# Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and
Kida’s Formula

Debanjana Kundu  Department of Mathematics
University of British Columbia
Vancouver BC, V6T 1Z2, Canada<EMAIL_ADDRESS>and  Anwesh Ray  Department
of Mathematics
University of British Columbia
Vancouver BC, V6T 1Z2, Canada<EMAIL_ADDRESS>
###### Abstract.

This paper aims at studying the Iwasawa $\lambda$-invariant of the $p$-primary
Selmer group. We study the growth behaviour of $p$-primary Selmer groups in
$p$-power degree extensions over non-cyclotomic $\mathbb{Z}_{p}$-extensions of
a number field. We prove a generalization of Kida’s formula in such a case.
Unlike the cyclotomic $\mathbb{Z}_{p}$-extension, where all primes are
finitely decomposed; in the $\mathbb{Z}_{p}$-extensions we consider, primes
may be infinitely decomposed. In the second part of the paper, we study the
relationship for Iwasawa invariants with respect to congruences, obtaining
refinements of the results of R. Greenberg-V. Vatsal and K. Kidwell. As an
application, we provide an algorithm for constructing elliptic curves with
large anticyclotomic $\lambda$-invariant. Our results are illustrated by
explicit computation.

###### Key words and phrases:

$\lambda$-invariant, Kida’s formula

###### 2010 Mathematics Subject Classification:

Primary 11R23

##  1\. Introduction

The classical Riemann-Hurwitz formula describes the relationship of the Euler
characteristics of two surfaces when one is a ramified covering of the other.
Suppose $\pi\mathrel{\mathop{\ordinarycolon}}R_{1}\rightarrow R_{2}$ is an
$n$-fold covering of compact, connected Riemann surfaces and $g_{1},\ g_{2}$
are their respective genus. The classical Riemann-Hurwitz formula is the
statement

$2g_{2}-2=(2g_{1}-2)n+\sum\left(e(P_{2})-1\right)$

where the sum is over all points $P_{2}$ on $R_{2}$ and $e(P_{2})$ denotes the
ramification index of $P_{2}$ for the covering $\pi$ (see [33, Chapter II
Theorem 5.9]). An analogue of the above formula for algebraic number fields
was proven by Y. Kida in [20]. Kida’s formula describes the change of
(cyclotomic) Iwasawa $\lambda$-invariants in a $p$-extension in terms of the
degree and the ramification index. In [18], K. Iwasawa proved this formula
using the theory of Galois cohomology for extensions of $\mathbb{Q}$ which are
not necessarily finite. More precisely,

######  Theorem 1 ([18, Theorem 6]).

Let $p\geq 2$ and $K$ be a number field. Let $K^{\operatorname{cyc}}$ be the
cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and
$\mathcal{L}/K^{\operatorname{cyc}}$ be a cyclic extension of degree $p$,
unramified at every infinite place of $K_{\operatorname{cyc}}$. Assume that
the classical $\mu$-invariant, $\mu(K^{\operatorname{cyc}})=0$. Then

$\lambda(\mathcal{L})=p\lambda(K^{\operatorname{cyc}})+\sum_{w^{\prime}}\left(e(w^{\prime}|v^{\prime})-1\right)+(p-1)(h_{2}-h_{1}).$

The sum is over all primes $w^{\prime}$ of $\mathcal{L}$ (above $v^{\prime}$
in $K^{\operatorname{cyc}}$) not above $p$, $h_{i}$ is the rank of the abelian
group $H^{i}(\mathcal{L}/K_{\operatorname{cyc}},U(\mathcal{L}))$, and
$U(\mathcal{L})$ is the group of all units of $\mathcal{L}$.

In the study of rational points of abelian varieties (in particular, elliptic
curves), the Selmer group plays a crucial role. In [25], B. Mazur initiated
the study of the growth of the $p$-primary Selmer group in
$\mathbb{Z}_{p}$-extension of number fields. In [17], Y. Hachimori and K.
Matsuno proved an analogue of Kida’s formula for $p$-primary Selmer groups of
elliptic curves in $p$-power extensions of the _cyclotomic_
$\mathbb{Z}_{p}$-extension of a number field. This result was extended to a
general class of Galois representations including the case of $p$-ordinary
Hilbert modular forms and $p$-supersingular modular forms by R. Pollack and T.
Weston in [28]. In the first part of this article, we prove a Kida-like
formula in _more general_ $\mathbb{Z}_{p}$-extensions (see Theorem 4.3).

For ease of exposition, in the introduction, we state our main result in a
simplified setting. Let $p\geq 5$ be a fixed rational prime. Let
$E_{/\mathbb{Q}}$ be a non-CM elliptic curve with good ordinary reduction at
$p$ and $K$ be a fixed imaginary quadratic field. Let $K^{\operatorname{ac}}$
denote the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$, $L/K$ be a Galois
extension of $p$-power degree disjoint from $K^{\operatorname{ac}}$, and
$L_{\infty}$ be the compositum $L\cdot K^{\operatorname{ac}}$. Suppose that
the $p$-primary Selmer group over $K^{\operatorname{ac}}$ (and $L_{\infty}$)
is cofinitely generated as a $\mathbb{Z}_{p}$-module. Then our result relates
their respective $\lambda$-invariants.

######  Theorem 2.

With the setting as above,

$\lambda(E/L_{\infty})=[L_{\infty}\mathrel{\mathop{\ordinarycolon}}K^{\operatorname{ac}}]\lambda(E/K^{\operatorname{ac}})+\sum_{w^{\prime}\in\mathcal{P}_{1}}\left(e(w^{\prime}|v^{\prime})-1\right)+\sum_{w^{\prime}\in\mathcal{P}_{2}}2\left(e(w^{\prime}|v^{\prime})-1\right)$

where $e(w^{\prime}|v^{\prime})$ is the ramification index of $w^{\prime}$
above a prime $v^{\prime}$ (in $K^{\operatorname{ac}}$) and the sets
$\mathcal{P}_{1},\mathcal{P}_{2}$ are as in Definition 4.1.

We remark that the $p$-primary Selmer group need _not always_ be
$\Lambda$-cotorsion over the anticyclotomic $\mathbb{Z}_{p}$-extension,
$K^{\operatorname{ac}}$ (see [3, 10, 36]). However, by the work of V. Vatsal
[35], M. Bertolini-H. Darmon [4], and Pollack-Weston [29, Theorem 5.3], it is
known that if some very natural conditions are satisfied, then the $p$-primary
Selmer group is not only $\Lambda$-cotorsion, but also cofinitely generated as
a $\mathbb{Z}_{p}$-module. We show (in Corollary 4.10) that this property
propagates to all $p$-power extensions $L$ of $K$ (which are disjoint from
$K^{\operatorname{ac}}$) satisfying the additional condition that if $E$ has
split multiplicative reduction at $\ell$ and $v|\ell$ is inert in $K$, then
$v$ splits completely in $L$. We prove such results for
$\mathbb{Z}_{p}$-extensions of number fields.

Two elliptic curves $E_{1}$ and $E_{2}$ over $K$ are said to be $p$-congruent
if their associated residual representations are isomorphic, i.e., $E_{1}[p]$
and $E_{2}[p]$ are isomorphic as $\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muK}/K)$-modules. There is much interest in
investigating the behaviour of Iwasawa invariants for congruent Galois
representations. Such investigations were initiated by Greenberg and Vatsal in
[16]. They considered the case when both $E_{1}$ and $E_{2}$ are defined over
$\mathbb{Q}$, and showed how the (cyclotomic) Iwasawa $\mu$ and
$\lambda$-invariants for the $p$-primary Selmer groups of $E_{1}$ and $E_{2}$
are related. Let $\Sigma_{0}$ be the set of primes $v\nmid p$ of $F$ at which
$E_{1}$ or $E_{2}$ has bad reduction. Greenberg and Vatsal compare the
$\Sigma_{0}$-imprimitive Selmer groups of $E_{1}$ and $E_{2}$. These results
were conditionally generalized by K. Kidwell for $\mathbb{Z}_{p}$-extensions
of number fields (see [21, Theorem 6.1]). This requires a finiteness
assumption, which is satisfied if the set of primes $\Sigma_{0}$ are finitely
decomposed in $K_{\infty}$. We refine the results of Greenberg-Vatsal and
Kidwell in two ways. We show that the set of primes $\Sigma_{0}$ can be
replaced by a smaller set $\Omega_{0}$ (see Definition 6.1). Having a smaller
set of primes $\Omega_{0}$ to consider makes for a more refined relationship
between $\lambda$-invariants for the Selmer groups of $E_{1}$ and $E_{2}$

(1.1)
$\lambda(E_{1}/K_{\infty})-\lambda(E_{2}/K_{\infty})=\sum_{v\in\Omega_{0}}\left(\sigma_{E_{2}}^{(v)}-\sigma_{E_{1}}^{(v)}\right),$

where $\sigma_{E_{i}}^{(v)}$ is the corank of a certain locally defined
$\Lambda$-module (see Definition 6.13). We also clarify the assumptions on the
splitting of primes in $\mathbb{Z}_{p}$-extensions. These assumptions apply
only to the smaller set of primes $\Omega_{0}$ and not to the set of all
primes $\Sigma_{0}$. The approach used here is similar to that of the second
named author and R. Sujatha, who prove such results for the cyclotomic
$\mathbb{Z}_{p}$-extension in [30].

