begin{equation*} X[p^\infty ] \sim X'[p^\infty ], \end{equation*}
and in particular we have
$m_0(X) = m_0(X')$
.
In order to reach the second aim formulated above, we first have to formulate more precise asymptotic growth formulas. We will use a formula which has been obtained by the first-named author and Katharina Müller in [Reference Kleine and Müller34], generalizing work of Cuoco and Monsky from [Reference Cuoco and Monsky11]. In order to formulate the result, we have to introduce some notation from [Reference Cuoco and Monsky11].
In the asymptotic formula which we will use, the coinvariant modules
$X_{G_n}$
are replaced by certain smaller quotients, which are a bit involved to introduce. For the convenience of the reader, we provide a definition from scratch. Recall that
$\Lambda _d$
has been identified with
${\mathbb{Z}}_p[[T_1, \ldots , T_d]]$
by mapping a set of topological generators
$\gamma _1, \ldots , \gamma _d$
of
$G \cong {\mathbb{Z}}_p^d$
to the elements
$T_1 + 1, \ldots , T_d + 1$
. For any element
${\sigma \in G \setminus G^p}$
, we can introduce a variable
$S = \sigma - 1$
in
$\Lambda _d$
. We consider the polynomials
\begin{equation*} \omega _0(S) \,:\!=\, S, \quad \omega _n(S) \,:\!=\, (S+1)^{p^n}-1, \, n \ge 1, \quad \nu _n(S) \,:\!=\, \frac {\omega _{n+1}(S)}{\omega _n(S)}, \end{equation*}
where
$n$
is any natural number. Moreover, we also let, for two natural numbers
$n \ge m$
,
\begin{equation*} \nu _{n,m}(S) \,:\!=\, \frac {\omega _n(S)}{\omega _m(S)} = \nu _{n-1}(S) \cdot \nu _{n-2}(S) \cdot \ldots \cdot \nu _m(S). \end{equation*}
Following [Reference Cuoco and Monsky11, Definitions 4.1 and 4.6], we define
Definition 6.7.
Let
$M$
be a finitely generated torsion
$\Lambda _d$
-module. A structure
$\mathcal{S}$
on
$M$
consists of an integer
$v \ge 0$
and a finite set of tuples
$(\sigma _i, M_i)$
, where
$\sigma _i \in G \setminus G^p$
and
$M_i \subseteq M$
is a submodule for each
$i$
.
For each
$n \in {\mathbb{N}}$
let
$I_n \subseteq \Lambda _d$
be the ideal generated by
$\omega _n(T_1), \ldots , \omega _n(T_d)$
.
Given a structure
$\mathcal{S}$
on
$M$
, we define
$J_n = J_n(\mathcal{S})$
to be the submodule of
$M$
generated by
$I_n \cdot M$
and
$\sum _i \nu _{n,v}(S_i) \cdot M_i$
, where the sum runs over all pairs
$(\sigma _i, M_i)$
in
$\mathcal{S}$
and
$S_i = \sigma _i - 1$
for each
$i$
.
A structure
$\mathcal{S}$
is called admissible on
$M$
if
\begin{equation*} \text {rank}_{{\mathbb{Z}}_p}(M/J_n) = O(p^{(d-2)n}). \end{equation*}
Remark 6.8.
The quotients
$M_{G_n}$
of coinvariants considered in the asymptotic formula in Lemma 6.1 correspond to the quotients
$M/I_n$
, since
$I_n$
is just the augmentation ideal of the subgroup
$G_n = G^{p^n} \subseteq G$
in
$\Lambda _d$
. Therefore, these quotients correspond to the ‘empty structure’ on
$M$
, that is,
$J_n = I_n \cdot M$
and there are no additional submodules
$M_i$
of
$M$
. Unfortunately, this structure is not always admissible in the above sense.
Suppose now that we are given a finitely generated and torsion
$\Lambda _d$
-module
$M$
. Let us briefly describe how one can come up with an admissible structure on
$M$
.
Let
$E_M$
be an elementary torsion
$\Lambda _d$
-module attached to
$M$
. As in the introduction, we write
\begin{equation*} E_M = \bigoplus _{i = 1}^s \Lambda _d/(p^{m_i}) \oplus \bigoplus _{j = 1}^t \Lambda _d/(h_j^{n_j}), \end{equation*}
where the power series
$h_1, \ldots , h_t$
are irreducible and coprime with
$p$
. Recall that the characteristic power series
$f_M$
of
$M$
is just the same as
$p^{m_0(M)} \cdot \prod _{j = 1}^t h_j^{n_j}$
. It follows from [Reference Cuoco and Monsky11, Theorem 3.13] that the empty structure (see also Remark 6.8) is admissible on
$M$
as long as no irreducible element
$h_j$
is a special prime element. By this, Cuoco and Monsky mean the following:
Definition 6.9.
A prime element
$h \in \Lambda _d = {\mathbb{Z}}_p[[G]]$
is called special if there exists some
$\sigma \in G \setminus G^p$
such that for
$S = \sigma - 1$
we have either
-
(1)
$h = S$
or
-
(2)
$h = \nu _r(S)$
for some
$r \in {\mathbb{N}}$
.
It remains to define an admissible structure on
$M$
if some of the irreducible elements
$h_j$
dividing the characteristic power series
$f_M$
of
$M$
are special. Suppose first that
$M = E_M = \Lambda _d/(h^n)$
for some special prime
$h$
. We let
$\sigma$
be the corresponding group element, and we define
$\mathcal{S}$
to be the one element structure
$(v, \{ (\sigma , M)\})$
, where
$v = 0$
in case (1) and
$v = r$
in case (2). In other words, we define
\begin{equation*} J_n = I_n \cdot M + \nu _{n,v}(S) \cdot M, \end{equation*}
where the choice of
$v$
depends on the type of the special prime
$h$
, as described above. It is easy to see that
$\text {rank}_{{\mathbb{Z}}_p}(M/J_n) = O(p^{(d-2)n})$
with that choice of
$\mathcal{S}$
, that is, this structure is admissible on
$M = \Lambda _d/(h^n)$
.
Now suppose that
$M = E_M = \bigoplus _{i = 1}^s \Lambda _d/(p^{m_i}) \oplus \bigoplus _{j = 1}^t \Lambda _d/(h_j^{n_j})$
. In this case, we let (without loss of generality)
$M_1, \ldots , M_c$
be the cyclic submodules
$\Lambda _d/(h_i^{n_i})$
such that
$h_1, \ldots , h_c$
are special. For each such
$i$
, let
$S_i = \sigma _i - 1$
be the corresponding variable and choose
$r_i = 0$
or
$r_i = r$
depending on whether we have a first case special prime or a second case special prime, and let
$\hat {r} = \max (r_1, \ldots , r_c)$
. Note that
$\hat {r}$
is the smallest number such that
$\nu _{\hat {r}}(S_i)$
is coprime with each
$h_i \in {\mathbb{Z}}_p[S_i]$
. This is what makes the following structure admissible: We take
\begin{align} \mathcal{S} = (\hat {r}, \{ (\sigma _1, M_1), (\sigma _2, M_2), \ldots , (\sigma _c, M_c)\}), \end{align}
where
$M_i = \Lambda _d/(h_i^{n_i})$
, respectively.
If
$M$
is not an elementary
$\Lambda _d$
-module, then we proceed via a ‘dirty trick’, by applying the following result of Cuoco and Monsky (see [Reference Cuoco and Monsky11, Theorem 4.12]).
Proposition 6.10.
Let
$M$
and
$M'$
be two pseudo-isomorphic torsion
$\Lambda _d$
-modules, and let
$\mathcal{S}$
and
$\mathcal{S}'$
be any admissible structures on
$M$
and
$M'$
, with attached quotients
$M/J_n$
and
$M'/J_n'$
. Then
\begin{equation*} |v_p(|(M/J_n)[p^\infty ]|) - v_p(|(M'/J_n')[p^\infty ]|)| = O(p^{(d-1)n}). \end{equation*}
Since we will be interested in the growth of the orders of our quotients only up to an error of
$O(p^{(d-1)n})$
(see also Lemma 6.12 below), we can now proceed as follows: If
$M$
is not elementary, then we just replace
$M$
by
$E_M$
and choose the admissible structure on
$E_M$
which has been defined above. It follows from the above proposition that it suffices to study the growth of
$v_p(|E_M/J_n|)$
for this particular structure. In other words, we do not have to think too much about a particular choice of admissible structure on
$M$
if
$M$
is not elementary.
The following result of Cuoco and Monsky will allow us to simplify the definition of the structure as given in (8).
Proposition 6.11.
Let
$\mathcal{S}$
be an admissible structure on
$M$
, and let
$J_n(\mathcal{S}) \subseteq M$
be the corresponding submodules used in the quotients. Let
$\mathcal{S}'$
be the new structure which is obtained from
$\mathcal{S}$
by replacing
$M_1$
by
$M$
, and let
$J_n' \,:\!=\, J_n(\mathcal{S}')$
be the corresponding submodules. Then
$\mathcal{S}'$
is also admissible, and
\begin{equation*} |v_p(|(M/J_n)[p^\infty ]|) - v_p(|(M/J_n')[p^\infty ]|)| = O(p^{(d-1)n}). \end{equation*}
Proof.
It is obvious that
$\mathcal{S}'$
is again admissible, since we are making the quotients
$M/J_n$
even smaller. Now apply [Reference Cuoco and Monsky11, Lemma 4.9].
Applying this proposition iteratively, we can replace the structure
$\mathcal{S}$
from (8) by the easier structure
\begin{equation*} {S}' = (\hat {r}, \{(\sigma _1, M), (\sigma _2, M), \ldots , (\sigma _c, M)\}), \end{equation*}
which we will denote by
$\mathcal{S}$
again in what follows. Note that the quotients
$M/J_n$
with respect to this structure are just
$M/(\mathcal{I}_n \cdot M)$
, where
\begin{align} \mathcal{I}_n = \mathcal{I}_n(\mathcal{S}) = I_n + (\nu _{n,\hat {r}}(S_1), \ldots , \nu _{n,\hat {r}}(S_c)). \end{align}
Now we turn to the study of
$l_0$
-invariants. We can formulate the following asymptotic formula from [Reference Kleine and Müller34].
Lemma 6.12.
Let
$M$
be a finitely generated and torsion
$\Lambda _d$
-module, and consider the quotients
$M/J_n$
attached to an admissible structure
$\mathcal{S}$
on
$M$
, as described above. Then
\begin{equation*} \left |v_p(|M/J_n/p^n|) - \left ( m_0(M) \cdot p^{dn} + l_0(M) \cdot n p^{(d-1)n}\right )\right | = O(p^{(d-1)n}). \end{equation*}
Proof. It follows from [Reference Kleine and Müller34, Lemma 5.12] that
\begin{equation*} \left |v_p(|(M/J_n)[p^\infty ]/p^n|) - \left ( m_0(M) \cdot p^{dn} + l_0(M) \cdot n p^{(d-1)n}\right )\right | = O(p^{(d-1)n}). \end{equation*}
Since
$\mathcal{S}$
is admissible, we have
$\text {rank}_{{\mathbb{Z}}_p}(M/J_n)=O(p^{(d-2)n})$
, and therefore we may replace
$(M/J_n)[p^\infty ]$
by
$M/J_n$
in the above asymptotic formula.
The last ingredient which will be needed is the following control theorem. By an identical proof to [Reference Kleine and Müller34, Theorem 6.8], we have
Proposition 6.13.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension with intermediate number fields
$\mathbb{L}_n$
as above, let
$E$
be an elliptic curve with good ordinary reduction at
$p$
and suppose that no prime of
$S$
splits completely in
$\mathbb{L}_\infty$
. Then the kernels and the cokernels of the natural maps
\begin{equation*} s_n \colon X_{G_n} \longrightarrow X_n \end{equation*}
are finite. Moreover, the
$p$
-valuations of the orders of the kernels and cokernels are in
$O(p^{(d-1)n})$
.
Corollary 6.14.
In the setting of the previous proposition, let
$\mathcal{S}$
be an admissible structure on
$X = X^{str}(E/\mathbb{L}_\infty )$
, with corresponding quotients
$X/\mathcal{I}_n$
, where the ideals
$\mathcal{I}_n \subseteq \Lambda _d$
are defined as in (9). Then the kernels and cokernels of the induced maps
\begin{equation*} t_n \colon X/\mathcal{I}_n \longrightarrow X_n/\mathcal{I}_n\end{equation*}
are finite, and the
$p$
-valuations of their orders are in
$O(p^{(d-1)n})$
.
Note that
$I_n \subseteq \mathcal{I}_n$
operates trivially on
$X_n = X^{str}(E/\mathbb{L}_n)$
, that is,
\begin{equation*} X/\mathcal{I}_n = X_n/(\nu _{n,\hat {r}}(S_1), \ldots , \nu _{n,\hat {r}}(S_c)). \end{equation*}
Proof. Consider the commutative diagram
Let
$t$
be the number of generators of
$\mathcal{I}_n$
(note that each of these ideals has the same number of generators). Then it follows from the snake lemma that
\begin{equation*} |\textrm {coker}(t_n)| \le |\textrm {coker}(s_n)| \end{equation*}
and
\begin{equation*} |\ker (t_n)| \le |\ker (s_n)| \cdot |\textrm {coker}(r_n)|. \end{equation*}
Since
$\textrm {coker}(r_n) = \mathcal{I}_n \cdot \textrm {coker}(s_n)$
, we may conclude that
\begin{equation*} |\ker (t_n)| \le |\ker (s_n)| \cdot |\textrm {coker}(s_n)|^t.\end{equation*}
Therefore, the statement of the corollary follows from Proposition 6.13.
Now assume that the hypotheses
$(i)$
and
$(ii)$
from Lemma 6.5 are satisfied. Then Lemma 6.5 implies that
\begin{equation} |v_p(|X_n/p^n|) - v_p(|X^{str}(E_1[p^n]/\mathbb{L}_n)|)| = O(n p^{(d-2)n}). \end{equation}
If we assume that
$E_1[p^n] \cong E_2[p^n]$
for each
$n \in {\mathbb{N}}$
(here we will not be able to relax this assumption), then we may conclude as in Lemma 3.5 that
\begin{equation} X^{str}(E_1[p^n]/\mathbb{L}_n) \cong X^{str}(E_2[p^n]/\mathbb{L}_n). \end{equation}
Choose a structure
$\mathcal{S}$
which is admissible on both
$X$
and
$X'$
. Then we may use the same arguments as in the proof of Corollary 6.14 to derive from (10) and (11) that
\begin{equation*} \left | v_p(|(X_n/J_n)/p^n|) - v_p(|(X'_n/J_n)/p^n|)\right | = O(np^{(d-2)n}). \end{equation*}
Then it follows from Lemma 6.12 and Corollary 6.14 that
\begin{equation*} \left |\left ( m_0(X) p^{dn} + l_0(X) n p^{(d-1)n}\right ) - \left ( m_0(X') p^{dn} + l_0(X') n p^{(d-1)n}\right )\right | = O(p^{(d-1)n}).\end{equation*}
This (together with Theorem6.6) proves the following
Theorem 6.15.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension such that no prime of
$S$
splits completely in
$\mathbb{L}_\infty$
. We assume that the following hypotheses are satisfied.