As an application of our results on congruences, we provide an algorithm for
constructing elliptic curves whose Selmer group has a large
$\lambda$-invariant. Such elliptic curves are considered over the
anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic field. In
[24], this question has been studied for some small primes (i.e., $p\leq 7$ or
$p=13$) for elliptic curves with good _ordinary_ reduction at $p$, over the
cyclotomic $\mathbb{Z}_{p}$-extension (of $\mathbb{Q}$). In [22], elliptic
curves over $\mathbb{Q}$ with _supersingular_ reduction at $p=3$ and
arbitrarily large (plus/minus) $\lambda$-invariant are constructed. The latter
method is more amenable for generalization to our setting. In Section 7, we
fix the elliptic curve $E=32a2$ (Cremona label) and the prime $p=5$. Using the
work of K. Rubin and A. Silverberg (see [31]), we have an algorithm to produce
elliptic curves which are 5-congruent to $E$ with large anticyclotomic
$\lambda$-invariant over $K=\mathbb{Q}(\sqrt{-1})$. Here, we crucially use the
formula (1.1), which relates $\lambda$-invariants of the Selmer groups of
elliptic curves in Rubin-Silverberg family. We illustrate our algorithm via
explicit computation.

##  2\. Basic Notions

Let $p$ be an odd prime number and $K$ be a number field. Let $K_{\infty}$ be
any $\mathbb{Z}_{p}$-extension of $K$. Two examples of interest are as
follows:

  1. (1)

$K$ is a number field and $K_{\infty}=K^{\operatorname{cyc}}$ is the
_cyclotomic_ $\mathbb{Z}_{p}$-extension of $K$. This is the unique
$\mathbb{Z}_{p}$-extension of $K$ contained in $K(\mu_{p^{\infty}})$.

  2. (2)

$K$ is an imaginary quadratic field and $K_{\infty}=K^{\operatorname{ac}}$ is
the _anticyclotomic_ $\mathbb{Z}_{p}$-extension of $K$. This is a
$\mathbb{Z}_{p}$-extension of $K$ which is Galois and pro-dihedral over
$\mathbb{Q}$. More generally, if $K$ is any CM field, $K_{\infty}$ may be
chosen to be $K^{\operatorname{ac}}$.

If $K$ is any number field with signature $(r_{1},r_{2})$, the Leopoldt’s
conjecture predicts that the $\mathbb{Z}_{p}$-rank of the maximal abelian
pro-$p$ extension of $K$ unramified away from primes above $p$ is $1+r_{2}$.
In other words, it predicts that there are $1+r_{2}$ _independent_
$\mathbb{Z}_{p}$-extensions of $K$. For an imaginary quadratic field, the
cyclotomic and anticyclotomic extensions together generate a
$\mathbb{Z}_{p}^{2}$-extension, containing infinitely many
$\mathbb{Z}_{p}$-extensions that are unramified at all primes $v\nmid p$ and
not Galois over $\mathbb{Q}$.

Set $\Gamma\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(K_{\infty}/K)$
and
$\Gamma_{n}\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(K_{\infty}/K_{n})$.
The Iwasawa algebra $\Lambda$ is the completed group ring
$\varprojlim_{n}\mathbb{Z}_{p}\llbracket\Gamma/\Gamma_{n}\rrbracket$. Choose a
topological generator $\gamma\in\Gamma$. There is an isomorphism
$\Lambda\simeq\mathbb{Z}_{p}\llbracket T\rrbracket$ associating the formal
variable $T$ with $\gamma-1$.

For a discrete $p$-primary abelian group $M$, set
$M^{\vee}\mathrel{\mathop{\ordinarycolon}}=\operatorname{Hom}_{\operatorname{cnts}}(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$
to denote its _Pontryagin dual_. We say that $M$ is cofinitely generated
(resp. cotorsion) $\Lambda$-module if $M^{\vee}$ is finitely generated (resp.
torsion) as a $\Lambda$-module. For a cofinitely generated and cotorsion
$\Lambda$-module $M$, the _Structure Theorem of $\Lambda$-modules_ asserts
that $M^{\vee}$ is pseudo-isomorphic to a finite direct sum of cyclic
$\Lambda$-modules, i.e., there is a map of $\Lambda$-modules

$M^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\Lambda/(p^{m_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(h_{j}(T))\right)$

with finite kernel and cokernel. Here, $m_{i}>0$ and $h_{j}(T)$ is a
distinguished polynomial (i.e. a monic polynomial with non-leading
coefficients divisible by $p$). The _characteristic ideal_ of $M^{\vee}$ is
(up to a unit) generated by the characteristic element,

$f_{M}^{(p)}(T)\mathrel{\mathop{\ordinarycolon}}=p^{\sum_{i}m_{i}}\prod_{j}h_{j}(T).$

The $\mu$-invariant of $M$ is defined as the power of $p$ in $f_{M}^{(p)}(T)$.
More precisely,

$\mu(M)\mathrel{\mathop{\ordinarycolon}}=\begin{cases}0&\textrm{ if }s=0\\\
\sum_{i=1}^{s}m_{i}&\textrm{ if }s>0.\end{cases}$

The $\lambda$-invariant of $M$ is the degree of the characteristic element,
i.e.

$\lambda(M)\mathrel{\mathop{\ordinarycolon}}=\sum_{j=1}^{t}\deg h_{j}(T).$

######  Remark 2.1.

Write $\Sigma$ for the set of primes of $K$ that are finitely decomposed in
$K_{\infty}$. For $K_{\infty}=K^{\operatorname{cyc}}$, all primes are finitely
decomposed. In this case, $\Sigma$ consists of all primes.

Consider the case when $K$ is an imaginary quadratic field and
$K_{\infty}=K^{\operatorname{ac}}$. Let $\ell$ be a rational prime and
$v|\ell$ be a prime of $K$. Then, there are finitely many primes $w$ of
$K^{\operatorname{ac}}$ lying above $v$ if and only if either $\ell=p$ or
$\ell$ splits in $K$ (see for example [5]). Thus, it follows from the
Cheboratev density theorem that $\Sigma$ consists of $50\%$ of the primes of
$K$.

Henceforth, we fix a number field $K$ and a $\mathbb{Z}_{p}$-extension
$K_{\infty}$ over $K$. Let $E_{/K}$ be an elliptic curve with good ordinary
reduction at $p$. We now define our module of interest, i.e., the $p$-primary
Selmer group over $K_{\infty}$. Let $S$ be a finite set of primes $v$ of $K$
containing the primes above $p$ and the primes of bad reduction of $E$. Denote
by $K_{S}$ the maximal algebraic extension of $K$ which is unramified at the
primes $v\notin S$. For a field $F$ such that $K\subseteq F\subseteq K_{S}$
and a prime $v\in S$, define

$J_{v}(E/F)\mathrel{\mathop{\ordinarycolon}}=\prod_{w|v}H^{1}\left(F_{w},E\right)[p^{\infty}]$

where the product is over all primes $w$ of $F$ lying above $v$. The _$p$
-primary Selmer group over $F$_ is defined as follows

$\operatorname{Sel}(E/F)\mathrel{\mathop{\ordinarycolon}}=\ker\left\\{H^{1}\left(K_{S}/F,E[p^{\infty}]\right)\longrightarrow\bigoplus_{v\in
S}J_{v}(E/F)\right\\}.$

The $p$-primary Selmer group $\operatorname{Sel}(E/K_{\infty})$ is the direct
limit of $\operatorname{Sel}(E/F)$ as $F$ ranges over all finite extensions of
$K$ contained in $K_{\infty}$. The $\mu$ and $\lambda$-invariants of
$\operatorname{Sel}(E/K_{\infty})$ are denoted by $\mu(E/K_{\infty})$ and
$\lambda(E/K_{\infty})$. We make the following assumption.

######  Assumption 2.2.

$\operatorname{Sel}(E/K_{\infty})$ is a cotorsion $\Lambda$-module with
$\mu(E/K_{\infty})=0$.

######  Remark 2.3.

Such an assumption is expected to hold in the special cases of interest.

  1. (1)

Consider the case when $E$ is an elliptic curve defined over $\mathbb{Q}$ and
$K_{\infty}=K^{\operatorname{cyc}}$ for an abelian extension $K/\mathbb{Q}$.
The $p$-primary Selmer group $\operatorname{Sel}(E/K_{\infty})$ is a cotorsion
$\Lambda$-module; see [19]. When $E$ is an elliptic curve for which the
residual Galois representation
$\overline{\rho}\mathrel{\mathop{\ordinarycolon}}\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muK}/K)\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is irreducible, it is conjectured by Greenberg that the
$\mu(E/K^{\operatorname{cyc}})=0$ (see [13, Conjecture 1.11]).

  2. (2)

Consider the case when $E_{/\mathbb{Q}}$ is an elliptic curve, $K$ is an
imaginary quadratic number field, and $K_{\infty}=K^{\operatorname{ac}}$. Let
$N$ be the conductor of $E$. Write $N=N^{+}N^{-}$, where $N^{+}$ (resp.
$N^{-}$) is divisible by primes that split (resp. are inert) in $K$. Let
$\overline{\rho}\mathrel{\mathop{\ordinarycolon}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
be the residual Galois representation on the $p$-torsion points $E[p]$.
Suppose that the following conditions are satisfied

    1. (a)

$\overline{\rho}$ is surjective,

    2. (b)

if $q$ is a prime such that $q|N^{+}$ and $q\equiv 1\mod{p}$, then
$\overline{\rho}$ is ramified at $q$,

    3. (c)

the number of primes dividing $N^{-}$ is odd.