-
(a)
$E_1[p^n] \cong E_2[p^n]$
for every
$n \in {\mathbb{N}}$
. -
(b)
$E_1$
and
$E_2$
satisfy condition
$({\star})$
from the introduction for all
$v \in S$
. -
(c) The decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
have
${\mathbb{Z}}_p$
-ranks at least two for each
$v \in S_p$
.
Then
\begin{equation*} \text {rank}_{\Lambda _d}(X) = \text {rank}_{\Lambda _d}(X'). \end{equation*}
If these ranks are equal to 0, then
-
(i)
$m_0(X) = m_0(X')$
and
-
(ii)
$l_0(X) = l_0(X')$
.
Remark 6.16.
Hypothesis (a) in Theorem
6.15
implies that the
$p$
-adic Tate modules of
$E_1[p^\infty ]$
and
$E_2[p^\infty ]$
are isomorphic. Indeed, letting
$C_n$
be the set of
$G_K$
-isomorphisms between
$E_1[p^n]$
and
$E_2[p^n]$
,
$n \in {\mathbb{N}}$
, the assumption implies that each
$C_n$
is non-empty. Moreover, each
$C_n$
is finite because
$E_1[p^n]$
is finite. It follows from [Reference Ribes and Zalesskii51
, Proposition 1.1.4] that the inverse limit
$C = \varprojlim _n C_n$
, taken with regard to the restriction maps, is non-empty. Any element in
$C$
gives an isomorphism between the Tate modules.
By results of Faltings on the Tate conjecture, it follows that there exists an isogeny of degree prime to
$p$
between
$E_1$
and
$E_2$
.Footnote
1
But then the Iwasawa modules
$X$
and
$X'$
are isomorphic. Therefore, the conclusion of the theorem is not interesting under assumption (a).
7. Proof of the second main theorem
In this section, we will apply the up-down approach from the proof of Theorem1.4 to the investigation of
$l_0$
-invariants. This will enable us to prove the second main result (Theorem1.5) without assuming the strong hypothesis (a) from Theorem6.15. Note that this comes at the price of the additional hypotheses (i) and (ii) in Theorem1.5. These two assumptions, however, are natural and mild, as we pointed out in Section 1. On the other hand, we have seen in Remark 6.16 that the conclusion of the theorem is not interesting under hypothesis (a). As this is the best result that we can prove via Lim’s approach, we apply the up-down approach instead in the current section.
In what follows, we will work with Greenberg Selmer groups.
Let
$K_\infty \in \mathcal{E}$
be arbitrary, but fixed. The canonical restriction map
$\textrm {Gal}(\mathbb{L}_\infty /K) \twoheadrightarrow \textrm {Gal}(K_\infty /K)$
induces a surjective homomorphism
\begin{equation*} \pi \colon \Lambda _d \cong {\mathbb{Z}}_p[[\textrm {Gal}(\mathbb{L}_\infty /K)]] \longrightarrow \Lambda \cong {\mathbb{Z}}_p[[\textrm {Gal}(K_\infty /K)]]. \end{equation*}
We write the characteristic power series
$f_1, f_2 \in \Lambda _d$
of
$X^{Gr}(E_1/\mathbb{L}_\infty )$
and
$X^{Gr}(E_2/\mathbb{L}_\infty )$
as
\begin{equation*} f_1 = p^{m_{0,1}} \cdot g_1, \quad f_2 = p^{m_{0,2}} \cdot g_2. \end{equation*}
Then we have the following auxiliary result which will be one of the main ingredients in the proof of Theorem1.5.
Lemma 7.1.
In the above setting, suppose that
$\pi (g_1) \not \equiv 0 \pmod {p}$
. Then there exists a Greenberg neighborhood
$U = \mathcal{E}(K_\infty ,n)$
such that for any
$\tilde {K}_{\infty } \in U$
we have
$\tilde {\pi }(g_1) \not \equiv 0 \pmod {p}$
where
\begin{equation*}\tilde {\pi } \colon \Lambda _d \cong {\mathbb{Z}}_p[[\textrm {Gal}(\mathbb{L}_\infty /K)]] \longrightarrow \Lambda \cong {\mathbb{Z}}_p[[\textrm {Gal}(\tilde {K}_\infty /K)]]\end{equation*}
is the surjective homomorphism induced by the canonical restriction map
$\textrm {Gal}(\mathbb{L}_\infty /K) \twoheadrightarrow \textrm {Gal}(\tilde {K}_\infty /K)$
.
Proof.
We can choose the variables
$T_1, T_2,\ldots ,T_d$
of
$\Lambda _d$
such that the map
$\pi$
is given by
\begin{equation*} \pi (T_1) = \pi (T_2)=\cdots =\pi (T_{d-1})= 0, \quad \pi (T_d) = T \in \Lambda = {\mathbb{Z}}_p[[T]] \end{equation*}
(this choice of variables corresponds to a choice
$\gamma _1, \gamma _2,\ldots , \gamma _d$
of topological generators of
$G$
such that
$\textrm {Gal}(\mathbb{L}_\infty /K_\infty ) = \langle \gamma _1, \gamma _2,\ldots ,\gamma _{d-1} \rangle$
, and
$T_i = \gamma _i - 1$
for each
$i$
). Since
$\pi (g_1)\not \equiv 0 \pmod {p}$
, the image
$\pi (g_1) \in \Lambda$
is associated with a distinguished polynomial of degree, say,
$r$
by the Weierstraß Preparation Theorem. Now choose
$U = \mathcal{E}(K_\infty , 2s)$
, where
$s$
is large enough such that
$p^s \gt r$
. Then the proof of [Reference Babaĭcev3, Proposition 1.1] implies that
\begin{equation*} \tilde {\pi }(g_1) \equiv \pi (g_1) \mod {(p^s, T^{p^s})}\end{equation*}
for every
$\tilde {K}_\infty \in U$
. In particular,
$\tilde {\pi }(g_1) \not \equiv 0 \pmod {p}$
(in fact,
$\tilde {\pi }(g_1)$
is also associated with a distinguished polynomial of degree
$r$
).
Proposition 7.9 is a key result that allows us to use the up-down approach to prove Theorem1.5. In order to prove this key result, we make use of the following auxiliary lemma. We start with a finitely generated torsion
$\Lambda _d$
-module
$M$
. Consider a presentation
\begin{equation*} \Lambda _d^g \stackrel {A}{\longrightarrow } \Lambda _d^l \longrightarrow M \longrightarrow 0\end{equation*}
of the
$\Lambda _d$
-module
$M$
. Since
$M$
is torsion, we know that
$l \le g$
. Recall that the Fitting ideal of
$M$
over
$\Lambda _d$
is generated by the
$l \times l$
-minors of the matrix
$A = (a_{ij})$
, and that this ideal does not depend on the chosen presentation of
$M$
.
Lemma 7.2.
Let
$M$
be a torsion
$\Lambda _d$
-module with characteristic power series
$f$
, and let
$f_1, \ldots , f_k$
be any set of generators of the Fitting ideal of
$M$
.
-
(1) Then
$f = \gcd (f_1, \ldots , f_k)$
. -
(2) Let
$\pi \colon \Lambda _d \longrightarrow \Lambda$
be a surjective ring homomorphism, and suppose that
${M_\pi \,:\!=\, M/(\ker (\pi ) \cdot M)}$
is a torsion
$\Lambda = \Lambda _d/\ker (\pi )$
-module. Then the characteristic power series
$f_\pi \in \Lambda$
of
$M_\pi$
satisfies
\begin{equation*} f_\pi = \gcd (\pi (f_1), \ldots , \pi (f_k)). \end{equation*}
Proof.
Since
$\pi \colon \Lambda _d \longrightarrow \Lambda$
is surjective, the
$\Lambda$
-module
$M_\pi$
has a presentation
\begin{align} \Lambda ^g \stackrel {\pi (A)}{\longrightarrow } \Lambda ^l \longrightarrow M_\pi \longrightarrow 0, \end{align}
where we denote by
$\pi (A)$
the matrix with entries
$(\pi (a_{ij}))_{i,j}$
in
$\Lambda$
.
In order to prove assertion (1), we let
$h \in \Lambda _d$
be any irreducible element (e.g.
$h = p$
) and consider the prime ideal
$\mathfrak{p} = (h)$
in
$\Lambda _d$
. The localization
$(\Lambda _d)_{\mathfrak{p}}$
of
$\Lambda _d$
at
$\mathfrak{p}$
is a discrete valuation ring with maximal ideal
$\mathfrak{p}$
. Since
$M$
is a torsion
$\Lambda _d$
-module, we can attach to it an elementary torsion
$\Lambda _d$
-module and a short exact sequence
\begin{equation*} 0 \longrightarrow M_1 \longrightarrow M \longrightarrow \bigoplus _{i = 1}^s \Lambda _d/(p^{n_i}) \oplus \bigoplus _{j = 1}^t \Lambda _d/(h_j(T)^{l_j}) \longrightarrow M_2 \longrightarrow 0, \end{equation*}
where
$M_1$
and
$M_2$
are pseudo-null. The localized sequence remains exact, and
$(M_1)_{\mathfrak{p}} = (M_2)_{\mathfrak{p}} = \{0\}$
(Indeed, since the
$M_i$
are pseudo-null, we can choose an element
$a$
in
$\Lambda _d$
which is coprime with
$h$
and annihilates both
$M_1$
and
$M_2$
. But
$a$
becomes a unit in the localization
$(\Lambda _d)_{\mathfrak{p}}$
.). Therefore,
\begin{equation*} M_{\mathfrak{p}} \cong \bigoplus _{i = 1}^s (\Lambda _d)_{\mathfrak{p}}/(p^{n_i}) \oplus \bigoplus _{j = 1}^t (\Lambda _d)_{\mathfrak{p}}/(h_j(T)^{l_j}). \end{equation*}
Note that
$(\Lambda _d)_{\mathfrak{p}}/(\lambda ) = \{0\}$
for each
$\lambda \in \Lambda _d$
which is coprime with
$h$
. Therefore,
$M_{\mathfrak{p}}$
is of the form
\begin{equation*} M_{\mathfrak{p}} \cong \bigoplus _{i = 1}^k (\Lambda _d)_{\mathfrak{p}} / (h^{e_i})\end{equation*}
(this sum might be empty, i.e.
$k = 0$
is possible). In other words, the powers
$h^{e_i}$
are the principal divisors of
$M_{\mathfrak{p}}$
as a module over the principal ideal domain
$(\Lambda _d)_{\mathfrak{p}}$
.
Now we apply the following auxiliary result (see [Reference Bosch4, Theorem 2.9.6]):
Lemma 7.3.
Let
$N$
denote a finitely generated torsion module over a principal ideal domain
$R$
, with matrix of relations
$B$
, and suppose that we are given a presentation
\begin{equation*} R^g \stackrel {B}{\longrightarrow } R^l \longrightarrow N \longrightarrow 0.\end{equation*}
Then the product of the
$l$
principal divisors of
$N$
is equal to the greatest common divisor of the
$l \times l$
-minors of
$B$
.
We apply this lemma to the presentation
\begin{equation*} (\Lambda _d)_{\mathfrak{p}}^g \longrightarrow (\Lambda _d)_{\mathfrak{p}}^l \longrightarrow M_{\mathfrak{p}} \longrightarrow 0, \end{equation*}
with matrix of relations
$A$
(viewed as a matrix with entries in the larger ring
$(\Lambda _d)_{\mathfrak{p}}$
). We may deduce that
$e \,:\!=\, \sum _{i = 1}^k e_i$
is the exact exponent of the largest power of
$h$
which divides the greatest common divisor of the
$l \times l$
-minors of
$A$
. Since these minors generate the Fitting ideal of
$M$
, this also means that
$h^e$
is the exact power of
$h$
which divides the gcd of the generators
$f_1, \ldots , f_k$
chosen above. Note that
$h^e$
is also the exact power of
$h$
which divides the characteristic power series
$f$
of
$M$
. Since
$h$
was arbitrary, this proves assertion (1).
In order to prove assertion (2), we start from the presentation (12) with the matrix
$\pi (A)$
. Now we proceed as above: Choose any irreducible element
$h \in \Lambda$
, and localize at
$\mathfrak{p} = (h)$
. Then we get
\begin{equation*} (M_\pi )_{\mathfrak{p}} \cong \bigoplus _{i = 1}^k \Lambda _{\mathfrak{p}}/(h^{e_i})\end{equation*}
for suitable
$e_i \in {\mathbb{N}}$
, and the powers
$h^{e_i}$
are the principal divisors of
$(M_\pi )_{\mathfrak{p}}$
over
$\Lambda _{\mathfrak{p}}$
. An application of Lemma 7.3 then proves assertion (2).
We prove a corollary which will be used below.
Corollary 7.4.
Let
$M$
be a finitely generated torsion
$\Lambda _d$
-module, with characteristic power series
$f$
, and write
$f = p^{m_0(M)} \cdot g$
as usual. Let
$\pi \colon \Lambda _d \longrightarrow \Lambda$
be a surjective ring homomorphism, and write
$M_\pi = M/(\ker (\pi )\cdot M)$
(this is a finitely generated
$\Lambda$
-module).
If
$d \gt 2$
, then we assume that the maximal pseudo-null
$\Lambda _d$
-submodule
$M^\circ$
of
$M$
is finitely generated over
${\mathbb{Z}}_p$
. Then the following statements hold.
-
(a) If
$\pi (f) \ne 0$
, then
$M_\pi$
is a torsion
$\Lambda$
-module.
-
(b) Suppose that
$M_\pi$
is a torsion
$\Lambda$
-module. Then
$\mu (M_\pi ) = m_0(M)$
if and only if
$\pi (g) \not \equiv 0 \pmod {p}$
.
Proof.
Let
$E_M$
be an elementary
$\Lambda _d$
-module attached to
$M$
, and consider an exact sequence
\begin{equation*} 0 \longrightarrow M^\circ \longrightarrow M \longrightarrow E_M \longrightarrow P \longrightarrow 0, \end{equation*}
where
$P$
is a pseudo-null
$\Lambda _d$
-module. This sequence shows that
\begin{equation*} \text {Ann}(M^\circ ) \cdot \text {Ann}(E_M) \subseteq \text {Ann}(M), \end{equation*}
where we denote by
$\text {Ann}(Y)$
the annihilator ideal of any finitely generated
$\Lambda _d$
-module
$Y$
.