Then Assumption 2.2 holds; see [29, Theorem 5.3].

##  3\. Preliminary Results

As before, let $p$ be an odd prime number and $K$ be a number field. Let $L$
be a finite Galois extension of $K$ such that the order of the Galois group
$G=\operatorname{Gal}(L/K)$ is a power of $p$. Let $K_{\infty}$ be a
$\mathbb{Z}_{p}$-extension of $K$ and
$L_{\infty}\mathrel{\mathop{\ordinarycolon}}=K_{\infty}\cdot L$. The fields
are depicted in the diagram below

$\mathbb{Q}$.$K$$L$$K_{\infty}$$L_{\infty}$G$\mathbb{Z}_{p}$

In the two examples of special interest, we know the following.

  1. (1)

When $K$ is a number field and $K_{\infty}=K^{\operatorname{cyc}}$ is the
_cyclotomic_ $\mathbb{Z}_{p}$-extension of $K$, $L_{\infty}$ is identified
with the cyclotomic $\mathbb{Z}_{p}$-extension of $L$, namely
$L^{\operatorname{cyc}}$.

  2. (2)

When $K$ is an imaginary quadratic field and
$K_{\infty}=K^{\operatorname{ac}}$ is the _anticyclotomic_
$\mathbb{Z}_{p}$-extension of $K$, the $\mathbb{Z}_{p}$-extension,
$L_{\infty}$ over $L$ is a non-cyclotomic extension (but it is not an
anticyclotomic extension).

In this section, we record some preliminary results required in the proof of
Theorem 2. For the remainder of this discussion, we will assume the following.

######  Assumption 3.1.

  1. (1)

$E$ has good ordinary reduction at the primes of $K$ above $p$,

  2. (2)

at least one of the following hold

    1. (a)

$E$ does not have complex multiplication,

    2. (b)

$E(L)[p]=0$.

The assumption of $E(L)[p]=0$ is weaker than insisting that the residual
representation
$\overline{\rho}\mathrel{\mathop{\ordinarycolon}}\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muL}/L)\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
on $E[p]$ is irreducible.

######  Lemma 3.2.

Let $K$ be a number field and $E_{/K}$ be an elliptic curve without complex
multiplication. Let $\mathcal{L}/K$ be any algebraic extension which is Galois
over $K$. Then, either

  1. (a)

$E(\mathcal{L})[p^{\infty}]=E[p^{\infty}]$ _or_

  2. (b)

$E(\mathcal{L})[p^{\infty}]$ is finite.

###### Proof.

Set $M=E(\mathcal{L})[p^{\infty}]$ and $T_{p}(M)=\varprojlim_{n}M[p^{n}]$.
Note that
$V_{p}(M)\mathrel{\mathop{\ordinarycolon}}=T_{p}(M)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$
is a submodule of
$V_{p}(E)\mathrel{\mathop{\ordinarycolon}}=T_{p}(E)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$
where $T_{p}(E)$ is the (usual) Tate module of $E$. Since $E$ is an elliptic
curve without complex multiplication, by a theorem of Serre (see [32, IV,
Theorem 2.1(a)]) we know that $V_{p}(E)$ (hence $V_{p}(M)$) is irreducible.
Thus, $V_{p}(M)$ is either trivial or equal to $V_{p}(E)$. Since $\dim
V_{p}(M)=\operatorname{corank}_{\mathbb{Z}_{p}}M$, it follows that
$V_{p}(M)=0$ if and only if $M$ is finite. But, $E[p^{\infty}]$ is cofree, so
$V_{p}(M)=V_{p}(E)$ if and only if $M=E[p^{\infty}]$. This completes the proof
of the lemma. ∎

######  Corollary 3.3.

Let $K$ be a number field and $K_{\infty}$ be a $\mathbb{Z}_{p}$-extension of
$K$. Let $E$ be an elliptic curve over $K$. Assume that one of the following
conditions are satisfied

  1. (1)

$E$ does not have complex multiplication,

  2. (2)

$E(K)[p]=0$.

Then, $E(K_{\infty})[p^{\infty}]$ is finite.

###### Proof.

  1. (1)

When $E$ does not have complex multiplication, it follows from Serre’s big
image theorem that the field $K(E[p^{\infty}])$ is not a solvable extension of
$K$. Therefore, $E(K_{\infty})[p^{\infty}]\neq E[p^{\infty}]$. The assertion
follows from Lemma 3.2.

  2. (2)

Since $\Gamma$ is pro-$p$, it follows that $E(K_{\infty})[p]=0$ (see for
example [26, 1.6.13]). Hence, $E(K_{\infty})[p^{\infty}]=0$.

∎

######  Lemma 3.4.

Let $p\geq 3$. Suppose that $\operatorname{Sel}(E/K_{\infty})$ is
$\Lambda$-cotorsion and $E(K_{\infty})[p^{\infty}]$ is finite. Then the map

(3.1)
$H^{1}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)\longrightarrow\bigoplus_{v\in
S}J_{v}(E/K_{\infty})$

is surjective. Furthermore,
$H^{2}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)=0$.

###### Proof.

Recall the Poitou–Tate sequence

$\displaystyle 0\rightarrow\operatorname{Sel}(E/K_{\infty})$
$\displaystyle\rightarrow
H^{1}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)\xrightarrow{\theta}\bigoplus_{v\in
S}J_{v}(E/K_{\infty})$
$\displaystyle\rightarrow\left(\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})\right)^{\vee}\rightarrow
H^{2}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)\rightarrow 0.$

Here
$\mathfrak{S}_{p}(E/K_{n})\mathrel{\mathop{\ordinarycolon}}=\varprojlim_{m}\operatorname{Sel}_{p^{m}}(E/K_{n})$,
and the inverse limit is taken with respect to the maps induced by
multiplication by $p$. The inverse limit of $\mathfrak{S}_{p}(E/K_{n})$ is
taken with respect to the corestriction map. In order to prove the result, it
suffices to show that $\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})=0$.

Consider the exact sequence (see [9, Lemma 1.8])

$0\rightarrow
E(K_{n})[p^{\infty}]\rightarrow\mathfrak{S}_{p}(E/K_{n})\rightarrow\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}(E/K_{n})^{\vee},\mathbb{Z}_{p}\right)\rightarrow
0.$

Taking inverse limits, we have an exact sequence

$\varprojlim_{n}E(K_{n})[p^{\infty}]\rightarrow\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})\rightarrow\varprojlim_{n}\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}(E/K_{n})^{\vee},\mathbb{Z}_{p}\right).$

Note that the inverse limit $\varprojlim_{n}E(K_{n})[p^{\infty}]$ is taken
with respect to norm maps. Since $E(K_{\infty})[p^{\infty}]$ finite, it
follows that $\varprojlim_{n}E(K_{n})[p^{\infty}]=0$, and therefore, we have
an injection

$\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})\hookrightarrow\varprojlim_{n}\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}(E/K_{n})^{\vee},\mathbb{Z}_{p}\right).$

By an application of the Control Theorem, (see [14, Theorem 4.1] or [15,
Theorem 1]) we have an injection

$\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})\hookrightarrow\varprojlim_{n}\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}(E/K_{\infty})^{\vee}_{\Gamma_{n}},\mathbb{Z}_{p}\right).$

For a $\Lambda$-module $M$, we shall denote by $M^{\iota}$ the
$\Lambda$-module with the same underlying $\mathbb{Z}_{p}$-module and inverse
$\Gamma$-action. By [27, §2 Lemme 4(i)]), there is an isomorphism

$\varprojlim_{n}\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}(E/K_{\infty})^{\vee}_{\Gamma_{n}},\mathbb{Z}_{p}\right)\simeq\operatorname{Hom}_{\Lambda}\left(\operatorname{Sel}(E/K_{\infty})^{\vee},\Lambda\right)^{\iota}.$

Since $\operatorname{Sel}(E/K_{\infty})$ is a cotorsion $\Lambda$-module, it
follows that
$\operatorname{Hom}_{\Lambda}\left(\operatorname{Sel}(E/K_{\infty})^{\vee},\Lambda\right)^{\iota}=0$.
Therefore, $\varprojlim_{n}\mathfrak{S}_{p}(E/K_{n})=0$. The assertion of the
lemma follows. ∎

##  4\. Generalizations of Kida’s formula

In this section, we state a generalization of Kida’s formula which applies to
more general $\mathbb{Z}_{p}$-extensions. Let $K$ be a number field and
$K_{\infty}$ be any $\mathbb{Z}_{p}$-extension over $K$ such that all primes
above $p$ are ramified in $K_{\infty}$. Let $L$ be a finite Galois extension
of $K$ with Galois group $G=\operatorname{Gal}(L/K)$ of $p$-power order. Let
$E_{/K}$ be an elliptic curve such that the Assumption 3.1 holds. Recall that
$S$ is any finite set of primes $v$ of $K$ _containing_ the primes above $p$
and the primes of bad reduction of $E$. For our discussion on generalized
Kida-type formula, we choose the set $S$ to be the primes $v$ in $K$
satisfying at least one of the following conditions

  1. (1)

$v|p$,

  2. (2)

$E$ has bad reduction at $v$,

  3. (3)

$v$ is ramified in $L$.