Now our assumption on
$M^\circ$
implies that there exists an element
$H \in \text {Ann}(M^\circ )$
such that
\begin{align*}\pi (H) \not \equiv 0 \pmod {p},\end{align*}
so in particular
$\pi (H) \ne 0$
(cf. the proof of [Reference Kleine and Matar35, Theorem 3.22]). Therefore,
$\pi (H) \cdot \pi (f)$
yields a nonzero element in
\begin{equation*} \pi (\text {Ann}(M)) \subseteq \text {Ann}(M_\pi ).\end{equation*}
Choosing an elementary
$\Lambda$
-module
$E_\pi$
of
$M_\pi$
, the natural inclusion map
$E_\pi \hookrightarrow M_\pi$
induces an inclusion
\begin{equation*} \text {Ann}(M_\pi ) \hookrightarrow \text {Ann}(E_\pi ) = (f_\pi ), \end{equation*}
where we denote by
$f_\pi$
the characteristic power series of
$M_\pi$
and
$E_\pi$
. Since
$\pi (f) \pi (H) \ne 0$
, we may conclude that
$f_\pi \ne 0$
, that is,
$M_\pi$
is a torsion
$\Lambda$
-module.
Now we turn to the proof of statement (b). Choose generators
$f_1, \ldots , f_k$
of the Fitting ideal of
$M$
. We know from Lemma 7.2 that
\begin{equation*} \pi (f) = \pi (\gcd (f_1, \ldots , f_k)) \, \Big | \, \gcd (\pi (f_1), \ldots , \pi (f_k)) = f_\pi . \end{equation*}
On the other hand, the first part of the proof shows that
$f_\pi \mid (\pi (f) \cdot \pi (H))$
. Summarizing, the chain of divisibilities
\begin{equation*} \pi (f) \; \Big | \; f_\pi \; \Big | \; (\pi (f) \pi (H))\end{equation*}
shows that
$\mu (M_\pi ) \gt m_0$
if and only if
$\pi (g)$
is divisible by
$p$
(recall that
$\pi (H) \not \equiv 0 \pmod {p}$
).
Now we turn to the study of
$l_0$
-invariants and move on towards our key proposition. Let
$X = X^{Gr}(E/\mathbb{L}_\infty )$
for some elliptic curve
$E$
which is defined over
$K$
and has good ordinary reduction at all primes of
$K$
above
$p$
. Let
$f = p^{m_0(X)} \cdot g$
be the characteristic power series of
$X$
. If
$l_0(g) \gt 0$
, then there exists some element
$\gamma \in \textrm {Gal}(\mathbb{L}_\infty /K)$
which is not a
$p$
-th power, and such that the coset of
$T \,:\!=\, \gamma - 1$
in
$\Omega _d = \Lambda _d/p$
divides the coset of
$g$
. If
$\pi \colon \Lambda _d \longrightarrow \Lambda$
maps
$T$
to 0, then
$\pi (g) \equiv 0 \pmod {p}$
. For brevity, we introduce the following notion.
Definition 7.5.
Recall that
$\mathcal{E}_p \cap \mathcal{E}_{ns}$
denotes the set of all
${\mathbb{Z}}_p$
-subextensions
$K_\infty \subseteq \mathbb{L}_\infty$
of
$K$
such that each prime above
$p$
ramifies in
$K_\infty$
and no prime of
$S$
splits completely in
$K_\infty$
. We let
$\mathcal{E}_\mu = \mathcal{E}_\mu (X)$
denote the subset of
$\mathcal{E}_p \cap \mathcal{E}_{ns}$
which consists exactly of the
${\mathbb{Z}}_p$
-extensions
$K_\infty /K$
satisfying that
$X^{Gr}(E/K_\infty )$
is
$\Lambda$
-torsion and
\begin{equation*} \mu (X^{Gr}(E/K_\infty )) = m_0(X). \end{equation*}
Remark 7.6.
Let
$K_\infty \in \mathcal{E}$
, let
$\pi \colon \Lambda _d \longrightarrow \Lambda$
be the canonical surjection induced by the restriction map
\begin{equation*} \textrm {Gal}(\mathbb{L}_\infty /K) \longrightarrow \textrm {Gal}(K_\infty /K), \end{equation*}
and write
$X_\pi = X/(\ker (\pi )\cdot X)$
as usual. We assume that
$X^{Gr}(E/K_\infty )$
is a torsion
$\Lambda$
-module. In view of Corollary 3.7
, this means that
$X_\pi$
is also
$\Lambda$
-torsion and that we have
\begin{equation*} \mu (X_\pi ) = \mu (X^{Gr}(E/K_\infty ))\end{equation*}
as soon as
$K_\infty \in \mathcal{E}_{ns}$
. Therefore, any such
$K_\infty$
is contained in
$\mathcal{E}_\mu$
if and only if
\begin{equation*} \mu (X_\pi ) = m_0(X). \end{equation*}
In fact, since
$\pi (f)$
divides the characteristic power series of
$X_\pi$
in view of Lemma 7.2
, we have that
$\mu (X_\pi ) \ge m_0(X)$
for any
${\mathbb{Z}}_p$
-extension
$K_\infty$
such that
$X_\pi$
is
$\Lambda$
-torsion. Moreover, equality in this equation can hold only if
\begin{equation*} \pi (g) \not \equiv 0 \pmod {p}. \end{equation*}
In general, the latter condition is not sufficient for ensuring
$\mu (X_\pi ) = m_0(X)$
. We have shown in Corollary 7.4 above that it is also sufficient if hypothesis (ii) from Theorem
1.5
is satisfied.
Proposition 7.7.
Suppose that
$\mathcal{E}_p$
and
$\mathcal{E}_{ns}$
are non-empty and that
$X = X^{Gr}(E/\mathbb{L}_\infty )$
is
$\Lambda _d$
-torsion. Then the subset
$\mathcal{E}_\mu \subseteq \mathcal{E}$
is open and dense.
Proof.
First note that
$\mathcal{E}_p \cap \mathcal{E}_{ns} \subseteq \mathcal{E}$
is an open subset with respect to Greenberg’s topology. Indeed, suppose that
$K_\infty \in \mathcal{E}_p \cap \mathcal{E}_{ns}$
. Let
$m \in {\mathbb{N}}$
be large enough such that
-
(i) each prime of
$K$
above
$p$
ramifies in the
$m$
-th layer
$K_m$
of
$K_\infty$
, and -
(ii) no prime in
$S$
splits completely in
$K_m/K$
.
Then it is easy to see that
$\mathcal{E}(K_\infty ,m) \subseteq \mathcal{E}_p \cap \mathcal{E}_{ns}$
.
Now fix
$K_\infty \in \mathcal{E}_\mu$
, with homomorphism
$\pi$
. In particular, we have that
$E(K_\infty )[p^\infty ]$
is finite (see [Reference Greenberg17, Prop. 3.2(ii)]). Then taking into account Propositions 2.2 and 2.3(c), it follows from [Reference Kleine32, Theorem 4.11] that there exists a neighborhood
$\mathcal{E}(K_\infty ,n)$
which is contained in
$\mathcal{E}_p \cap \mathcal{E}_{ns}$
such that for each
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
, with homomorphism
$\tilde {\pi }$
, we have that
$X^{Gr}(E/\tilde {K}_\infty )$
is
$\Lambda$
-torsion and
\begin{equation*} \mu (X_{\tilde {\pi }}) = \mu (X^{Gr}(E/\tilde {K}_\infty )) \le \mu (X^{Gr}(E/K_\infty )) = \mu (X_\pi ) = m_0(X). \end{equation*}
Since it is a general fact that
$\mu (X_{\tilde {\pi }}) \ge m_0(X)$
(see also Remark 7.6), we may conclude that
\begin{equation*} \tilde {K}_\infty \in \mathcal{E}_\mu \end{equation*}
for each
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
, that is,
$\mathcal{E}_\mu$
is open with respect to Greenberg’s topology.
Let us now prove denseness. We will apply the following
Lemma 7.8.
Let
$\mathcal{E}$
be the set of
${\mathbb{Z}}_p$
-extensions of
$K$
which are contained in some fixed
${\mathbb{Z}}_p^d$
-extension
$\mathbb{L}_\infty /K$
. Suppose we are given finitely many
${\mathbb{Z}}_p^{d-1}$
-subextensions
$\mathbb{L}^{(1)}, \ldots , \mathbb{L}^{(r)} \subseteq \mathbb{L}_\infty$
of
$K$
. Let
$\tilde {\mathcal{E}} \subseteq \mathcal{E}$
denote the subset of
${\mathbb{Z}}_p$
-extensions of
$K$
which are not contained in any of the
$\mathbb{L}^{(i)}$
.
Then each Greenberg open neighborhood
$U = \mathcal{E}(K_\infty ,n)$
,
$K_\infty \in \mathcal{E}$
, and
$n \in {\mathbb{N}}$
, contains an element of
$\tilde {\mathcal{E}}$
, that is,
$\tilde {\mathcal{E}} \subseteq \mathcal{E}$
lies dense with respect to Greenberg’s topology.
Proof.
Let
$K_\infty \in \mathcal{E}$
,
$n \in {\mathbb{N}}$
be arbitrary. First assume that
$K_\infty$
itself is contained in
$\tilde {\mathcal{E}}$
. Then we choose
$n \in {\mathbb{N}}$
large enough to ensure that the
$n$
-th layer
$K_n$
of
$K_\infty$
is not contained in
$\mathbb{L}^{(i)}$
for any
$i$
, and we let
$U = \mathcal{E}(K_\infty ,n)$
. In this case, we actually have
$\mathcal{E}(K_\infty ,n) \subseteq \tilde {\mathcal{E}}$
.
Now suppose that
$K_\infty \not \in \tilde {\mathcal{E}}$
. Then
$K_\infty$
is contained in at least one of the
${\mathbb{Z}}_p^{d-1}$
-extensions
$\mathbb{L}^{(1)}, \ldots , \mathbb{L}^{(r)}$
of
$K$
.
Choose any
$F \in \tilde {\mathcal{E}}$
, and consider the
${\mathbb{Z}}_p^2$
-extension
$K_\infty \cdot F$
of
$K$
(note that
$K_\infty \ne F$
since we assumed that
$K_\infty \not \in \tilde {\mathcal{E}}$
). Then
$F \cap \mathbb{L}^{(i)}$
is a finite extension of
$K$
for each
$i \in \{ 1, \ldots , r\}$
. Therefore,
$K_\infty \cdot F$
contains at most
$r$
different
${\mathbb{Z}}_p$
-extensions of
$K$
which are not contained in
$\tilde {\mathcal{E}}$
: Indeed, if
$M^{(1)}, \ldots , M^{(r+1)} \subseteq K_\infty F$
are not contained in
$\tilde {\mathcal{E}}$
, then there exist numbers
$i \ne j \in \{1, \ldots , r+1\}$
and
$s \in \{1, \ldots , r\}$
such that both
$M^{(i)}$
and
$M^{(j)}$
are contained in
$\mathbb{L}^{(s)}$
. But then
$M^{(i)} \cdot M^{(j)} \subseteq \mathbb{L}^{(s)}$
. Since we necessarily have
$M^{(i)} \cdot M^{(j)} = K_\infty \cdot F$
, this contradicts the fact that
$(F \cap \mathbb{L}^{(s)})/K$
is finite.
Since
$K_\infty F \cap \mathcal{E}(K_\infty ,n)$
is infinite for each
$n \in {\mathbb{N}}$
, this concludes the proof of Lemma 7.8.
This also concludes the proof of Proposition 7.7. Indeed, since
$\mathcal{E}_p$
and
$\mathcal{E}_{ns}$
are non-empty, we know that each
${\mathbb{Z}}_p$
-extension which is not contained in
$\mathcal{E}_p \cap \mathcal{E}_{ns}$
is contained in a finite set of
${\mathbb{Z}}_p^{d-1}$
-extensions (or smaller multiple
${\mathbb{Z}}_p$
-extensions of
$K$
), namely, the inertia and decomposition subfields of
$\mathbb{L}_\infty$
of the primes above
$p$
(respectively, the primes in
$S$
). Since
$X$
is
$\Lambda _d$
-torsion, we know from Lemma 5.1 that
$X^{Gr}(E/K_\infty )$
is
$\Lambda$
-torsion and that
$\mu (X^{Gr}(E/K_\infty )) = m_0(X)$
for all
${\mathbb{Z}}_p$
-extensions
$K_\infty$
of
$K$
outside of a certain finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
. Therefore, we can consider
$\tilde {\mathcal{E}} = \mathcal{E}_\mu (X)$
in Lemma 7.8.
Now we can prove our key auxiliary result. Recall that we write the characteristic power series of
$X$
as
$f = p^{m_0(X)} \cdot g$
.
Proposition 7.9.
Let
$X = X^{Gr}(E/\mathbb{L}_\infty )$
, let
$\gamma \in \textrm {Gal}(\mathbb{L}_\infty /K)$
be an element which is not a
$p$
-th power, and set
$T = \gamma -1$
. We let
$M = \mathbb{L}_\infty ^{\langle T+1\rangle }$
, and we suppose that
$\overline {T}^e \mid \overline {g}$
for some
$e \in {\mathbb{N}}$
(note that
$e = 0$
is possible – in that case assertion (a) below is empty).
-
(a) Let
$K_\infty \subseteq M$
be any
${\mathbb{Z}}_p$
-extension of
$K$
contained in
$\mathcal{E}_{ns}$
. Then there exists a constant
$C$
such that for
$n \gg 0$
for each
\begin{equation*} \lambda (X^{Gr}(E/\tilde {K}_\infty )) \ge e \cdot p^n - C\end{equation*}
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
which is contained in
$\mathcal{E}_\mu$
.
-
(b) If
$d \gt 2$
, then we assume that
-
•
$X = X^{Gr}(E/\mathbb{L}_\infty )$
is
$\Lambda _d$
-torsion,
-
• the maximal pseudo-null submodule of
$X$
is finitely generated over
${\mathbb{Z}}_p$
, -
•
$\mathcal{E}_p \cap \mathcal{E}_{ns} \subseteq \mathcal{E}$
is non-empty, and
-
•
$\overline {T}^{e+1} \nmid \overline {g}$
.
Then there exists a
${\mathbb{Z}}_p^2$
-extension
$N$
of
$K$
and a
${\mathbb{Z}}_p$
-extension
$K_\infty \in \mathcal{E}_{\mu }$
such that
$K_\infty \subseteq N \cap M$
and there exists a constant
$C'$
such that for
$n \gg 0$
for each
\begin{equation*} \lambda (X^{Gr}(E/\tilde {K}_\infty )) \le e \cdot p^n + C'\end{equation*}
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n) \setminus \mathcal{E}(K_\infty , n+1)$
which is a subextension of
$N$
and contained in
$\mathcal{E}_\mu$
.
In particular, if
$\overline {T}$
does not divide
$\overline {g}$
, that is,
$e = 0$
, then for
$n \gg 0$
\begin{equation*} \lambda (X^{Gr}(E/\tilde {K}_\infty )) = O(1). \end{equation*}
-
Proof.
Let
$K_\infty \subseteq M$
be any
${\mathbb{Z}}_p$
-extension of
$K$
contained in
$\mathcal{E}_{ns}$
, say,
$K_\infty = \mathbb{L}_\infty ^{\langle T+1, T_2+1, \ldots , T_{d-1}+1\rangle }$
for suitable variables
$T_2, \ldots , T_{d-1}$
, and write
$T_i + 1 = \gamma _i$
for each
$i$
(in other words,
$\gamma , \gamma _2, \ldots , \gamma _{d-1}$
are topological generators of
$\textrm {Gal}(\mathbb{L}_\infty /K_\infty )$
). Let
$\pi \colon \Lambda _d \longrightarrow \Lambda$
be the canonical epimorphism corresponding to
$K_\infty$
. For the proof of (a), we will assume that
$e \ge 1$
. Since
$\overline {T}^e \mid \overline {g}$
by assumption, we know that
\begin{align*}\pi (g) \equiv 0 \pmod {p}.\end{align*}
Let now
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
be arbitrary, and choose topological generators
$\tilde {\gamma }_1, \ldots , \tilde {\gamma }_{d-1}$
of
$\textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
. We let
$\tilde {\pi }$
denote the corresponding surjection onto
$\Lambda$
, that is,
$\tilde {\pi }(\tilde {\gamma }_i) = 1$
for each
$i$
. Suppose that
$\tilde {\gamma }_d$
denotes a topological generator of
$\textrm {Gal}(\tilde {K}_\infty /K)$
. Let
$\tilde {K}_n \subseteq \tilde {K}_\infty$
be the unique intermediate field of degree
$p^n$
over
$K$
. Since
\begin{equation*} K_\infty \cap \tilde {K}_\infty \supseteq \tilde {K}_n = \mathbb{L}_\infty ^{\langle \tilde {\gamma }_1, \tilde {\gamma }_2, \ldots , \tilde {\gamma }_{d-1}, \tilde {\gamma }_d^{p^n}\rangle }, \end{equation*}
we may conclude that
$\gamma , \gamma _2, \ldots , \gamma _{d-1} \in \langle \tilde {\gamma }_1, \tilde {\gamma }_2, \ldots , \tilde {\gamma }_{d-1}, \tilde {\gamma }_d^{p^n} \rangle$
. Since
$\tilde {\pi }(\tilde {\gamma }_i) = 1$
for each
$i$
, it follows that
$\tilde {\pi }(\gamma ) \in \textrm {Gal}(\tilde {K}_\infty /K)^{p^n}$
and
\begin{equation*} \tilde {\pi }(\gamma _i) \in \textrm {Gal}(\tilde {K}_\infty /K)^{p^n} \end{equation*}
for
$i \in \{2, \ldots , d-1\}$
. Writing
${\mathbb{Z}}_p[[\textrm {Gal}(\tilde {K}_\infty /K)]] = {\mathbb{Z}}_p[[S]]$
, we may conclude that the reduced degree of
$\tilde {\pi }(T)$
(i.e. the degree of an element in its coset in the quotient ring
$\Omega = \Lambda /(p)$
) is at least
$p^n$
if
${\tilde {\pi }(T) \not \equiv 0 \pmod {p}}$
, and the same holds for the reduced degree of
$\tilde {\pi }(T_i)$
for each
$i$
.
Recall that
$\overline {T}^e \mid \overline {g}$
. Therefore, the above means that either
$\tilde {\pi }(g) \equiv 0 \pmod {p}$
or the reduced degree of
$\tilde {\pi }(g)$
is at least
$e \cdot p^n$
.
Now suppose that
$\tilde {K}_\infty \in \mathcal{E}_\mu$
. Then we know (see Remark 7.6) that
$\tilde {\pi }(g)$
is not divisible by
$p$
. This means that
$\tilde {\pi }(g) \in \Lambda$
is associated with a distinguished polynomial of degree at least
$e \cdot p^n$
.
Choose generators
$f_1, \ldots , f_k$
of the Fitting ideal of
$X$
. Then it follows from Lemma 7.2 that
\begin{align} \tilde {\pi }(f) = \tilde {\pi }(\gcd (f_1, \ldots , f_k)) \; \Big | \; \gcd (\tilde {\pi }(f_1), \ldots , \tilde {\pi }(f_k)) = f_{\tilde {\pi }}, \end{align}
where we denote by
$f_{\tilde {\pi }} \in \Lambda$
the characteristic power series of
$X_{\tilde {\pi }} = X/(\ker (\tilde {\pi } \cdot X))$
. It follows from the above that
$\tilde {\pi }(f) = p^{m_0(X)} \cdot \tilde {\pi }(g)$
is a multiple of a distinguished polynomial of degree at least
$e \cdot p^n$
. In view of Corollary 3.7, we know that for
$n \gg 0$
\begin{equation*} | \lambda (X_{\tilde {\pi }}) - \lambda (X^{Gr}(E/\tilde {K}_\infty ))| \end{equation*}
is bounded on
$\mathcal{E}(K_\infty ,n)$
. Therefore, assertion (a) of the proposition follows.
Now we turn to the proof of assertion (b). We decompose the coset
$\overline {g}$
of
$g$
in
$\Omega _d = \Lambda _d/p$
into a product of irreducible factors:
\begin{equation*} \overline {g} = \overline {T}^e \cdot \overline {g_2}^{e_2} \cdot \ldots \cdot \overline {g_t}^{e_t}, \end{equation*}
where the
$\overline {g_i}$
are not divisible by
$\overline {T}$
, and pairwise coprime in
$\Omega _d$
. Let
$s \le t$
be such that, without loss of generality,
$\overline {T}, \overline {g_2}, \ldots , \overline {g_s}$
are the factors of
$\overline {g}$
which contribute to
$l_0(X)$
. Choose lifts
$g_i$
of
$\overline {g_i}$
in
$\Lambda _d$
. Our aim is to choose a
${\mathbb{Z}}_p$
-extension
$K_\infty \subseteq M$
of
$K$
such that the homomorphism
$\pi \colon \Lambda _d \longrightarrow \Lambda$
which corresponds to
$K_\infty$
maps each
$g_i$
,
$2 \le i \le t$
, to an element of
$\Lambda$
which is not divisible by
$p$
(and in particular will be nonzero).
Since
$X$
is
$\Lambda _d$
-torsion and
$\mathcal{E}_p \cap \mathcal{E}_{ns} \ne \emptyset$
by assumption, it follows from Proposition 7.7 that
$\mathcal{E}_\mu \subseteq \mathcal{E}$
is a dense subset so in particular,
$\mathcal{E}_{\mu } \neq \emptyset$
. Choose
$K_\infty \in \mathcal{E}_{\mu }$
. Then
$\pi (g) \not \equiv 0 \pmod {p}$
(see Remark 7.6). Therefore,
$\pi (g_i) \not \equiv 0 \pmod {p}$
for each
$i$
. This means that for each
$i$
the power series
$\pi (g_i)$
is associated with a distinguished polynomial.
Let
$d$
denote the degree of the distinguished polynomial associated with
$\pi (g_2^{e_2} \cdot \ldots \cdot g_t^{e_t})$
, and choose
$n$
large enough such that
$p^{(n-1)/2} \gt d$
. Since
$\mathcal{E}_\mu \subseteq \mathcal{E}$
is open by Proposition 7.7, we may assume that
$\mathcal{E}(K_\infty ,n) \subseteq \mathcal{E}_\mu$
. Let
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
be an arbitrary element, and let
$\tilde {\pi }$
be the corresponding homomorphism. Write
$n = 2m$
if
$n$
is even, and
$n = 2m +1$
if
$n$
is odd. Then in both cases
$\mathcal{E}(K_\infty ,n) \subseteq \mathcal{E}(K_\infty , 2m)$
and
$p^m \gt d$
. Therefore, as in the proof of Lemma 7.1,
\begin{equation*}\deg \big(\tilde {\pi }\big(g_2^{e_2} \cdot \ldots \cdot g_t^{e_t}\big)\big) = \deg \big(\pi \big(g_2^{e_2} \cdot \ldots \cdot g_t^{e_t}\big)\big) = d \end{equation*}
does not depend on the choice of
$\tilde {K}_\infty$
. This shows that the contribution of the divisors
$\overline {g_i}$
of
$\overline {g}$
to the degree of the distinguished polynomial attached to
$\tilde {\pi }(g)$
is bounded as
$\tilde {K}_\infty$
runs over the elements in
$\mathcal{E}(K_\infty ,n)$
.
As in the proof of (a), we let
$\gamma , \gamma _2, \ldots , \gamma _{d-1}$
denote a set of topological generators of
$\textrm {Gal}(\mathbb{L}_\infty /K_\infty )$
. Now let
$N = \mathbb{L}_\infty ^{\langle \gamma _2, \ldots , \gamma _{d-1}\rangle }$
and suppose that
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n) \setminus \mathcal{E}(K_\infty , n+1)$
. Let
$\tilde {\gamma }_1, \ldots , \tilde {\gamma }_{d-1}$
denote a set of topological generators of
$\textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
. In fact, we will assume from now on that
$\tilde {\gamma }_i = \gamma _i$
for each
$i \in \{2, \ldots , d-1\}$
. Note this just means that we are restricting to
${\mathbb{Z}}_p$
-extensions
$\tilde {K}_\infty$
of
$K$
which are contained in
$N$
, as in the statement of the proposition.
Moreover, we denote by
$\tilde {\gamma }_d$
a topological generator of
$\textrm {Gal}(\tilde {K}_\infty /K)$
. Then, as in the proof of (a),
$\gamma = T+1$
can be written in the form
\begin{align} \tilde {\gamma }_1^{a_1} \cdot \ldots \cdot \tilde {\gamma }_{d-1}^{a_{d-1}} \cdot \tilde {\gamma }_d^{a_d}, \end{align}
where
$a_1, \ldots , a_{d} \in {\mathbb{Z}}_p$
and
$a_d$
is divisible by
$p^e$
. Since
$\tilde {K}_\infty \not \in \mathcal{E}(K_\infty , n+1)$
, we know that at least one of the topological generators
$\gamma , \gamma _2, \ldots , \gamma _{d-1}$
of
$\textrm {Gal}(\mathbb{L}_\infty /K_\infty )$
is not contained in
$\langle \tilde {\gamma }_1, \tilde {\gamma }_2, \ldots , \tilde {\gamma }_{d-1}, \tilde {\gamma }_d^{p^{n+1}} \rangle$
. Since
$\tilde {K}_\infty \subseteq N$
by assumption, we may conclude that the exponent
$a_d$
from (14) is not divisible by
$p^{n+1}$
. In fact, the topological generator
$\tilde {\gamma }_d$
of
$\textrm {Gal}(\tilde {K}_\infty /K)$
can be chosen such that
$a_d = p^n$
. Therefore,
\begin{equation*} \deg (\tilde {\pi }(T^e)) = e \cdot p^n.\end{equation*}
Summarizing, we have shown that
\begin{equation*}\deg (\tilde {\pi }(g)) = e \cdot p^n + O(1)\end{equation*}
on
$\mathcal{E}(K_\infty ,n) \setminus \mathcal{E}(K_\infty , n+1)$
. Since the maximal pseudo-null submodule
$X^\circ$
of
$X$
is finitely generated as a
${\mathbb{Z}}_p$
-module in view of our hypotheses, we may, as in the proof of Corollary 7.4, choose an element
$H \in \text {Ann}(X^\circ )$
such that
$\pi (H) \not \equiv 0 \pmod {p}$
. Then
$\deg (\tilde {\pi }(H)) = O(1)$
on
$\mathcal{E}(K_\infty ,n)$
by the same argument as for
$g_2^{e_2} \cdot \ldots \cdot g_t^{e_t}$
above (it might be necessary to increase
$n$
appropriately). Moreover,
\begin{equation*} \tilde {\pi }(f) \mid f_{\tilde {\pi }} \mid \tilde {\pi }(f) \tilde {\pi }(H) \end{equation*}
for each
$\tilde {K}_\infty \in \mathcal{E}(K_\infty ,n)$
, as in the proof of Corollary 7.4 (here
$f_{\tilde {\pi }}$
denotes the characteristic power series of
$X_{\tilde {\pi }}$
, as in the proof of (a)). We may thus conclude that
\begin{equation*} \lambda (X_{\tilde {\pi }}) = e \cdot p^n + O(1).\end{equation*}
In view of Corollary 3.7, we know that for
$n \gg 0$
\begin{equation*} | \lambda (X_{\tilde {\pi }}) - \lambda (X^{Gr}(E/\tilde {K}_\infty ))| \end{equation*}
is bounded on
$\mathcal{E}(K_\infty ,n)$
. Therefore,
\begin{equation*} \lambda (X^{Gr}(E/\tilde {K}_\infty )) = e \cdot p^n + O(1). \end{equation*}
The following result is the last auxiliary result which will be used in the proof of Theorem1.5.
Proposition 7.10.
Let
$E$
be
$E_1$
or
$E_2$
. If
$d \gt 2$
, suppose that the maximal pseudo-null submodule of
$X = X^{Gr}(E/\mathbb{L}_{\infty })$
is finitely generated over
$\mathbb{Z}_p$
. Let
$K_\infty \in \mathcal{E}$
. Assume that no prime in
$S_p$
splits completely in
$K_{\infty }/K$
and that
$X^{Gr}(E/K_{\infty })$
is
$\Lambda$
-torsion.
We write the characteristic power series of
$X$
as
$f = p^{m_0(X)} \cdot g$
. Then the following are equivalent:
-
(1)
$\pi (g) \not \equiv 0 \pmod {p}$
, where
$\pi$
corresponds to
$K_{\infty }$
. -
(2)
$\lambda (X^{Gr}(E/\tilde {K}_{\infty }))$
is bounded as
$\tilde {K}_{\infty }$
runs over the elements in a sufficiently small neighborhood of
$K_{\infty }$
.
Proof.
[Reference Kleine and Matar35, Lemma 3.21] proves the analogous statement for the classical Selmer group. Using Corollary 3.7, the proof of [Reference Kleine and Matar35, Lemma 3.21] can be formulated for the Greenberg Selmer group in an identical way. Taking this into account, we see that the equivalence in the statement of the proposition will follow if we show the existence of the element
$H \in \Lambda _d$
in loc. cit. As the maximal pseudo-null submodule of
$X^{Gr}(E/\mathbb{L}_{\infty })$
is finitely generated over
$\mathbb{Z}_p$
, its annihilator ideal is not contained in any prime ideal of height at most
$d-1$
. Therefore, from the proof of [Reference Kleine and Matar35, Theorem 3.22] such an element
$H \in \Lambda _d$
exists.
Now, we can finally prove Theorem1.5.
Proof of Theorem
1.5
. Assume that all the hypotheses of Theorem1.5 are satisfied. By Propositions 2.2 and 2.3, it suffices to prove the theorem with
$X(E_j/\mathbb{L}_{\infty })$
replaced by
$X^{Gr}(E_j/\mathbb{L}_{\infty })$
for
$j=1,2$
.