Let $L_{\infty}$ denote the composite $K_{\infty}\cdot L$. We define two sets
of primes $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ in $L_{\infty}$, which will
play a crucial role in the generalization of Kida’s formula.

######  Definition 4.1.

For $i=1,2$, the set $\mathcal{P}_{i}$ consists of primes $w^{\prime}\nmid p$
in $L_{\infty}$ such that

  1. (1)

$w^{\prime}\in\mathcal{P}_{1}$ if $E$ has split multiplicative reduction at
$w^{\prime}$,

  2. (2)

$w^{\prime}\in\mathcal{P}_{2}$ if _all_ of the following conditions are
satisfied

    1. (a)

$w^{\prime}$ is ramified above a prime $v^{\prime}$ in $K_{\infty}$,

    2. (b)

$E$ has good reduction at $w^{\prime}$

    3. (c)

$E(L_{\infty,w^{\prime}})$ possesses a point of order $p$.

We introduce an important assumption.

######  Assumption 4.2.

Assume that each prime $w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$ lies
above a prime $v\in\Sigma$.

Note that any prime $w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$ must lie
above a prime $v\in S\cap\Sigma$. The above assumption guarantees that the
sets $\mathcal{P}_{i}$ are finite for $i=1,2$. Our choice of $\mathcal{P}_{1}$
is the same as that in [17], and $\mathcal{P}_{2}$ bears the additional
condition that the primes $w^{\prime}\in\mathcal{P}_{2}$ are ramified over
$K_{\infty}$. This does not make an actual difference in terms in the formula
relating the $\lambda$-invariants.

For any algebraic extension $\mathcal{L}$ of $K$, and a set of primes $T$ of
$K$, we will write $T(\mathcal{L})$ to denote the set of primes of $w$ of
$\mathcal{L}$ such that $w|v$ for primes $v\in T$. For a prime $v$ of $K$, we
write $v(\mathcal{L})$ for the set of primes $w|v$ of $\mathcal{L}$.

We record our first main result; its proof will occupy this section and the
next.

######  Theorem 4.3.

For each prime $w^{\prime}$ of $L_{\infty}$, write
$e_{w^{\prime}}(L_{\infty}/K_{\infty})$ for the ramification index of
$w^{\prime}$ for the extension $L_{\infty}/K_{\infty}$. The
$\lambda$-invariant of $\operatorname{Sel}(E/L_{\infty})$ is given by the
formula

$\lambda(E/L_{\infty})=[L_{\infty}\mathrel{\mathop{\ordinarycolon}}K_{\infty}]\lambda(E/K_{\infty})+\sum_{w^{\prime}\in\mathcal{P}_{1}}\left(e_{w^{\prime}}-1\right)+\sum_{w^{\prime}\in\mathcal{P}_{2}}2\left(e_{w^{\prime}}-1\right).$

Since $G$ is a $p$-group, it is solvable. Let $L_{0}\subset
L_{1}\subset\cdots\subset L_{M}$ be a filtration of $L$ such that $L_{i+1}$ is
Galois over $L_{i}$ with Galois group
$\operatorname{Gal}(L_{i+1}/L_{i})\simeq\mathbb{Z}/p\mathbb{Z}$. It is easy to
show that it suffices to prove Theorem 4.3 when $\\#G=p$. We assume from here
on in without loss of generality that $G\simeq\mathbb{Z}/p\mathbb{Z}$.

######  Lemma 4.4.

Let $F$ be a number field and $F_{\infty}$ be any $\mathbb{Z}_{p}$-extension
of $F$. Then $F_{\infty}$ is unramified at all primes $v\nmid p$.

###### Proof.

Let $F_{\infty}^{\prime}\subseteq F_{\infty}$ be the maximal subextension
which is unramified at all primes $v\nmid p$ and let
$H=\operatorname{Gal}(F_{\infty}/F_{\infty}^{\prime})$. Since
$\operatorname{Gal}(F_{\infty}/F)\simeq\mathbb{Z}_{p}$, the subgroup $H$ is
either trivial or a finite index subgroup. The latter case can not hold.
Indeed, if it is true then $F_{\infty}^{\prime}$ is a number field and
$F_{\infty}$ with a $\mathbb{Z}_{p}$-extension which is unramified at the
primes $v|p$. A direct application of class field theory shows that this is
not possible. Therefore, $F_{\infty}=F_{\infty}^{\prime}$ is unramified at all
primes $v\nmid p$. ∎

######  Lemma 4.5.

Let $F$ be a number field and $F_{\infty}$ be a $\mathbb{Z}_{p}$-extension of
$F$. Let $v\nmid p$ be a prime of $F$ and $v^{\prime}|v$ a prime of
$F_{\infty}$. The following assertions hold

  1. (1)

if $v$ is finitely decomposed in $F_{\infty}$, then the localization
$F_{\infty,v^{\prime}}=F_{v}^{\operatorname{cyc}}$.

  2. (2)

if there exists a Galois $p$-extension of $F_{v}$, then $F_{v}$ contains
$\mu_{p}$.

###### Proof.

  1. (1)

Since $v$ is finitely decomposed in $F_{\infty}$, we see that
$F_{\infty,v^{\prime}}$ is a $\mathbb{Z}_{p}$-extension of $F$. Lemma 4.4
asserts that $v$ is unramified in $F_{\infty}$. By local class field theory,
there is a unique unramified pro-extension of $F_{v}$ and therefore
$F_{\infty,v^{\prime}}=F_{v}^{\operatorname{cyc}}$.

  2. (2)

This statement is a direct consequence of local class field theory.

∎

Let $v$ (resp. $w$) be primes above $\ell$ in $K$ (resp. $L$) such that $w|v$.
Let $w^{\prime}|w$ be a prime of $L_{\infty}$ and
$v^{\prime}\mathrel{\mathop{\ordinarycolon}}=w^{\prime}_{\restriction
K_{\infty}}$. The diagram below depicts the labelling of primes

$\ell.$$v$$w$$v^{\prime}$$w^{\prime}$

######  Lemma 4.6.

Let $v\in S\setminus S_{p}$ and
$w^{\prime}\notin\mathcal{P}_{1}\cup\mathcal{P}_{2}$ be a prime of
$L_{\infty}$ above $v$, set $v^{\prime}$ and $w$ be as in the diagram above.
Then at least one of the following assertions hold.

  1. (1)

$L_{\infty,w^{\prime}}=K_{\infty,v^{\prime}}$,

  2. (2)

$E(L_{\infty,w^{\prime}})[p^{\infty}]=0$.

###### Proof.

Suppose that $L_{\infty,w^{\prime}}\neq K_{\infty,v^{\prime}}$, then it
follows that $L_{w}$ is a nontrivial $p$-extension of $K_{v}$ which is Galois
over $K_{v}$. Therefore by local class field theory, $\mu_{p}$ is contained in
$K_{v}$. Since $v\in S$, we have that either $v$ is ramified in $L$ or $E$ has
bad reduction at $v$. Consider the case when $E$ has good reduction at $v$,
thus $v$ is ramified in $L$. Note that since $v\nmid p$, it is unramified in
$K_{\infty}$ (see Lemma 4.4). Therefore,
$L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is a ramified extension. Recall
that $\mathcal{P}_{2}$ is the set of primes $w^{\prime}$ of $L_{\infty}$ at
which all of the following assumptions are satisfied:

  1. (1)

$L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is a ramified extension,

  2. (2)

$E$ has good reduction at $w^{\prime}$ and
$E(L_{\infty,w^{\prime}})[p^{\infty}]\neq 0$.

Note that since $E$ has good reduction at $v$, it follows that it has good
reduction at $w^{\prime}$. Since $L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$
is a ramified extension and $w^{\prime}\notin\mathcal{P}_{2}$, it follows that
$E(L_{\infty,w^{\prime}})[p^{\infty}]=0$.

Next consider the case when $E$ has bad reduction at $v$. Then there are two
subcases

  1. (i)

$E$ has good reduction at $w^{\prime}$,

  2. (ii)

$E$ has bad reduction at $w^{\prime}$.

First assume that $E$ has good reduction at $w^{\prime}$. In this case, it is
easy to see from the Neron Ogg Shafarevich criterion that
$L_{\infty,w^{\prime}}/K_{v}$ is ramified. Since $v$ is unramified in
$K_{\infty}$, it follows that $L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is
a ramified extension. In this setting the same argument as in the previous
case applies.

One is left with the case when $E$ has bad reduction at $w^{\prime}$. Since
$w^{\prime}\notin\mathcal{P}_{1}$, we have by assumption that $E$ has either
nonsplit multiplicative reduction or additive reduction at $w^{\prime}$. In
this setting, [17, Proposition 5.1 (iii)] applies and it follows that
$E(L_{\infty,w^{\prime}})[p^{\infty}]=0$. Note that this result may be applied
since $\mu_{p}$ is contained in $K_{v}$, which is one of the required
hypotheses. This has been shown to hold in the beginning of the proof. ∎

######  Lemma 4.7.

Let $v$ be any prime is $S$ which is not above $p$. Then the kernel of the map

$\gamma_{v}\mathrel{\mathop{\ordinarycolon}}J_{v}(E/K_{\infty})\rightarrow
J_{v}(E/L_{\infty})$

is finite.