Suppose that
$X(E_1/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module and
$E_1[p^i] \cong E_2[p^i]$
for some integer
$i \gt m_{0,1}$
. From Theorem1.4, we get that
$X(E_2/\mathbb{L}_{\infty })$
is a
$\Lambda _d$
-torsion module with
$m_{0,2}=m_{0,1}$
.
In order to conclude the proof of assertion (1), we assume that
$\widehat {l_{0,1}}=0$
. Then for every
$K_{\infty } \in \mathcal{E}$
, we have that
$\pi (g_1) \not \equiv 0 \pmod {p}$
, where
$\pi$
corresponds to
$K_{\infty }$
. From Corollaries 7.4 and 3.7, we get that
$X^{Gr}(E_1/K_{\infty })$
is
$\Lambda$
-torsion. Note that in view of hypothesis (ii) and (iii) from Theorem1.5, we can apply Proposition 7.10 to each
$K_\infty \in \mathcal{E}$
. Therefore from Proposition 7.10,
$\lambda (X^{Gr}(E_1/\tilde {K}_{\infty }))$
is bounded as
$\tilde {K}_{\infty }$
runs over the elements in a sufficiently small neighborhood
$V$
of
$\tilde {K}_{\infty }$
. Since
$\pi (g_1) \not \equiv 0 \pmod {p}$
for every
$\pi$
, it follows from Corollaries 7.4 and 3.7 that
$X^{Gr}(E_1/\tilde {K}_{\infty })$
is
$\Lambda$
-torsion with
$\mu (X^{Gr}(E_1/\tilde {K}_{\infty }))=\alpha$
for all
$\tilde {K}_{\infty } \in V$
.
By Lemma 4.7, the differences
$|\lambda (X^{Gr}(E_1/\tilde {K}_\infty )) - \lambda (X^{Gr}(E_2/\tilde {K}_\infty ))|$
are bounded on
$V$
. Therefore,
$\lambda (X^{Gr}(E_2/\tilde {K}_{\infty }))$
is bounded as
$\tilde {K}_{\infty }$
runs over the elements of
$V$
. So by Proposition 7.10,
${\pi (g_2) \not \equiv 0 \pmod {p}}$
. Since
$K_\infty$
was arbitrary, we have thus shown that
$\widehat {l_{0,2}}=0$
. The proof in the reverse direction is the same. This proves (1).
Now we prove assertion (2) of the theorem. We show that
$l_{0,2} \le l_{0,1}$
(the reverse inequality will follow from a symmetric argument). To this purpose, suppose that
$\overline {T}^e \mid \overline {g}$
but
$\overline {T}^{e+1} \nmid \overline {g}$
for some variable
$T$
(here
$g_1 = g(X^{Gr}(E_1/\mathbb{L}_\infty ))$
, as in the first part of the proof). Note that we allow
$e = 0$
here. Let
$K_\infty$
be any
${\mathbb{Z}}_p$
-extension of
$K$
contained in
$\mathcal{E}_{\mu }$
as in Proposition 7.9(b). Then we can choose a sequence of
${\mathbb{Z}}_p$
-extensions
$\tilde {K}_\infty ^{(n)}$
of
$K$
such that
\begin{align} \lambda (X^{Gr}(E/\tilde {K}_\infty ^{(n)})) \le e \cdot p^n + C' \end{align}
and
$\tilde {K}_\infty ^{(n)} \in \mathcal{E}(K_\infty ,n) \setminus \mathcal{E}(K_\infty , n+1)$
for each
$n \in {\mathbb{N}}$
and some fixed constant
$C'$
. As in the first part of the proof, the differences
$|\lambda (X^{Gr}(E_1/\tilde {K}_\infty ^{(n)})) - \lambda (X^{Gr}(E_2/\tilde {K}_\infty ^{(n)}))|$
are bounded for these
$\tilde {K}_\infty ^{(n)} \in \mathcal{E}_\mu$
.
Now suppose that
$\overline {g_2}$
,
$g_2 = g(X^{Gr}(E_2/\mathbb{L}_\infty ))$
, is divisible by a higher power of
$\overline {T}$
. Then Proposition 7.9(a) implies that for
$m \gg 0$
\begin{equation*} \lambda (X^{Gr}(E_2/\tilde {K}_\infty ^{(m)})) \ge (e+1) \cdot p^m\end{equation*}
for each
$\tilde {K}_\infty ^{(m)} \in \mathcal{E}(K_\infty ,m)$
which is contained in
$\mathcal{E}_\mu$
. For any
$m$
which is strictly larger than
$n$
and the sum of the constant
$C'$
from equation (15) and the constant bounding the above difference of
$\lambda$
-invariants, this yields a contradiction.
Since
$T = \gamma - 1$
and
$e \in {\mathbb{N}}$
had been chosen arbitrarily, this shows that indeed
$l_{0,2} \le l_{0,1}$
.
8. The
$\mathfrak{M}_H(G)$
-property
Finally, we mention an application to the
$\mathfrak{M}_H(G)$
-property. This property, motivated by the
$\mathfrak{M}_H(G)$
-conjecture (see [Reference Coates and Sujatha9]), has been studied before in [Reference Kleine, Matar and Sujatha36]. We adapt the following notation from [Reference Kleine, Matar and Sujatha36] to our purposes.
Definition 8.1.
We let
$ {\mathcal{H} \subseteq \mathcal{E}}$
denote the subset of
${\mathbb{Z}}_p$
-extensions
$L \subseteq \mathbb{L}_\infty$
which satisfy the following hypotheses:
-
(a)
$L \in \mathcal{E}_{p} \cap \mathcal{E}_{ns}$
, that is, each prime of
$K$
above
$p$
ramifies in
$L$
, and no prime in
$S$
splits completely in
$L/K$
, -
(b)
$X(E_1/L)$
and
$X(E_2/L)$
are both
$\Lambda$
-torsion.
Definition 8.2.
Let
$L \in \mathcal{H}$
, and write
$G = \textrm {Gal}(\mathbb{L}_\infty /K)$
and
$H = \textrm {Gal}(\mathbb{L}_\infty /L)$
. Then
$H \cong {\mathbb{Z}}_p^{d-1}$
, and we say that the
$\mathfrak{M}_H(G)$
-property holds for
$X(E_i/\mathbb{L}_\infty )$
if the quotient
\begin{equation*} X_f(E_i/\mathbb{L}_\infty ) \,:\!=\, X(E_i/\mathbb{L}_\infty )/X(E_i/\mathbb{L}_\infty )[p^\infty ] \end{equation*}
is finitely generated as a
${\mathbb{Z}}_p[[H]]$
-module.
Before stating the main results in this section, we need two lemmas.
Lemma 8.3.
Let
$i \in \{1,2\}$
,
$E=E_i$
,
$g=g_i$
, and
$K_{\infty } \in \mathcal{H}$
. Suppose that we have
$\pi (g) \not \equiv 0 \pmod {p}$
, where
$\pi$
corresponds to
$K_{\infty }$
as in the beginning of Section 7. Furthermore, if
$d \gt 2$
, suppose that the maximal pseudo-null submodule of
$X(E/\mathbb{L}_{\infty })$
is finitely generated over
$\mathbb{Z}_p$
.
Let
$H=\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })$
. Then
$X_f(E/\mathbb{L}_\infty )$
is finitely generated as a
${\mathbb{Z}}_p[[H]] \cong \Lambda _{d-1}$
-module.
Proof.
For simplicity, we denote
$X_f(E/\mathbb{L}_{\infty })$
by
$X_f$
. We have an exact sequence
\begin{equation*}0 \longrightarrow A \longrightarrow X_f \longrightarrow E\end{equation*}
where
$A$
is a pseudo-null
$\Lambda _d$
-module and
$E$
is an elementary
$\Lambda _d$
-module. Assume that
$\pi (g) \not \equiv 0 \pmod {p}$
.
If
$d=2$
, then as
$A$
is pseudo-null it has Krull dimension at most one. Multiplication by
$p$
is injective on
$X_f$
by definition. Therefore,
$A/p$
has Krull dimension zero by [Reference Atiyah and Macdonald1, Corollary 11.9], that is,
$A/p$
is finite whence
$A$
is a finitely generated
${\mathbb{Z}}_p$
-module.
Since
$X(E/\mathbb{L}_\infty )[p^\infty ] = X(E/\mathbb{L}_\infty )[p^n]$
for some
$n \in {\mathbb{N}}$
, we have an isomorphism
\begin{equation*} X_f \cong p^n \cdot X(E/\mathbb{L}_\infty ) \subseteq X(E/\mathbb{L}_\infty ).\end{equation*}
Therefore, if
$d \gt 2$
, the maximal pseudo-null
$\Lambda _d$
-submodule of
$X_f$
is finitely generated as a
${\mathbb{Z}}_p$
-module by assumption. Since this former statement is also true for
$d=2$
by the above, we see from the above exact sequence that
$X_f$
will be finitely generated over
${\mathbb{Z}}_p[[H]]$
if we can show that
$E$
is finitely generated over
${\mathbb{Z}}_p[[H]]$
.
To this end, we first note that
$E_H$
is a finitely generated torsion
${\mathbb{Z}}_p[[\Gamma ]] \cong \Lambda$
-module, where
$\Gamma = \textrm {Gal}(K_\infty /K)$
. To see this, we first remark that since characteristic power series are multiplicative in exact sequences (this can be shown by localizations at height one primes ideals of
$\Lambda _d$
), the characteristic power series of
$X_f$
(and hence
$E$
) is
$g$
. The assumption
$\pi (g) \not \equiv 0 \pmod {p}$
certainly implies that
$\pi (g) \neq 0$
. As
$g$
annihilates
$E$
, we see that
$E_H$
is a finitely generated torsion
$\Lambda$
-module.
Now we claim that
$\mu (E_H)=0$
. This can be seen from the following sequence of equivalences. Assume that
$E=\bigoplus _{j=1}^t \Lambda _d/(h_j^{n_j})$
so that
$g=\prod h_j^{n_j}$
. Let
$\Omega _d\,:\!=\,\Lambda _d/p$
be the ‘mod
$p$
’ Iwasawa algebra. We have
\begin{align*} &\mu (E_H)=0 \\ \Longleftrightarrow \,&\mu ((\Lambda _d/(h_j^{n_j}))_H)=0 \text{ for }j=1,\ldots ,t\\ \Longleftrightarrow \, &(\Omega _d)_H/(\pi (\overline {h_j}^{n_j})) \text{ is finite for }j=1,\ldots ,t\\ \Longleftrightarrow \, &\pi (h_j^{n_j}) \not \equiv 0 \pmod {p} \text{ for }j=1,\ldots ,t\\ \Longleftrightarrow \, &\pi (g) \not \equiv 0 \pmod {p}. \end{align*}
Therefore, we see that
$\mu (E_H)=0$
. So,
$E_H$
is a finitely generated torsion
$\Lambda$
-module with
$\mu =0$
. Therefore,
$E_H$
is finitely generated over
$\mathbb{Z}_p$
. This proves the lemma.
Lemma 8.4.
Let
$E$
be
$E_1$
or
$E_2$
,
$K_{\infty } \in \mathcal{H}$
and
$H=\textrm {Gal}(\mathbb{L}_{\infty }/K_{\infty })$
. Assume that
$X_f(E/\mathbb{L}_{\infty })$
is finitely generated over
${\mathbb{Z}_p}[[H]]$
and that
$E(\mathbb{L}_{\infty })[p^{\infty }]$
is finite. Then
$\mu (X(E/K_{\infty }))=m_0(X(E/\mathbb{L}_{\infty }))$
.
Proof.
Since
$X_f(E/\mathbb{L}_{\infty })$
is finitely generated over
${\mathbb{Z}_p}[[H]]$
, it is easy to see that
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda$
-torsion. Since
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion and
$E(\mathbb{L}_{\infty })[p^{\infty }]$
is finite, it follows from [Reference Hachimori and Venjakob25, Theorem 7.2] that
$H^2(G_S(\mathbb{L}_{\infty }), E[p^{\infty }])=0$
and that the map
\begin{equation*}H^1(G_S(\mathbb{L}_{\infty }), E[p^{\infty }]) \longrightarrow \bigoplus _{v \in S} J_v(E/\mathbb{L}_{\infty })\end{equation*}
is surjective. Here,
$J_v(E/\mathbb{L}_{\infty })=\mathop {\varinjlim }\limits \bigoplus _{w|v} H^1(F_w,E)[p^{\infty }]$
where the direct limit runs over the finite extensions
$F$
of
$K$
contained in
$\mathbb{L}_{\infty }$
.
Consider the map
\begin{equation*}s_{K_{\infty }}\,:\, X(E/\mathbb{L}_{\infty })_H \longrightarrow X(E/K_{\infty })\end{equation*}
whose dual is induced by restriction. Since
$K_{\infty } \in \mathcal{H} \subseteq \mathcal{E}_p \cap \mathcal{E}_{ns}$
, it follows from [Reference Kleine and Matar35, Lemma 3.14, Corollary 3.15] that
$\ker s_{K_{\infty }}$
and
$\textrm {coker} s_{K_{\infty }}$
are finitely generated torsion
$\Lambda$
-modules with
$\mu =0$
and therefore are finitely generated over
$\mathbb{Z}_p$
.
The previous results mentioned allow us to use Lim’s result [Reference Lim38, Proposition 4.7]:
\begin{equation} m_0(X(E/\mathbb{L}_{\infty }))=\mu (X(E/K_{\infty })) + \sum _{i=0}^{d-2} (-1)^{i+1} \mu (H_i(H, X_f(E/\mathbb{L}_{\infty }))). \end{equation}
Note that Lim proved this result when
$K_{\infty }$
is the cyclotomic
$\mathbb{Z}_p$
-extension of
$K$
. The same proof works for arbitrary
$K_{\infty } \in \mathcal{H}$
because as mentioned above
$\ker s_{K_{\infty }}$
and
$\textrm {coker} s_{K_{\infty }}$
are finitely generated over
$\mathbb{Z}_p$
.
By assumption,
$X(E/\mathbb{L}_\infty )_f$
is finitely generated over
$\Lambda (H) = {\mathbb{Z}}_p[[H]]$
. Therefore for any
$i \geq 0$
, we have that
$H_i(H, X_f(E/\mathbb{L}_{\infty }))$
is finitely generated over
$\mathbb{Z}_p$
(see the proof of [Reference Howson26, Theorem 1.1]). So from (16), we have
$\mu (X(E/K_{\infty }))=m_0(X(E/\mathbb{L}_{\infty }))$
. This completes the proof.
The following criterion for the
$\mathfrak{M}_H(G)$
-property generalizes part of the main result from [Reference Kleine, Matar and Sujatha36] from the setting of
${\mathbb{Z}}_p^2$
-extensions to
${\mathbb{Z}}_p^d$
-extensions,
$d \ge 2$
.