###### Proof.

We begin by recalling that $v\nmid p$ may be infinitely decomposed in
$K_{\infty}/K$, i.e., the set $v(K_{\infty})$ may be infinite. At each prime
$v^{\prime}\in v(K_{\infty})$, choose a prime $w^{\prime}$ of $L_{\infty}$
such that $w^{\prime}|v^{\prime}$. It follows from the inflation-restriction
sequence that

(4.1) $\ker\gamma_{v}\subseteq\prod_{v^{\prime}\in
v(K_{\infty})}H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})\right)[p^{\infty}],$

the right hand side of which may possibly be an infinite product. Since
$v^{\prime}\nmid p$, we know that (see for example [14, Theorem 2.4(i)])

$E(K_{\infty,v^{\prime}})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}=0.$

The following isomorphism is immediate from the Kummer sequence,

(4.2)
$H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})\right)[p^{\infty}]\simeq
H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})[p^{\infty}]\right).$

Since $E(L_{\infty,w^{\prime}})[p^{\infty}]$ is a cofinite
$\mathbb{Z}_{p}$-module and
$G\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(L/K)$ is finite, the
right hand side of (4.2) is a finite group. To prove the lemma, we consider
two cases
_Case 1:_ $v\in\Sigma$. By definition of $\Sigma$, the set of primes
$v(K_{\infty})$ is finite. The result is therefore immediate.
_Case 2:_ $v\notin\Sigma$. Assumption 4.2 implies that
$w^{\prime}\notin\mathcal{P}_{1}\cup\mathcal{P}_{2}$. But,

(4.3)
$H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})[p^{\infty}]\right)=0$

by Lemma 4.6. Therefore, $\ker\gamma_{v}$ is zero when $v\nmid\Sigma$.

∎

######  Lemma 4.8.

Let $v\in S_{p}$ and consider the restriction map

$\gamma_{v}\mathrel{\mathop{\ordinarycolon}}J_{v}(E/K_{\infty})\rightarrow
J_{v}(E/L_{\infty}).$

The kernel of $\gamma_{v}$ is finite.

###### Proof.

Recall the assumption that each of the primes $v\in S_{p}$ is assumed to be
ramified in $K_{\infty}$. It follows that if $v^{\prime}|v$ for $v\in S_{p}$,
the extension $K_{\infty,v^{\prime}}/K_{v}$ is deeply ramified in the sense of
Coates and Greenberg (see [8, p. 130]). To prove the lemma, it suffices to
show that $\ker\gamma_{v}$ is finite for each $v|p$. At each prime
$v^{\prime}\in v(K_{\infty})$, choose a prime $w^{\prime}$ of $L_{\infty}$
such that $w^{\prime}|v^{\prime}$. We have the isomorphism

$\ker\gamma_{v}\simeq\bigoplus_{v^{\prime}\in
v(K_{\infty})}H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})\right)[p^{\infty}].$

In fact, we will show that
$H^{1}\left(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}},E(L_{\infty,w^{\prime}})\right)=0$
for $v^{\prime}\in v(K_{\infty})$.

Let $\mathcal{F}$ denote the formal group attached to $E$. Let
$\mathfrak{m}_{w^{\prime}}$ denote the maximal ideal of (the ring of integers
of) $L_{\infty,w^{\prime}}$ and $l_{w^{\prime}}$ be the residue field. Let
$\widetilde{E}$ denote the reduction of $E$ modulo
$\mathfrak{m}_{w^{\prime}}$. Consider the short exact sequence of $G$-modules
(see [33, p. 124])

$0\rightarrow\mathcal{F}(\mathfrak{m}_{w^{\prime}})\rightarrow
E(L_{\infty,w^{\prime}})\rightarrow\widetilde{E}(l_{w^{\prime}})\rightarrow
0.$

Since $K_{\infty,v^{\prime}}$ is a deeply ramified extension, it follows from
[8, Theorem 3.1] that

$H^{i}\left(G,\mathcal{F}(\mathfrak{m}_{w^{\prime}})\right)=0\quad\textrm{ for
}i=1,2.$

A connected commutative algebraic group over a finite field is cohomologically
trivial (see result of S. Lang in [25, p. 204]). Therefore,

$H^{i}\left(G,\widetilde{E}(l_{w^{\prime}})\right)=0\quad\textrm{ for }i=1,2.$

The lemma is now immediate. ∎

######  Proposition 4.9.

With setting as above, the kernel and cokernel of the restriction map

$\alpha\mathrel{\mathop{\ordinarycolon}}\operatorname{Sel}(E/K_{\infty})\rightarrow\operatorname{Sel}(E/L_{\infty})^{G}$

are finite.

###### Proof.

The map $\alpha$ fits into a diagram

${0}$${\operatorname{Sel}(E/K_{\infty})}$${H^{1}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)}$${\bigoplus_{v\in
S}J_{v}(E/K_{\infty})}$${0}$${0}$${\operatorname{Sel}(E/L_{\infty})^{G}}$${H^{1}\left(K_{S}/L_{\infty},E[p^{\infty}]\right)^{G}}$${\bigoplus_{v\in
S}J_{v}(E/L_{\infty}).}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

The map $\gamma$ decomposes into a direct sum of local maps

$\gamma_{v}\mathrel{\mathop{\ordinarycolon}}J_{v}(E/K_{\infty})\rightarrow
J_{v}(E/L_{\infty}).$

By the snake lemma, it suffices to show the following

  1. (1)

the kernel and cokernel of $\beta$ are both finite,

  2. (2)

the kernel of each map $\gamma_{v}$ is finite (resp. trivial) when $v|\ell$
such that $\ell$ is finitely decomposed (resp. $\ell$ splits completely) in
$K_{\infty}/K$.

By the inflation-restriction sequence, the map $\beta$ fits in an exact
sequence

$\begin{split}0&\rightarrow
H^{1}\left(G,E(L_{\infty})[p^{\infty}]\right)\rightarrow
H^{1}\left(K_{S}/K_{\infty},E[p^{\infty}]\right)\xrightarrow{\beta}H^{1}\left(K_{S}/L_{\infty},E[p^{\infty}]\right)^{G}\\\
&\rightarrow
H^{2}\left(G,E(L_{\infty})[p^{\infty}]\right)\rightarrow\dots\end{split}.$

Since $E(L_{\infty})[p^{\infty}]$ is of cofinite type, the cohomology groups
$H^{i}\left(G,E(L_{\infty})[p^{\infty}]\right)$ are finite for $i=1,2$.
Therefore, the kernel and cokernel of $\beta$ are both finite. The result
follows from the analysis of the local kernel (see Lemmas 4.7 and 4.8). ∎

######  Corollary 4.10.

Let $E_{/K}$ be an elliptic curve such that $\operatorname{Sel}(E/K_{\infty})$
is a $\Lambda$-cotorsion module with $\mu(E/K_{\infty})=0$. If $L/K$ is a
Galois extension such that Lemma 4.9 holds, then
$\operatorname{Sel}(E/L_{\infty})$ is a $\Lambda$-cotorsion module with
$\mu(E/L_{\infty})=0$.

###### Proof.

Note that $\operatorname{Sel}(E/K_{\infty})$ is a cotorsion $\Lambda$-module
with $\mu=0$ if and only if it is cofinitely generated as a
$\mathbb{Z}_{p}$-module. It follows from Lemma 4.9 that
$\operatorname{Sel}(E/L_{\infty})^{G}$ is also a cofinitely generated
$\mathbb{Z}_{p}$-module. Equivalently, the $G$-coinvariant of
$\operatorname{Sel}(E/L_{\infty})^{\vee}$ is finitely generated as a
$\mathbb{Z}_{p}$-module. Since $G\simeq\mathbb{Z}/p\mathbb{Z}$, the group ring
$\mathbb{Z}_{p}[G]$ is local. By Nakayama’s Lemma,
$\operatorname{Sel}(E/L_{\infty})^{\vee}$ is finitely generated as a
$\mathbb{Z}_{p}[G]$-module and hence finitely generated as a
$\mathbb{Z}_{p}$-module. This completes the proof of the lemma. ∎

Since Assumption 2.2 holds, it follows from Proposition 4.9 that

$\displaystyle\lambda(E/K_{\infty})$
$\displaystyle=\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}(E/K_{\infty})=\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}(E/L_{\infty})^{G},$
$\displaystyle\lambda(E/L_{\infty})$
$\displaystyle=\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}(E/L_{\infty}).$

######  Definition 4.11.

Let $G$ be a $p$-group, and $M$ be a divisible $\mathbb{Z}_{p}[G]$-module of
cofinite type. Then the _Herbrand quotient_ of $M$ is defined as

$h_{G}(M)=\frac{\\#H^{2}(G,M)}{\\#H^{1}(G,M)}.$

When $G=\mathbb{Z}/p\mathbb{Z}$ and $M=\operatorname{Sel}(E/L_{\infty})$, it
is straightforward to see from the work of K. Iwasawa that (cf. [17, p. 589])

(4.4)
$\lambda(E/L_{\infty})=p\lambda(E/K_{\infty})+(p-1)\operatorname{ord}_{p}\left(h_{G}\left(\operatorname{Sel}(E/L_{\infty})\right)\right).$

##  5\. Herbrand Quotient Calculations

Let $E_{/K}$ be an elliptic curve. Throughout this section, we suppose that
Assumptions 2.2, 3.1 and 4.2 are satisfied. It follows from (4.4) that to
complete the proof of Theorem 4.3, we need to calculate the Herbrand quotient
$h_{G}\left(\operatorname{Sel}(E/L_{\infty})\right)$. We carry out that
calculation here, thereby completing the proof of the main theorem.