Lemma 8.5.
Let
$E$
be an elliptic curve defined over
$K$
, with good ordinary reduction at
$p$
. Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension such that
$E(\mathbb{L}_\infty )[p^\infty ]$
is finite. If
$d \gt 2$
, then we moreover assume that the maximal pseudo-null
$\Lambda _d$
-submodule of
$X(E/\mathbb{L}_\infty )$
is finitely generated over
${\mathbb{Z}}_p$
.
Write the characteristic power series of
$X(E/\mathbb{L}_\infty )$
as
$f = p^{m_0(X(E/\mathbb{L}_\infty ))} g$
. Let
$K_\infty \in \mathcal{H}$
,
$H = \textrm {Gal}(\mathbb{L}_\infty /K_\infty )$
and let
\begin{equation*} \pi \colon \Lambda _d \twoheadrightarrow \Lambda = {\mathbb{Z}}_p[[\textrm {Gal}(K_\infty /K)]]\end{equation*}
be the canonical surjection.
Then the following conditions are equivalent.
-
(a)
$X(E/\mathbb{L}_\infty )$
has the
$\mathfrak{M}_H(G)$
-property.
-
(b)
$\mu (X(E/K_\infty )) = m_0(X(E/\mathbb{L}_\infty ))$
. -
(c)
$\pi (g) \not \equiv 0 \pmod {p}$
.
Proof.
$(a) \Longrightarrow (b)$
: Lemma 8.4.
$(b) \Longrightarrow (c)$
: Let
$X = X(E/\mathbb{L}_\infty )$
for brevity, and write
$X_\pi = X/\ker (\pi )$
. Since
$K_\infty \in \mathcal{H}$
, we have that
$X_\pi$
is torsion as a
$\Lambda$
-module. Therefore, we can apply Corollary 7.4(b) to
$X$
and
$X_\pi$
.
Suppose now that
$\pi (g) \equiv 0 \pmod {p}$
. Then
$\mu (X_H) \gt m_0(X)$
by Corollary 7.4(b). But since
$K_\infty \in \mathcal{H}$
and therefore no prime in
$S$
is totally split in
$K_\infty /K$
, it follows from [Reference Kleine and Matar35, Corollary 3.15(b)] that
$\mu (X_H) = \mu (X(E/K_\infty ))$
.
$(c) \Longrightarrow (a)$
: This follows from Lemma 8.3.
Corollary 8.6.
In the setting of the previous lemma suppose that the equivalent conditions hold for some
$K_\infty \in \mathcal{H}$
. Then there exists a Greenberg neighborhood
$U$
of
$K_\infty$
such that the conditions are satisfied for each
$\tilde {K}_\infty \in U$
.
Proof.
We have seen in Lemma 7.1 that the validity of condition
$(c)$
from Lemma 8.5 implies that this condition is true for each
$\tilde {K}_\infty$
in a certain neighborhood
$U$
of
$K_\infty$
. Note that Lemma 7.1 has been proven for Greenberg Selmer groups. However, since
$K_\infty \in \mathcal{H} \subseteq \mathcal{E}_p \cap \mathcal{E}_{ns}$
, it follows from Proposition 2.2 and 2.3(c) that
$U$
can be chosen small enough to ensure that
$X^{Gr}(E/\tilde {K}_\infty ) = X(E/\tilde {K}_\infty )$
for each
$\tilde {K}_\infty \in U$
.
We can now prove the main result of this section. One should compare this result to results of Lim (see [Reference Lim39, Proposition 5.1.3] and the main result from [Reference Lim41]), which need additional hypotheses. Moreover, our approach will also yield Corollary 8.8.
Theorem 8.7.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension, and let
$K_\infty \in \mathcal{H}$
. Write
$H = \textrm {Gal}(\mathbb{L}_\infty /K_\infty )$
. We assume that
-
(i) both
$E_1(\mathbb{L}_\infty )[p^\infty ]$
and
$E_2(\mathbb{L}_\infty )[p^\infty ]$
are finite,
-
(ii)
$E_1[p^{i}] \cong E_2[p^{i}]$
as Galois modules, where
$i \gt m_{0,1} = m_0(X(E_1/\mathbb{L}_\infty ))$
.
If
$d \gt 2$
, then we moreover assume that the maximal pseudo-null
$\Lambda _d$
-submodules of
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
are finitely generated
${\mathbb{Z}}_p$
-modules.
If the
$\mathfrak{M}_H(G)$
-property holds for
$E_1$
, then it also holds for
$E_2$
.
Proof.
Let
$K_\infty = \mathbb{L}_\infty ^H$
. First, since
$K_\infty \in \mathcal{H}$
by assumption and thus
$X(E_i/K_\infty )$
are
$\Lambda$
-torsion,
$i = 1,2$
, it follows from [Reference Kleine and Matar35, Lemma 3.17] that both
$X(E_i/\mathbb{L}_\infty )$
,
$i = 1,2$
, are finitely generated and
$\Lambda _d$
-torsion. Moreover, since
$H \in \mathcal{H}$
, it follows from [Reference Kleine32, Theorem 4.11] that we can choose a neighbourhood
$U$
of
$K_\infty$
such that
$X(E_i/\tilde {K}_\infty )$
is
$\Lambda$
-torsion for both
$i = 1,2$
, and
\begin{equation*} \mu (X(E_1/\tilde {K}_\infty )) \le \mu (X(E_1/K_\infty )), \quad \mu (X(E_2/\tilde {K}_\infty )) \le \mu (X(E_2/K_\infty ))\end{equation*}
hold for each
$\tilde {K}_\infty \in U$
. In addition, we may assume that
$U \subseteq \mathcal{E}_p \cap \mathcal{E}_{ns}$
(indeed, it suffices if
$U = \mathcal{E}(K_\infty ,n)$
has been chosen small enough such that each prime of
$K$
above
$p$
ramifies in the
$n$
-th layer
$K_n$
of
$K_\infty$
and such that no prime of
$S$
is totally split in
$K_n/K$
. Both conditions can be ensured for large
$n$
since
$K_\infty \in \mathcal{H}$
).
Now suppose that the
$\mathfrak{M}_H(G)$
-property holds for
$E_1$
. Then it follows from Lemma 8.5 and Corollary 8.6 that
\begin{equation*} \mu (X(E_1/\tilde {K}_\infty )) = m_{0,1}\end{equation*}
for each
$\tilde {K}_\infty \in U$
(it might be necessary here to make
$U$
slightly smaller).
Since
$U \subseteq \mathcal{E}_p \cap \mathcal{E}_{ns}$
, we have by Propositions 2.2 and 2.3 that
$X(E_i/\tilde {K}_{\infty })=X^{Gr}(E_i/\tilde {K}_{\infty })$
for each
$\tilde {K}_{\infty } \in U$
. Therefore, Theorem4.3(b) implies that for any
$\tilde {K}_\infty \in U$
we have a pseudo-isomorphism
\begin{equation*} X(E_1/\tilde {K}_\infty )[p^\infty ] \longrightarrow X(E_2/\tilde {K}_\infty )[p^\infty ]. \end{equation*}
In particular,
$\mu (X(E_1/\tilde {K}_\infty )) = \mu (X(E_2/\tilde {K}_\infty )) = m_{0,1}$
for each
$\tilde {K}_\infty \in U$
.
As we have seen in the proof of Lemma 5.1, we have
$\mu (X(E_2/F)) = m_{0,2}$
for each
$F \in \mathcal{E}_p \cap \mathcal{E}_{ns}$
which is not contained in a finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions of
$K$
(here we again apply Propositions 2.2 and 2.3). Therefore from Lemma 7.8, we see that
$U$
contains a
${\mathbb{Z}}_p$
-extension
$F$
of
$K$
such that
$\mu (X(E_2/F)) = m_{0,2} \,:\!=\, m_0(X(E_2/\mathbb{L}_\infty ))$
. This concludes the proof of the theorem. Indeed, it follows that
\begin{equation*} \mu (X(E_2/\tilde {K}_\infty )) = \mu (X(E_2/F)) = m_{0,2}\end{equation*}
for each
$\tilde {K}_\infty \in U$
, that is, the
$\mathfrak{M}_{\tilde {H}}(G)$
-property holds for
$E_2$
and every
$\tilde {H}= \textrm {Gal}(\mathbb{L}_\infty /\tilde {K}_\infty )$
in view of Lemma 8.5. In particular, it holds for
$\tilde {K}_\infty = K_\infty$
.
In the above proof, we also showed the following
Corollary 8.8.
Let
$\mathbb{L}_\infty$
be as in Theorem
8.7
. We assume that
$\mathcal{H} \ne \emptyset$
. Then the
$\mathfrak{M}_H(G)$
-property holds for a dense subset of
$\mathcal{E}$
.
Proof.
If
$\mathcal{H} \ne \emptyset$
, then the decomposition field of any
$v \in S$
is contained in some
${\mathbb{Z}}_p^{d-1}$
-extension of
$K$
, and the inertia subfield
$\mathbb{L}^{(v)} \subseteq \mathbb{L}_\infty$
of any
$v \mid p$
is contained in a
${\mathbb{Z}}_p^{d-1}$
-extension of
$K$
. Moreover, it follows from [Reference Kleine and Matar35, Lemma 3.17] that both
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
are finitely generated and
$\Lambda _d$
-torsion. Therefore, [Reference Kleine and Matar35, Lemma 3.1 and Corollary 3.15] (see also the proof of Lemma 5.1) imply that
$X(E_i/K_\infty )$
is
$\Lambda$
-torsion and
$\mu (X(E_i/K_\infty )) = m_0(X(E_i/\mathbb{L}_\infty ))$
for both
$i$
as long as
$K_\infty$
is not contained in a certain finite number of
${\mathbb{Z}}_p^{d-1}$
-extensions.
The assertion now follows from Lemma 7.8.
9. Pseudo-null submodules
Now we prove two sufficient criteria for the hypothesis (ii) of Theorem1.5 on the maximal pseudo-null submodules of
$X(E_1/\mathbb{L}_\infty )$
and
$X(E_2/\mathbb{L}_\infty )$
. In fact, we describe two settings where these pseudo-null submodules are even trivial.
Lemma 9.1.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension,
$d \ge 3$
, and let
$E$
be an elliptic curve defined over
$K$
which has good ordinary reduction at
$p$
. We assume that the following conditions are met.
-
(i) Every prime
$v \in S_p$
ramifies in
$\mathbb{L}_{\infty }/K$
, -
(ii)
$E(\mathbb{L}_\infty )[p^\infty ]$
is finite,
-
(iii)
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion,
-
(iv) The decomposition subgroups
$D_v(\mathbb{L}_\infty /K) \subseteq \textrm {Gal}(\mathbb{L}_\infty /K)$
are open for each
$v \in S_p$
, and
-
(v)
$E$
has good reduction everywhere.
Then
$X(E/\mathbb{L}_\infty )$
does not contain any nontrivial pseudo-null submodules.
Proof.
In view of the hypotheses (ii) and (iii), it follows from [Reference Hachimori and Venjakob25, Theorem 7.2] that the weak Leopoldt conjecture holds for
$E$
over
$\mathbb{L}_\infty$
, that is, that
$H^2(\textrm {Gal}(K_S/\mathbb{L}_\infty ), E[p^\infty ]) = 0$
. Condition (i) implies by Propositions 2.2 and 2.3 that
$X(E/\mathbb{L}_{\infty })=X^{str}(E/\mathbb{L}_{\infty })$
. Therefore, we prove the desired result for
$X^{str}(E/\mathbb{L}_{\infty })$
.
Since the elliptic curve
$E$
has good reduction everywhere, the assertion of the lemma follows from [Reference Lim41, Proposition 5.1]. Indeed, since
$E$
has good reduction everywhere, we can take the set
$S$
to be the primes of
$K$
above
$p$
. Let
$S_{\infty }$
be the set of primes of
$\mathbb{L}_{\infty }$
above those in
$S$
. By assumption (iv), the set
$S_{\infty }$
is finite. By [Reference Hachimori and Venjakob25, Theorem 7.2], we have an exact sequence
\begin{equation*} 0 \longrightarrow \textrm {Sel}^{str}(E/\mathbb{L}_\infty ) \longrightarrow H^1(G_S(\mathbb{L}_\infty ), E[p^\infty ]) \longrightarrow \bigoplus _{w \in S_{\infty }} H^1(\mathbb{L}_{\infty ,w}, \tilde {E}[p^{\infty }])\longrightarrow 0.\end{equation*}
Here,
$\tilde {E}$
denotes the reduction of
$E$
over the residue field.
Now we conclude as in the proof of [Reference Lim41, Proposition 5.1]: Taking Pontryagin duals of the above exact sequence, we get
\begin{equation*} 0 \longrightarrow \left ( \bigoplus _{w \in S_{\infty }} H^1(\mathbb{L}_{\infty ,w}, \tilde {E}[p^{\infty }]) \right )^\vee \longrightarrow H^1(G_S(\mathbb{L}_\infty ), E[p^\infty ])^\vee \longrightarrow X^{str}(E/\mathbb{L}_\infty ) \longrightarrow 0. \end{equation*}
The first module in this exact sequence is reflexive by [Reference Ochi and Venjakob48, Lemma 5.4] (this again uses assumption
$(iv)$
). It follows from the validity of the weak Leopoldt conjecture that
$H^1(G_S(\mathbb{L}_\infty ), E[p^\infty ])^\vee$
does not contain any nonzero pseudo-null
$\Lambda _d$
-submodules (see [Reference Ochi and Venjakob48, Theorem 4.7]). Therefore, the same holds true for
$X^{str}(E/\mathbb{L}_\infty )$
by [Reference Hachimori and Ochiai24, Proposition 3.5].
Lemma 9.2.
Let
$\mathbb{L}_\infty /K$
be a
${\mathbb{Z}}_p^d$
-extension,
$d \ge 2$
, and let
$E$
be an elliptic curve defined over
$K$
which has good ordinary reduction at
$p$
. We assume that the following conditions are met.
-
(i) Every prime of
$K$
above
$p$
ramifies in
$\mathbb{L}_{\infty }/K$
and no prime in
$S$
splits completely in
$\mathbb{L}_{\infty }/K$
, -
(ii)
$E(K)[p]=0$
, -
(iii)
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion.
Then
$X(E/\mathbb{L}_\infty )$
does not contain any nontrivial pseudo-null submodules.
Proof.
The proof is based on the work of Greenberg [Reference Greenberg19]. Condition (i) implies by Propositions 2.2 and 2.3 that
$X(E/\mathbb{L}_{\infty })=X^{str}(E/\mathbb{L}_{\infty })$
. Therefore, we prove the desired result for
$X^{str}(E/\mathbb{L}_{\infty })$
.