###  5.1. Simplification of the Herbrand quotient

Corollary 4.10 asserts that $\operatorname{Sel}(E/L_{\infty})$ is
$\Lambda$-cotorsion with $\mu(E/L_{\infty})=0$. It follows from Corollary 3.3
that $E(L_{\infty})[p^{\infty}]$ is finite. Thus, by an argument identical to
Lemma 3.4, the Poitou–Tate sequence becomes

$0\rightarrow\operatorname{Sel}(E/L_{\infty})\rightarrow
H^{1}\left(K_{S}/L_{\infty},E[p^{\infty}]\right)\rightarrow\bigoplus_{v\in
S}J_{v}(E/L_{\infty})\rightarrow 0.$

It follows that

$h_{G}\left(\operatorname{Sel}(E/L_{\infty})\right)=\frac{h_{G}\left(H^{1}(K_{S}/L_{\infty},E[p^{\infty}])\right)}{\prod_{v\in
S}h_{G}\left(J_{v}(E/L_{\infty})\right)}.$

Let $v\in S\setminus S_{p}$, and write $v^{*}(L_{\infty})$ for the set of
primes $w^{\prime}\in v(L_{\infty})$ such that both conditions hold

  1. (1)

$w^{\prime}$ is a ramified prime above $v^{\prime}\in v(K_{\infty})$.

  2. (2)

$w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$.

It follows from Assumption 4.2 that $v^{*}(L_{\infty})$ is finite. Note that
since $L_{\infty}$ is a Galois extension of $K$, it follows that either
$v^{*}(L_{\infty})=\emptyset$ or $v^{*}(L_{\infty})=v(L_{\infty})$.

######  Lemma 5.1.

The following assertions hold.

  1. (1)

The Herbrand quotient

$h_{G}\left(H^{1}\left(K_{S}/L_{\infty},E[p^{\infty}]\right)\right)=h_{G}\left(E(L_{\infty})[p^{\infty}]\right)=1.$

  2. (2)

The local Herbrand quotient is expressed as follows

$h_{G}\left(J_{v}(E/L_{\infty})\right)=\prod_{w^{\prime}\in
v^{*}(L_{\infty})}h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right).$

In the above, the empty product is taken to be $1$.

###### Proof.

  1. (1)

This assertion follows from [17, Lemma 4.1].

  2. (2)

By [17, Lemmas 4.2 and 4.3],

$h_{G}\left(J_{v}(E/L_{\infty})\right)=\prod_{w^{\prime}\in
v(L_{\infty})}h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right).$

If $w^{\prime}|p$, $h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right)=1$
by [17, Lemma 4.3], which generalizes verbatim to our setting. Next, assume
that $v\in S\setminus S_{p}$. To prove the claim, we show that
$h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right)=1$ for
$w^{\prime}\notin v^{*}(L_{\infty})$. Since $w^{\prime}\notin
v^{*}(L_{\infty})$, at least one of the following conditions is satisfied:

    1. (a)

$L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is unramified,

    2. (b)

$w^{\prime}\notin\mathcal{P}_{1}\cup\mathcal{P}_{2}$.

Note that if $w^{\prime}\notin\mathcal{P}_{1}\cup\mathcal{P}_{2}$, then it
follows from Lemma 4.6 that

$h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right)=0.$

Therefore assume that $w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$, and
thus it follows from the above discussion that
$L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is unramified. By assumption
4.2, since $w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$, the prime
$v\in\Sigma$ and is therefore finitely decomposed in $K_{\infty}$. Since
$v\nmid p$, it follows that
$K_{\infty,v^{\prime}}=K_{v}^{\operatorname{cyc}}$, the unique unramified pro-
extension of $K_{v}$. Since $L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}}$ is
unramified, it follows that

$L_{\infty,w^{\prime}}=K_{\infty,v^{\prime}}=K_{v}^{\operatorname{cyc}}.$

In this case, it is clear that the Herbrand quotient

$h_{G}\left(E(L_{\infty,w^{\prime}})[p^{\infty}]\right)=0$

since the group
$\operatorname{Gal}(L_{\infty,w^{\prime}}/K_{\infty,v^{\prime}})$ is trivial.

∎

######  Corollary 5.2.

We have that

$\operatorname{ord}_{p}\left(h_{G}\left(\operatorname{Sel}(E/L_{\infty})\right)\right)=-\sum_{v\in
S}\left(\sum_{w^{\prime}\in
v^{*}(L_{\infty})}\operatorname{ord}_{p}\left(h_{G}(E(L_{\infty,w^{\prime}})[p^{\infty}])\right)\right).$

###### Proof of Theorem 4.3.

Recall that is suffices to prove the theorem when $\\#G=p$. It is known that
(see [17, Corollary 5.2])

$\operatorname{ord}_{p}\left(h_{G}(E(L_{\infty,w^{\prime}})[p^{\infty}])\right)=\begin{cases}-1\text{
if }w^{\prime}\in\mathcal{P}_{1},\\\ -2\text{ if
}w^{\prime}\in\mathcal{P}_{2}.\end{cases}$

The result follows from (4.4) and Corollary 5.2. ∎

##  6\. Elliptic Curves with Congruent Galois representations

In this section, we study the $p$-primary Selmer groups of $p$-congruent
elliptic curves over (general) $\mathbb{Z}_{p}$-extensions. We show that for
two elliptic curves which are $p$-congruent, the $p$-primary Selmer group of
one is a finitely generated $\mathbb{Z}_{p}$-module if and only if the same is
true for the other elliptic curve (see Theorem 6.11 for the precise
statement). Using this result, we can compare the $\lambda$-invariants of
these Selmer groups (see Proposition 6.16). This is accomplished by
introducing _imprimitive Selmer groups_ for general
$\mathbb{Z}_{p}$-extensions. Proposition 6.16 is the key tool which allows us
to construct elliptic curves with large $\lambda$-invariants in Section 7.

Assume throughout this section that $p\geq 5$. Consider elliptic curves
$E_{1}$ and $E_{2}$ defined over _any_ number field $K$ such that

  1. (1)

both $E_{1}$ and $E_{2}$ have good ordinary reduction at the primes $v\in
S_{p}$,

  2. (2)

the elliptic curves are $p$-congruent, i.e., there is a
$\operatorname{Gal}({}\mkern 3.0mu\overline{\mkern-3.0muK}/K)$-equivariant
isomorphism $E_{1}[p]\simeq E_{2}[p]$,

  3. (3)

$E_{1}(K)[p]=0$ (or equivalently $E_{2}(K)[p]=0$). Recall that this condition
is automatically satisfied when the residual representation is irreducible.

Since $E_{i}[p]=0$, by an application of Nakayama’s lemma,
$E_{i}(K_{\infty})[p^{\infty}]=0$. For $i=1,2$, let $N_{i}$ be the conductor
of $E_{i}$ and let ${}\mkern 3.0mu\overline{\mkern-3.0muN}_{i}$ be the Artin
conductor of the residual representation $E_{i}[p]$. Note that ${}\mkern
3.0mu\overline{\mkern-3.0muN}_{i}$ divides $N_{i}$ for $i=1,2$. Since $E_{1}$
and $E_{2}$ are $p$-congruent, we have that ${}\mkern
3.0mu\overline{\mkern-3.0muN}_{1}={}\mkern 3.0mu\overline{\mkern-3.0muN}_{2}$.
Let $K_{\infty}$ be _any_ $\mathbb{Z}_{p}$-extension of $K$ and for $i=1,2$
denote by $\operatorname{Sel}(E_{i}/K_{\infty})$ the Selmer group over
$K_{\infty}$.

######  Definition 6.1.

Let $\Omega(E_{i})$ be the set of primes $v\notin S_{p}$ at which both of the
following conditions are satisfied

  1. (1)

$v$ divides $N_{i}/{}\mkern 3.0mu\overline{\mkern-3.0muN}_{i}$,

  2. (2)

if $K_{v}$ contains $\mu_{p}$, then $E_{i}$ has split multiplicative
reduction.

Let $\Omega_{0}$ denote the set $\Omega(E_{1})\cup\Omega(E_{2})$, $\Omega$ be
the set of primes $\Omega_{0}\cup S_{p}$, and $\widetilde{\Omega}$ the set of
primes $v$ for which $v\in S_{p}$ or one of the elliptic curves $E_{1}$ or
$E_{2}$ has bad reduction at $v$. Note that $\widetilde{\Omega}$ contains the
set $\Omega$.

Recall that $\Sigma$ is the set of primes of $K$ that are finitely decomposed
in $K_{\infty}$. For the remainder of the section we make the following
assumption for $E_{1}$ and $E_{2}$.

######  Assumption 6.2.

For $i=1,2$, assume that $\Omega(E_{i})\subset\Sigma$.

Throughout, we write $E$ to denote either of the elliptic curves, $E_{1}$ or
$E_{2}$.

######  Lemma 6.3.

Suppose that $\operatorname{Sel}(E/K_{\infty})$ is $\Lambda$-cotorsion and
that $E(K_{\infty})[p^{\infty}]$ is finite. Then
$H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}])$ has no proper
$\Lambda$-submodules of finite index.

###### Proof.

The proof of this result in the cyclotomic case is due to Greenberg (see [13,
Proposition 4.9]), which crucially uses the surjectivity of the restriction
map (3.1). The argument generalizes verbatim. ∎

We recall an equivalent definition of the Selmer group due to Greenberg. The
$p$-primary Selmer group can also be defined as the kernel of the following
_global-to-local_ map

$H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}])\rightarrow\prod_{v\in\widetilde{\Omega}}\mathcal{H}_{v}(E/K_{\infty}).$

Here for each finite prime $v\in\widetilde{\Omega}\setminus S_{p}$, the local
term is defined as follows

$\mathcal{H}_{v}(E/K_{\infty})=\prod_{v^{\prime}|v}H^{1}\left({K_{\infty}}_{,v^{\prime}},E[p^{\infty}]\right),$

where $K_{\infty,v^{\prime}}$ is the union of all completions of number fields
contained in $K_{\infty}$. The local condition at primes $v\in S_{p}$ is more
subtle. Since $E$ has good ordinary reduction at all primes $v\in S_{p}$, its
$p$-adic Tate module $T$ fits into a short exact sequence of
$\operatorname{G}_{K_{v}}$-modules

$0\rightarrow T^{+}\rightarrow T\rightarrow T^{-}\rightarrow 0$

such that $T^{+}$ and $T^{-}$ are free of rank $1$ over $\mathbb{Z}_{p}$, and
$T^{-}$ is unramified. We identify $E[p^{\infty}]$ with
$T\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ and let
$D\mathrel{\mathop{\ordinarycolon}}=T^{-}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$.
For $v\in S_{p}$, define

$\mathcal{H}_{v}(E/K_{\infty})=\bigoplus_{v^{\prime}|v}H^{1}\left({K_{\infty}}_{,v^{\prime}},E[p^{\infty}]\right)/L_{v^{\prime}}$

with

$L_{v^{\prime}}=\ker\left(H^{1}\left({K_{\infty}}_{,v^{\prime}},E[p^{\infty}]\right)\rightarrow
H^{1}\left(I_{v^{\prime}},D\right)\right).$

Here $I_{v^{\prime}}$ denotes the inertia group at $v^{\prime}$. This
definition of the Selmer group is more useful to work with when proving
results about congruent Galois representations. The following result shows
that the above definition of the Selmer group (due to Greenberg) matches the
usual Selmer group. Thus, we do not distinguish between them in this paper.

######  Lemma 6.4.

Let $v\in\widetilde{\Omega}$, then

$\mathcal{H}_{v}(E/K_{\infty})=J_{v}(E/K_{\infty}).$

###### Proof.

First, consider the case when $v\notin S_{p}$. Let $\ell\neq p$ be a rational
prime and $L$ be any algebraic extension of $\mathbb{Q}_{\ell}$. Then [13,
Proposition 2.1] can be applied, and it asserts that
$E(L)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}=0$. In particular, we have that
$E(K_{\infty,v^{\prime}})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}=0$, for
$v^{\prime}|v$. Thus, $\mathcal{H}_{v}(E/K_{\infty})=J_{v}(E/K_{\infty})$.
When $v\in S_{p}$, the result follows from [13, Proposition 2.4]. ∎

We introduce the $\Omega_{0}$-imprimitive Selmer group to be the Selmer group
obtained by imposing conditions only at primes
$v\in\widetilde{\Omega}\setminus\Omega_{0}$.

######  Definition 6.5.

The _imprimitive Selmer group_ is defined as follows

$\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})=\ker\left(H^{1}\left(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}]\right)\xrightarrow{\theta_{0}}\bigoplus_{v\in\widetilde{\Omega}\setminus\Omega_{0}}\mathcal{H}_{v}(E/K_{\infty})\right).$

######  Lemma 6.6.

Let $v\in\Omega_{0}$, then $\mathcal{H}_{v}(E/K_{\infty})$ is a cofinitely
generated and cotorsion $\Lambda$-module with $\mu$-invariant equal to $0$.
Equivalently, it is a cofinitely generated $\mathbb{Z}_{p}$-module.

###### Proof.

We refer the reader to the standard argument on [16, pp. 37-38]. ∎

Analogous to the classical and imprimitive Selmer group, we define the
_reduced_ classical and imprimitive Selmer groups which we denote by
$\operatorname{Sel}(E[p]/K_{\infty})$ and
$\operatorname{Sel}^{\Omega_{0}}(E[p]/K_{\infty})$, respectively. For
$v\in\widetilde{\Omega}\setminus S_{p}$ set

$\mathcal{H}_{v}(E[p]/K_{\infty})\mathrel{\mathop{\ordinarycolon}}=\prod_{v^{\prime}|v}H^{1}\left({K_{\infty}}_{,v^{\prime}},E[p]\right),$

and for $v\in S_{p}$, set

$\mathcal{H}_{v}(E[p]/K_{\infty})\mathrel{\mathop{\ordinarycolon}}=\bigoplus_{v^{\prime}|v}H^{1}(K_{\infty,v^{\prime}},E[p])/\overline{L}_{v^{\prime}},$

where

$\overline{L}_{v^{\prime}}\mathrel{\mathop{\ordinarycolon}}=\ker\left(H^{1}\left({K_{\infty}}_{,v^{\prime}},E[p]\right)\rightarrow
H^{1}\left(I_{v^{\prime}},D[p]\right)\right).$

######  Definition 6.7.

The residual imprimitive Selmer group is defined as follows

$\operatorname{Sel}^{\Omega_{0}}(E[p]/K_{\infty})\mathrel{\mathop{\ordinarycolon}}=\operatorname{ker}\left(H^{1}\left(K_{\widetilde{\Omega}}/K_{\infty},E[p]\right)\xrightarrow{\overline{\theta}_{0}}\bigoplus_{v\in\widetilde{\Omega}\setminus\Omega_{0}}\mathcal{H}_{v}(E[p]/K_{\infty})\right).$

######  Proposition 6.8.

Let $E$ be either $E_{1}$ or $E_{2}$ and assume that $E(K)[p]=0$. Then,

$\operatorname{Sel}^{\Omega_{0}}(E[p]/K_{\infty})\simeq\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})[p].$

###### Proof.

Recall that $\widetilde{\Omega}$ consists of a finite set of primes containing
the primes above $p$ and the primes at which either $E_{1}$ or $E_{2}$ has bad
reduction. In particular, $\Omega_{0}$ is contained in $\widetilde{\Omega}$.
Consider the diagram relating the two Selmer groups

${0}$${\operatorname{Sel}^{\Omega_{0}}(E[p]/K_{\infty})}$${H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p])}$${\operatorname{im}\overline{\theta}_{0}}$${0}$${0}$${\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})[p]}$${H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}])[p]}$${\left(\operatorname{im}\theta_{0}\right)[p]}$${0,}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$

where the vertical maps are induced by the Kummer sequence. Note that

$0=H^{0}(K,E[p])=H^{0}(K_{\infty},E[p])^{\Gamma}.$

Since $K_{\infty}/K$ is a pro-$p$ extension, it follows that
$H^{0}(K_{\infty},E[p^{\infty}])=0$. Therefore $g$ (and hence $f$) is
injective. On the other hand, it clear that $g$ is surjective.

By an application of the snake lemma, it suffices to show that $h$ is
injective. For $v\in\widetilde{\Omega}\setminus\Omega_{0}$, denote by
$\iota_{v}$ the natural map

$\iota_{v}\mathrel{\mathop{\ordinarycolon}}\mathcal{H}_{v}(E[p]/K_{\infty})\rightarrow\mathcal{H}_{v}(E[p^{\infty}]/K_{\infty})[p].$

When $v\in S_{p}$, consider the commutative square with injective horizontal
maps

${\mathcal{H}_{v}(E[p]/K_{\infty})}$${\bigoplus_{v^{\prime}|v}H^{1}\left(I_{v^{\prime}},D[p]\right)}$${\mathcal{H}_{v}(E[p^{\infty}]/K_{\infty})[p]}$${\bigoplus_{v^{\prime}|v}H^{1}\left(I_{v^{\prime}},D\right)[p].}$$\scriptstyle{\iota_{v}}$$\scriptstyle{j_{v}}$

Since $D$ is unramified at all primes $v\in S_{p}$, it follows that
$H^{0}(I_{v^{\prime}},D)=D$ is divisible. Therefore,

$\ker(j_{v})=H^{0}(I_{v^{\prime}},D)/p=0.$

Thus, $j_{v}$ is injective, and so is $\iota_{v}$.