Let us denote the absolute Galois group of
$K$
by
$G_K$
, recall that
$G\,:\!=\,\textrm {Gal}(\mathbb{L}_{\infty }/K)$
, and
$\Lambda _d={\mathbb{Z}_p}[[G]]$
. Let
$\Psi \,:\, G_K \twoheadrightarrow G \hookrightarrow \Lambda _d^{\times }$
be the natural character and let
$\Lambda _d(\Psi ^{-1})$
be the free rank one
$\Lambda _d$
-module with
$G_K$
action by
$\Psi ^{-1}$
. Denote the
$p$
-adic Tate module of
$E$
by
$T_p(E)$
. We define
\begin{equation*}\mathcal{T}\,:\!=\,T_p(E) \otimes _{{\mathbb{Z}_p}} \Lambda _d(\Psi ^{-1}) \quad \text{and} \quad \mathcal{D}\,:\!=\,\mathcal{T} \otimes _{\Lambda _d} \Lambda _d^{\vee }, \end{equation*}
where
$\Lambda _d^{\vee }$
is the Pontryagin dual of
$\Lambda _d$
. We let
$G_K$
act diagonally on
$\mathcal{T}$
and with left action on
$\mathcal{D}$
.
Now let
$v$
be a prime of
$K$
above
$p$
. Let
$\overline {K}_v$
be an algebraic closure of
$K_v$
and
$k_v$
the residue field. We let
$\tilde {E}(k_v)$
be the reduction of
$E$
over the residue field and denote by
$E^1(\overline {K}_v)$
the kernel of the reduction map
$E(\overline {K}_v) \longrightarrow \tilde {E}(k_v)$
. We let
$T^+_{v,p}(E)$
and
$T^-_{v,p}(E)$
to be the Tate modules of
$E^1(\overline {K}_v)$
and
$\tilde {E}(k_v)$
, respectively. Then similarly to the above, we define
\begin{equation*}\mathcal{T}_v^{\pm }\,:\!=\,T^{\pm }_{v,p}(E) \otimes _{{\mathbb{Z}_p}} \Lambda _d(\Psi ^{-1}) \quad \text{and} \quad \mathcal{D}_v^{\pm }\,:\!=\,\mathcal{T}_v^{\pm } \otimes _{\Lambda _d} \Lambda _d^{\vee }. \end{equation*}
It follows from Shapiro’s lemma (see [Reference Skinner and Urban58, Prop. 3.4]) that
$\textrm {Sel}^{str}(E/\mathbb{L}_{\infty })$
is
$G$
-isomorphic to the kernel of the map
\begin{equation*}H^1(G_S(K), \mathcal{D}) \longrightarrow \prod _{v \in S \setminus S_p} H^1(K_v, \mathcal{D}) \times \prod _{v \in S_p} H^1(K_v, \mathcal{D}_v^-). \end{equation*}
Let
$v \in S_p$
. It follows from [Reference Neukirch, Schmidt and Wingberg47, Theorem 7.1.8(i)] and Shapiro’s lemma that
$H^2(K_v, \mathcal{D}_v^+)=0$
. Therefore, if we let
\begin{equation*} L(K_v, \mathcal{D})=\textrm {img}(H^1(K_v, \mathcal{D}_v^+) \longrightarrow H^1(K_v, \mathcal{D})), \end{equation*}
we see that
$H^1(K_v, \mathcal{D}_v^-)=H^1(K_v, \mathcal{D})/L(K_v, \mathcal{D})$
.
For any
$v \in S \setminus S_p$
define
$L(K_v, \mathcal{D})=0$
. Let
$P(K, \mathcal{D})=\prod _{v \in S} H^1(K_v, \mathcal{D})$
and
$L(K, \mathcal{D})=\prod _{v \in S} L(K_v, \mathcal{D})$
. With these definitions let
\begin{equation*} Q_{\mathcal{L}}(K, \mathcal{D})=P(K, \mathcal{D})/L(K, \mathcal{D}).\end{equation*}
From the above, we see that
$\textrm {Sel}(E/\mathbb{L}_{\infty })$
is
$G$
-isomorphic to
$S_{\mathcal{L}}(K, \mathcal{D})$
which is defined as:
\begin{equation*}0 \longrightarrow S_{\mathcal{L}}(K, \mathcal{D}) \longrightarrow H^1(G_S(K), \mathcal{D}) \longrightarrow Q_{\mathcal{L}}(K, \mathcal{D}). \end{equation*}
We now list various hypotheses in Greenberg’s paper:
-
•
$\text{RFX}(\mathcal{D})$
: The module
$\mathcal{T}$
is a reflexive
$\Lambda _d$
-module. -
•
$\text{LOC}_v^{(1)}(\mathcal{D})\,:\, (\mathcal{T}^{\vee })^{G_{K_v}}=0$
for
$v \in S$
. -
•
$\text{LOC}_v^{(2)}(\mathcal{D})$
: The
$\Lambda _2$
-module
$\mathcal{T}^{\vee }/(\mathcal{T}^{\vee })^{G_{K_v}}$
is reflexive for
$v \in S$
. -
•
$\text{LEO}(\mathcal{D})$
: The discrete, co-finitely generated
$\Lambda _d$
-moduleis cotorsion.
\begin{equation*}\mathrm{III}(K, S, \mathcal{D})=\ker \Big (H^2(G_S(K), \mathcal{D}) \longrightarrow \prod _{v \in S} H^2(K_v, \mathcal{D}) \Big )\end{equation*}
-
•
$\text{CRK}(\mathcal{D}, \mathcal{L})$
: We have
\begin{equation*}\textrm {corank}_{\Lambda _d}(H^1(G_S(K), \mathcal{D}))=\textrm {corank}_{\Lambda _d}(S_{\mathcal{L}}(K, \mathcal{D}))+\textrm {corank}_{\Lambda _d}(Q_{\mathcal{L}}(K, \mathcal{D})).\end{equation*}
Greenberg calls a discrete
$\Lambda _d$
-module
$M$
almost divisible if
$\Pi M=M$
for almost all height one prime ideals
$\Pi$
in
$\text{Spec}(\Lambda _d)$
. This is equivalent to the Pontryagin dual of
$M$
having no nonzero pseudo-null submodules. Using the notation from Greenberg’s paper [Reference Greenberg19], we say that
$\mathcal{L}$
is almost
$\Lambda _d$
-divisible if
$L(K_v, \mathcal{D})$
is almost
$\Lambda _d$
-divisible for each
$v \in S$
.
Let
$\mathfrak{m}$
be the maximal ideal of
$\Lambda _2$
. Proposition 4.1.1 of Greenberg’s paper [Reference Greenberg19] states that
$S_{\mathcal{L}}(K, \mathcal{D})$
is almost divisible if the following are met:
-
(a)
$\text{RFX}(\mathcal{D})$
,
$\text{LEO}(\mathcal{D})$
, and
$\text{CRK}(\mathcal{D}, \mathcal{L})$
are satisfied. -
(b)
$\mathcal{L}$
is almost
$\Lambda _d$
-divisible. -
(c)
$\text{LOC}_v^{(2)}(\mathcal{D})$
is satisfied for all
$v \in S$
, and there exists a nonarchimedean prime
$v \in S$
such that
$\text{LOC}_v^{(1)}(\mathcal{D})$
is satisfied. -
(d)
$\mathcal{D}$
is a co-free
$\Lambda _d$
-module, and
$\mathcal{D}[\mathfrak{m}]$
has no quotient isomorphic to
$\mu _p$
for the action of
$G_K$
.
We now check each of the conditions above. Clearly,
$\text{RFX}(\mathcal{D})$
is true. Since
$E(K)[p]=0$
, we have
$E(\mathbb{L}_{\infty })[p^{\infty }]=0$
and so
$\text{LEO}(\mathcal{D})$
and
$\text{CRK}(\mathcal{D}, \mathcal{L})$
follow from [Reference Hachimori and Venjakob25, Theorem 7.2]. This takes care of (a). Condition (b) is true by [Reference Greenberg19, Prop. 4.3.2]. Since no prime
$v \in S$
splits completely in
$\mathbb{L}_{\infty }/K$
, we get by [Reference Greenberg18, Lemma 5.2.2] that
$\text{LOC}_v^{(1)}(\mathcal{D})$
is satisfied for all
$v \in S$
. This implies that
$\text{LOC}_v^{(2)}(\mathcal{D})$
is satisfied for all
$v \in S$
. Therefore, (c) is true. Finally, as explained on pg. 48 of [Reference Greenberg19], properties of the Weil pairing
$E[p] \times E[p] \longrightarrow \mu _p$
imply that if
$E(K)[p]=0$
, then (d) is satisfied. This completes the proof.
10. Examples for Theorems 1.4 and 1.5
In this final section, we show two examples that satisfy the conditions of Theorems1.4 and 1.5. First, we fix some notation. As in the introduction, let
$p$
be a an odd prime. If
$F$
is a number field, let
$F_{cyc}$
denote the cyclotomic
$\mathbb{Z}_p$
-extension of
$\mathbb{Q}$
. If
$E$
is an elliptic curve over
$\mathbb{Q}$
and
$L/{\mathbb{Q}}$
is an algebraic extension, we let
$\textrm {Sel}(E/L)$
denote the
$p$
-primary classical Selmer group for
$E/L$
and we let
$X(E/L)$
denote its Pontryagin dual.
Now let
$E/F\,:\, y^2=x^3+ax+b$
be an elliptic curve. If
$d$
is a non-square in
$F$
, then the quadratic twist of
$E$
by
$d$
, denoted
$E^d$
, is the elliptic curve defined by the equation
$E^d\,:\, y^2=x^3+d^2ax+bd^3$
. We have the following well-known result.
Lemma 10.1.
Let
$F$
be a number field,
$E$
be an elliptic curve over
$F$
and
$d$
a non-square in
$F$
. Let
$L=F(\sqrt {d})$
. Then we have an isomorphism
\begin{equation*}\textrm {Sel}(E/L) \cong \textrm {Sel}(E/F) \oplus \textrm {Sel}(E^d/F). \end{equation*}
Proof. See the proof of [Reference Ono, Papanikolas and Peters49, Lemma 3.1].
In the setup of the previous lemma, we note that since
$p$
is odd, the extensions
$F_{cyc}/F$
and
$L/F$
are linearly disjoint over
$F$
. Therefore, we may identify the Galois groups
$\textrm {Gal}(L_{cyc}/L)=\textrm {Gal}(F_{cyc}/F)=\Gamma _{cyc}$
. The proof of loc. cit, shows that the map defining the isomorphism of the previous lemma commutes with
$\Gamma _{cyc}$
. Therefore, we have
Lemma 10.2.
Let
$F$
be a number field,
$E$
be an elliptic curve over
$F$
and
$d$
a non-square in
$F$
. Let
$L=F(\sqrt {d})$
. Then we have an isomorphism of
${\mathbb{Z}_p}[[\Gamma _{cyc}]]$
-modules
\begin{equation*}X(E/L_{cyc}) \cong X(E/F_{cyc}) \oplus X(E^d/F_{cyc}). \end{equation*}
For one of our examples, we need a similar result to the previous lemma but for quartic twists. First, we define these. Let
$F$
be a number field and let
\begin{equation} E\,:\, y^2=x^3+ax \end{equation}
be an elliptic curve over
$F$
with
$j$
-invariant
$1728$
. If
$d \in F^{\times }$
, then the quartic twist of
$E$
by
$d$
, denoted
$E_d$
, is the elliptic curve defined by the equation
\begin{equation*} E_d\,:\, y^2=x^3+dax. \end{equation*}
Let
$\zeta$
be a primitive fourth root of unity in
$\bar {{\mathbb{Q}}}$
.
Lemma 10.3.
Let
$F$
be a number field containing
$\zeta$
,
$E$
an elliptic curve over
$F$
with
$j$
-invariant
$1728$
and
$d \in F^{\times }$
. Let
$L=F(\sqrt [4]{d})$
and assume that
$[L:F]=4$
. Then we have an isomorphism
\begin{equation*}\textrm {Sel}(E/L) \cong \textrm {Sel}(E/F) \oplus \textrm {Sel}(E_d/F) \oplus \textrm {Sel}(E_{d^2}/F) \oplus \textrm {Sel}(E_{d^3}/F). \end{equation*}
Proof.
Let
$G=\textrm {Gal}(L/F)$
and choose
$c \in L$
with
$c^4=d$
. Note that
$G$
is a cyclic group of order 4. Let
$\sigma$
be a generator of
$G$
such that
$\sigma (c)=\zeta c$
. Let
$[\zeta ] \in \textrm {Aut}(E)$
be defined as
$[\zeta ](x,y)=(\zeta ^2x, \zeta y)$
. This makes
${\textrm {Sel}_p}(E/L)$
a
${\mathbb{Z}}[\mu ]$
-module, where
$\mu$
is the group of fourth roots of unity. We define
\begin{equation*}\textrm {Sel}(E/L)^i=\{s \in \textrm {Sel}(E/L) \, | \, \sigma (s)=[\zeta ^i]_*(s)\}. \end{equation*}
We can write
$\textrm {Sel}(E/L)$
as a direct sum of eigenspaces
\begin{equation*}\textrm {Sel}(E/L)=\textrm {Sel}(E/L)^0 \oplus \textrm {Sel}(E/L)^1 \oplus \textrm {Sel}(E/L)^2 \oplus \textrm {Sel}(E/L)^3. \end{equation*}
Let
$E\,:\, y^2=x^3+ax$
, so that
$E_{d^i}\,:\, y^2=x^3+d^iax$
. Then we have an isomorphism defined over
$L$
:
\begin{align*}\phi ^i\,:\, E \longrightarrow E_{d^i},\end{align*}
defined as
$\phi ^i(x,y)=(c^{2i}x, c^{3i}y)$
. Then we have
\begin{equation*}\sigma (\phi ^i(x,y))=\sigma (c^{2i}x,c^{3i}y)=(\zeta ^{2i}c^{2i}\sigma (x), \zeta ^{3i}c^{3i}\sigma (y))=\phi ^i([\zeta ^{-i}]\sigma (x,y)).\end{equation*}
This shows that for
$0 \leq i \leq 3$
, we have an isomorphism
\begin{equation*}\textrm {Sel}(E/L)^i \cong \textrm {Sel}(E_{d^i}/F)^G. \end{equation*}
The restriction map induces a map
$\psi ^i\,:\, \textrm {Sel}(E_{d^i}/F) \longrightarrow \textrm {Sel}(E_{d^i}/F)^G$
. Using the snake lemma and the inflation restriction sequence (see e.g. the proof of [Reference Hachimori and Matsuno22, Lemma 3.3]), we see that the kernel and cokernel of the map
$\psi ^i$
are both annihilated by
$p$
and
$4$
. Since
$p$
is odd, the map
$\psi ^i$
is an isomorphism. By putting the above facts together, we get the desired isomorphism in the statement of the lemma.
If
$L/F$
is an extension of number fields with
$[L\,:\,F]=4$
, then since
$p$
is odd, the extensions
$F_{cyc}/F$
and
$L/F$
are linearly disjoint over
$F$
. Therefore, we may identify the Galois groups
$\textrm {Gal}(L_{cyc}/L)=\textrm {Gal}(F_{cyc}/F)=\Gamma _{cyc}$
. All the maps in the previous lemma commute with
$\Gamma _{cyc}$
. Therefore, we get:
Lemma 10.4.
Let
$F$
be a number field containing
$\zeta$
,
$E$
an elliptic curve over
$F$
with
$j$
-invariant
$1728$
and
$d \in F^{\times }$
. Let
$L=F(\sqrt [4]{d})$
and assume that
$[L:F]=4$
. Then we have an isomorphism of
${\mathbb{Z}_p}[[\Gamma _{cyc}]]$
-modules
\begin{equation*}X(E/L_{cyc}) \cong X(E/F_{cyc}) \oplus X(E_d/F_{cyc}) \oplus X(E_{d^2}/F_{cyc}) \oplus X(E_{d^3}/F_{cyc}). \end{equation*}
We would like to prove a similar result to the above in the case when
$\zeta \not \in L$
. To do this, we use a trick similar to [Reference Kim30, Proposition 4.2] using [Reference Sato56, Corollary 3.5].
Lemma 10.5.
Let
$F$
be a number field,
$E$
an elliptic curve over
$F$
with
$j$
-invariant
$1728$
and
$d \in F^{\times }$
. Let
$L=F(\sqrt [4]{d})$
and assume that
$[L\,:\,F]=4$
and
$\zeta \not \in L$
. Then we have the following relations of Iwasawa invariants of
$\Lambda \,:\!=\,{\mathbb{Z}_p}[[\Gamma _{cyc}]]$
-modules.
-
(1)
\begin{equation*}\textrm {rank}_{\Lambda }(X(E/L_{cyc}))=\sum _{i=0}^3 \textrm {rank}_{\Lambda }(X(E_{d^i}/F_{cyc})). \end{equation*}
-
(2) If the
$\Lambda$
-ranks in (1) are all zero, then
\begin{equation*}\mu (X(E/L_{cyc}))=\sum _{i=0}^3 \mu (X(E_{d^i}/F_{cyc})). \end{equation*}
Proof.
Let
$L'=L(\zeta )$
and
$F'=F(\zeta )$
. Then
$[L':F']=4$
and
$\zeta \in F'$
. Therefore by Lemma 10.4, we have
\begin{equation} X(E/L'_{cyc}) \cong X(E/F'_{cyc}) \oplus X(E_d/F'_{cyc}) \oplus X(E_{d^2}/F'_{cyc}) \oplus X(E_{d^3}/F'_{cyc}). \end{equation}
In order to get a result over
$L_{cyc}$
and
$F_{cyc}$
, we use the following observation: If
$E$
is an elliptic curve over a field
$K$
with
$j$
-invariant
$1728$
and
$\zeta \not \in K$
, then its quadratic twist
$E^{-1}$
is equal to
$E$
(recall that
$b = 0$
in equation (17)) and hence from Lemma 10.2 we have
\begin{equation*}X(E/K_{cyc}(\zeta )) \cong X(E/K_{cyc})\oplus X(E/K_{cyc}). \end{equation*}
Using this with each of the terms in the isomorphism (18), we obtain the desired result.
For the remainder of this section, let
$\mathbb{L}_{\infty }$
be a fixed
$\mathbb{Z}_p^d$
-extension of
$K$
that contains
$K_{cyc}$
. Theorems1.4 and 1.5 require that we know
$m_{0,1}$
. As we will explain later, we use the Iwasawa main conjecture to find the value of
$\mu (X(E/K_{cyc}))$
. The following proposition then allows us to get an upper bound on
$m_{0,1}$
.
Proposition 10.6.
Let
$E$
be an elliptic curve defined over
$K$
. Assume that
$X(E/K_{cyc})$
is
$\Lambda$
-torsion. Then we have
-
(1)
$X(E/\mathbb{L}_{\infty })$
is
$\Lambda _d$
-torsion,
-
(2)
$m_0(X(E/\mathbb{L}_{\infty })) \leq \mu (X(E/K_{cyc}))$
.
Proof.
Let
$H=\textrm {Gal}(\mathbb{L}_{\infty }/K_{cyc})$
. By [Reference Kleine and Matar35, Corollary 3.15], we have
\begin{equation*}\textrm {rank}_{\Lambda }(X(E/K_{cyc}))=\textrm {rank}_{\Lambda }(X(E/\mathbb{L}_{\infty }))_H).\end{equation*}
By [Reference Lim40, Lemma 4.7], the hypothesis that
$X(E/K_{cyc})$
is
$\Lambda$
-torsion therefore implies that
\begin{equation*}\textrm {rank}_{\Lambda _d}(X(E/\mathbb{L}_\infty )) =0.\end{equation*}
Therefore, we get (1).
The proof of Corollary 7.4 shows that the image of the characteristic power series of
$X(E/\mathbb{L}_{\infty })$
under the canonical surjection
$\pi \,:\, \Lambda _d \longrightarrow \Lambda$
divides the characteristic power series of
$X(E/\mathbb{L}_{\infty })_H$
. Since no prime splits completely in
$K_{cyc}/K$
, loc. cit. shows that
$\mu (X(E/K_{cyc}))=\mu (X(E/\mathbb{L}_{\infty }))_H)$
. We then deduce (2) from these facts.
Let
$\Gamma _{cyc}=\textrm {Gal}({\mathbb{Q}}_{cyc}/{\mathbb{Q}})$
and fix the Iwasawa algebra
$\Lambda \,:\!=\,{\mathbb{Z}_p}[[\Gamma _{cyc}]]={\mathbb{Z}_p}[[T]]$
. Let
$E$
be an elliptic curve defined over
$\mathbb{Q}$
with good ordinary reduction at
$p$
. Let
$\mathcal{L}_p(E)$
be the
$p$
-adic L-function of
$E$
defined by Mazur and Swinnerton-Dyer [Reference Mazur and Swinnerton-Dyer43]. By results of Kato [Reference Kato27] and Rohrlich [Reference Rohrlich52],
$X(E/{\mathbb{Q}}_{cyc})$
is a torsion
$\Lambda$
-module. Let
$f_{cyc} \in {\mathbb{Z}_p}[[T]]$
be the characteristic polynomial of
$X(E/{\mathbb{Q}}_{cyc})$
. The Iwasawa main conjecture is the following
Conjecture 10.7.
We have an equality of ideals in
${\mathbb{Z}_p}[[T]]$
:
$(f_{cyc})=(\mathcal{L}_p(E))$
.
Assuming that
$\textrm {Gal}({\mathbb{Q}}(E[p])/{\mathbb{Q}})=GL_2({\mathbb{F}_p})$
, Kato [Reference Kato27, Theorem 17.4] shows the inclusion of ideals
$(\mathcal{L}_p(E)) \subseteq (f_{cyc})$
. Assuming that there is a prime
$q$
where
$E$
has multiplicative reduction and which ramifies in
${\mathbb{Q}}(E[p])/{\mathbb{Q}}$
, Skinner and Urban [Reference Skinner and Urban58] establish the reverse inclusion thus proving Conjecture 10.7. Other cases that establish Conjecture 10.7 are Kim-Kim-Sun [Reference Kim, Kim and Sun29] and Wan [Reference Wan59]. For our examples, we need two remaining cases listed below.
Theorem 10.8.
If
$E$
has complex multiplication, then Conjecture 10.7 is true.
Proof. This is proven by Rubin [Reference Rubin53, Theorem 12.3].
If
$E$
has an isogeny of degree
$p$
defined over
$\mathbb{Q}$
, we have an isogeny character
$\phi \,:\, G_{{\mathbb{Q}}} \longrightarrow {\mathbb{F}_p}^{\times }$
. This is the character describing the action of
$G_{{\mathbb{Q}}}=\textrm {Gal}(\bar {{\mathbb{Q}}}/{\mathbb{Q}})$
on the kernel
$C \subseteq E[p]$
of the underlying
$p$
-isogeny. Let
$\chi \,:\, G_{{\mathbb{Q}}} \longrightarrow {\mathbb{F}_p}^{\times }$
be the mod-
$p$
cyclotomic character and let
$G_p \subseteq G_{{\mathbb{Q}}}$
be a decomposition group at
$p$
. We have the following theorem.
Theorem 10.9.
Assume that
$E$
has an isogeny of degree
$p$
defined over
$\mathbb{Q}$
. Assume that
$\phi |_{G_p}\neq 1, \chi$
. Then Conjecture 10.7 is true.
Proof. This is proven by Castella, Grossi, and Skinner [Reference Castella, Grossi and Skinner5, Theorem A].
We first show an example for Theorem1.4. Let
$E_1\,:\, y^2+xy+y=x^3+296x+1702$
be the elliptic curve 110c2 with Cremona labeling [Reference Cremona10]. Let
$K={\mathbb{Q}}(\sqrt {-7})$
and
$\mathbb{L}_{\infty }$
be the
$\mathbb{Z}_3^2$
-extension of
$K$
so that
$p=3$
. The quadratic twist of
$E_1$
by
$-7$
is
$(E_1)^{-7}\,:\, y^2 + xy = x^3 + x^2 + 14528x - 569344$
(5390s2). Both
$E_1$
and
$(E_1)^{-7}$
have good ordinary reduction at
$3$
. Computations in SAGE [55] and data from the LMFDB database [42] show that
$E_1$
and
$(E_1)^{-7}$
have isogenies of degree
$3$
with characters of their isogenies satisfying Theorem10.9. Also we have
$\mu (\mathcal{L}(E_1))=1$
and
$\mu (\mathcal{L}((E_1)^{-7}))=0$
. Therefore by Theorem10.9, we have
$\mu (X(E_1/{\mathbb{Q}}_{cyc}))=1$
and
$\mu (X((E_1)^{-7}/{\mathbb{Q}}_{cyc}))=0$
. By Lemma 10.2, this implies that
$\mu (X(E_1/K_{cyc}))=1$
. As
$K/{\mathbb{Q}}$
is an abelian extension results of Kato [Reference Kato27] and Rohrlich [Reference Rohrlich52] imply that
$X(E_1/K_{cyc})$
is
$\Lambda$
-torsion. Lemma 10.2 then implies that
$\mu (X(E_1/K_{cyc}))=1$
. Proposition 10.6 in turn implies that
$X(E_1/\mathbb{L}_{\infty }))$
is a torsion
$\Lambda _2$
-module with
$m_{0,1} \leq 1$
.
Now consider the elliptic curve
\begin{equation*}E_2\,:\, y^2+xy+y=x^3-41279995994x+3252655872050692 \; \; \text{(60610m2)}.\end{equation*}
From the tables in Tom Fisher’s paper [Reference Fisher13],
$E_1$
and
$E_2$
are
$9$
-congruent. Since
$3$
is unramified in
$K/{\mathbb{Q}}$
, it is clear that condition
$(\Delta )$
in the introduction is satisfied (note that by Remark 1.6 we actually do not have to check condition
$(\Delta )$
here because
$E_1$
and
$E_2$
are defined and congruent over
$\mathbb{Q}$
). Finally since
$K_{cyc} \subseteq \mathbb{L}_{\infty }$
and
$K_{cyc} \in \mathcal{E}_p \cap \mathcal{E}_{ns}$
, this data satisfies the conditions of Theorem1.4.
Now we show an example for Theorem1.5. Let
$E_1\,:\, y^2=x^3+x$
be the elliptic curve 64a4. Let
$K={\mathbb{Q}}(\sqrt [4]{-13})$
and let
$\mathbb{L}_{\infty }$
be the
$\mathbb{Z}_5^3$
-extension of
$K$
so that
$p=5$
. Now consider the three quartic twists of
$E_1$
:
$(E_1)_{-13}\,:\, y^2=x^3-13x$
(5408e1),
$(E_1)_{169}\,:\, y^2=x^3+169x$
(10816bb1), and
$(E_1)_{-2197}\,:\, y^2=x^3-2197x$
(5408k1).
$E_1$
and its three quartic twists have good ordinary reduction at
$5$
. In the LMFDB database, we see that the
$\mu$
-invariants of the 5-adic L-functions of all four curves are zero. As all of these curves have complex multiplication, Theorem10.8 shows that
$\mu (X(A/{\mathbb{Q}}_{cyc}))=0$
where
$A$
is any of the four curves. By the results of Kato [Reference Kato27] and Rohrlich [Reference Rohrlich52]
$X(A/{\mathbb{Q}}_{cyc})$
is
$\Lambda$
-torsion for each curve
$A$
. Therefore by Lemma 10.5 we have that
$X(E_1/K_{cyc})$
is
$\Lambda$
-torsion with
$\mu$
-invariant zero. Then by Proposition 10.6
$X(E_1/\mathbb{L}_{\infty })$
is
$\Lambda _3$
-torsion with
$m_{0,1}=0$
.
Now consider the elliptic curve
\begin{equation*}E_2\,:\, y^2 = x^3 - 262140x^2 + 638353031830801x - 134326647870701706899652.\end{equation*}
This elliptic curve is the curve with
$D=-1$
and
$t=4$
described in [Reference Rubin and Silverberg54, Theorem 5.3]. Loc. cit. shows that
$E_1$
and
$E_2$
are
$5$
-congruent which implies that condition (iv) of Theorem1.5 is satisfied. Let
$S$
be the set of primes of
$K$
above
$p$
and where either
$E_1$
or
$E_2$
have bad reduction. Our computations in SAGE reveal that both
$E_1$
and
$E_2$
satisfy condition
$({\star})$
for all
$v \in S$
. The prime
$5$
is inert in
$K/{\mathbb{Q}}$
which implies that condition (iii) is satisfied. Also we have
$E_1(K)[5]=E_2(K)[5]=0$
. This implies that
$E_1(\mathbb{L}_{\infty })[5^{\infty }]=E_2(\mathbb{L}_{\infty })[5^{\infty }]=0$
so condition (i) is satisfied. Since
$K_{cyc} \subseteq \mathbb{L}_{\infty }$
and
$K_{cyc} \in \mathcal{E}_p \cap \mathcal{E}_{ns}$
, we also see that the conditions of Lemma 9.2 are satisfied for both
$E_1$
and
$E_2$
(note that
$X(E_2/\mathbb{L}_{\infty })$
is
$\Lambda _3$
-torsion because
$X(E_1/\mathbb{L}_{\infty })$
is
$\Lambda _3$
-torsion and Theorem1.4 applies). Thus condition (ii) is satisfied. Finally since
$5$
is unramified in
$K/{\mathbb{Q}}$
, it is clear that condition
$(\Delta )$
in the introduction is satisfied.
Acknowledgements
We would like to thank Tom Fisher, Sam Frengley, and Meng Fai Lim for helpful correspondences during the preparation of this article. Moreover, we are grateful to the anonymous referee and to the editor Alex Bartel for providing valuable comments and suggestions.
Competing interests
The authors declare none.
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