Next, consider the case when
$v\in\widetilde{\Omega}\setminus(S_{p}\cup\Omega_{0})$. Recall that
$\Omega_{0}=\Omega(E_{1})\cup\Omega(E_{2})$. Since $v\notin\Omega_{0}$, we
have that $v\notin\Omega(E_{i})$. It follows that either of the following
conditions are true

  1. (1)

$\mu_{p}\subset K_{v}$ and $E_{i}$ has bad additive reduction or non-split
multiplicative reduction at $v$, or,

  2. (2)

$v\nmid N_{i}/{}\mkern 3.0mu\overline{\mkern-3.0muN}_{i}$.

It follows from the Kummer sequence that $\iota_{v}$ is injective if
$H^{0}(K_{\infty,v^{\prime}},E[p^{\infty}])/p=0$ for every prime
$v^{\prime}|v$. In other words, we need to show that
$E(K_{\infty,v^{\prime}})[p^{\infty}]$ is $p$-divisible. In the first case,
the result follows from [17, Proposition 5.1] and in the latter case, from
[12, Lemma 4.1.2]. ∎

######  Lemma 6.9.

The isomorphism $E_{1}[p]\simeq E_{2}[p]$ of Galois modules induces an
isomorphism of residual Selmer groups

$\operatorname{Sel}^{\Omega_{0}}(E_{1}[p]/K_{\infty})\simeq\operatorname{Sel}^{\Omega_{0}}(E_{2}[p]/K_{\infty}).$

###### Proof.

Let $T_{i}$ be the $p$-adic Tate module of $E_{i}$ and identify
$E_{i}[p^{\infty}]$ with $T_{i}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$. Note
that $E_{i}[p]$ fits into a short exact sequence

$0\rightarrow C_{i}[p]\rightarrow E_{i}[p]\rightarrow D_{i}[p]\rightarrow 0,$

where $C_{i}=T_{i}^{+}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ and
$D_{i}=T_{i}^{-}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$. The action of
$\operatorname{G}_{\mathbb{Q}_{p}}$ on $C_{i}$ is via $\chi\gamma_{i}$, where
$\gamma_{i}$ is an unramified character and $\chi$ is the $p$-adic cyclotomic
character. On the other hand, the action on $D_{i}$ is via the unramified
character $\gamma_{i}^{-1}$.

Let $\Phi\mathrel{\mathop{\ordinarycolon}}E_{1}[p]\xrightarrow{\sim}E_{2}[p]$
be a choice of isomorphism of Galois modules. Clearly, $\Phi$ induces an
isomorphism
$H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E_{1}[p])\xrightarrow{\sim}H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E_{2}[p])$.
It suffices to show that for $v\in\widetilde{\Omega}\setminus\Omega_{0}$, the
isomorphism $\Phi$ induces an isomorphism

$\mathcal{H}_{v}(E_{1}[p]/K_{\infty})\xrightarrow{\sim}\mathcal{H}_{v}(E_{2}[p]/K_{\infty}).$

This is clear for $v\neq p$. For $v=p$, this follows from the fact that $\Phi$
induces an isomorphism $D_{1}[p]\xrightarrow{\sim}D_{2}[p]$. ∎

######  Corollary 6.10.

The isomorphism $E_{1}[p]\simeq E_{2}[p]$ of Galois modules induces an
isomorphism

$\operatorname{Sel}^{\Omega_{0}}(E_{1}/K_{\infty})[p]\simeq\operatorname{Sel}^{\Omega_{0}}(E_{2}/K_{\infty})[p].$

######  Theorem 6.11.

Let $E_{1}$ and $E_{2}$ be $p$-congruent elliptic curves defined over $K$ such
that

  1. (1)

$E_{1}(K)[p]=0$, or equivalently, $E_{2}(K)[p]=0$.

  2. (2)

Assumption 6.2 is satisfied.

Then,

$\begin{split}&\operatorname{Sel}(E_{1}/K_{\infty})\text{ is
}\Lambda\text{-cotorsion with }\mu=0\\\
\Leftrightarrow&\operatorname{Sel}(E_{2}/K_{\infty})\text{ is
}\Lambda\text{-cotorsion with }\mu=0.\\\ \end{split}$

###### Proof.

We have a left exact sequence

$0\rightarrow\operatorname{Sel}(E_{i}/K_{\infty})\rightarrow\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})\rightarrow\bigoplus_{v\in\Omega_{0}}\mathcal{H}_{v}(E_{i}/K_{\infty})$

for $i=1,2$. By Lemma 6.6, we know that for $v\in\Omega_{0}$, the cohomology
group $\mathcal{H}_{v}(E_{i}/K_{\infty})$ is a cotorsion $\Lambda$-module with
$\mu=0$. Therefore,

$\begin{split}&\operatorname{Sel}(E_{i}/K_{\infty})\text{ is
}\Lambda\text{-cotorsion with }\mu=0\\\
\Leftrightarrow&\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})\text{ is
}\Lambda\text{-cotorsion with }\mu=0.\\\ \end{split}$

On the other hand,

$\begin{split}&\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})\text{ is
}\Lambda\text{-cotorsion with }\mu=0\\\
\Leftrightarrow&\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})[p]\text{ is
finite.}\\\ \end{split}$

The above assertion can be proven in the same way as [23, Proposition 3.10].
By Corollary 6.10, we have the isomorphism

$\operatorname{Sel}^{\Omega_{0}}(E_{1}/K_{\infty})[p]\simeq\operatorname{Sel}^{\Omega_{0}}(E_{2}/K_{\infty})[p].$

Thus, the assertion follows. ∎

For the rest of this section we assume that the conditions of Theorem 6.11 are
satisfied for both $E_{1}$ and $E_{2}$. Furthermore, we impose the following
assumption.

######  Assumption 6.12.

The Selmer group $\operatorname{Sel}(E_{1}/K_{\infty})$ (or equivalently,
$\operatorname{Sel}(E_{2}/K_{\infty})$) is a finitely generated
$\mathbb{Z}_{p}$-module.

In the rest of the section, the goal is to show that $\lambda$-invariants of
$\operatorname{Sel}^{\Omega_{0}}(E_{1}/K_{\infty})$ and
$\operatorname{Sel}^{\Omega_{0}}(E_{2}/K_{\infty})$ coincide. As before, we
write $E$ to denote either $E_{1}$ or $E_{2}$. Recall that in Lemma 3.4, we
showed that the map defining the (classical) Selmer group is surjective. It
follows that

(6.1)
$\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})/\operatorname{Sel}(E/K_{\infty})\simeq\bigoplus_{v\in\Omega_{0}}\mathcal{H}_{v}(E/K_{\infty}).$

For $v\in\Omega_{0}$, it follows from Assumption 6.2 that there are finitely
many primes $v^{\prime}$ of $K_{\infty}$ such that $v^{\prime}|v$. Therefore,
for $v\in\Omega_{0}$, we have that $\mathcal{H}_{v}(E/K_{\infty})$ is a direct
sum over a finite set of primes.

######  Definition 6.13.

Let $\sigma_{E}^{(v)}$ denote the $\mathbb{Z}_{p}$-corank of
$\mathcal{H}_{v}(E/K_{\infty})$ for $v\in\Omega_{0}$.

Let $\lambda^{\Omega_{0}}(E/K_{\infty})$ be the $\lambda$-invariant of
$\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})$. It follows from the structure
theory of $\Lambda$-modules that

$\lambda^{\Omega_{0}}(E/K_{\infty})=\operatorname{corank}_{\mathbb{Z}_{p}}\left(\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})\right).$

Thus by (6.1),

(6.2)
$\lambda^{\Omega_{0}}(E/K_{\infty})=\lambda(E/K_{\infty})+\sum_{v\in\Omega_{0}}\sigma_{E}^{(v)}.$

######  Lemma 6.14.

Let $E$ be either $E_{1}$ or $E_{2}$. Then, $\mathcal{H}_{p}(E/K_{\infty})$ is
$\Lambda$-cofree.

###### Proof.

The reader is referred to [16, p. 23] for the argument. ∎

######  Proposition 6.15.

Let $E$ be $E_{1}$ or $E_{2}$, and suppose that

  1. (1)

Assumptions 6.2 and 6.12 hold,

  2. (2)

$E(K)[p]=0$.

Then, the Selmer group $\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})$
contains no proper finite index $\Lambda$-submodules.

###### Proof.

We adapt the proof of [13, Proposition 4.14], which is due to Greenberg.
Recall that $\widetilde{\Omega}$ is a finite set of primes containing $S_{p}$
and the primes at which $E_{1}$ or $E_{2}$ has bad reduction. Consider the
strict and relaxed Selmer groups defined as follows:

$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})\mathrel{\mathop{\ordinarycolon}}=\operatorname{ker}\left\\{H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}])\rightarrow\bigoplus_{v\in\widetilde{\Omega}\setminus(\Omega_{0}\cup
S_{p})}\mathcal{H}_{v}(K_{\infty},E[p^{\infty}])\right\\},$
$\operatorname{Sel}^{\operatorname{str}}(E/K_{\infty})\mathrel{\mathop{\ordinarycolon}}=\operatorname{ker}\left\\{\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})\rightarrow\bigoplus_{v\in\Omega_{0}\cup
S_{p}}\prod_{v^{\prime}|v}H^{1}(K_{\infty,v^{\prime}},E[p^{\infty}])\right\\}.$

Lemma 3.4 asserts that the restriction map

$H^{1}(K_{\widetilde{\Omega}}/K_{\infty},E[p^{\infty}])\rightarrow\bigoplus_{v\in\widetilde{\Omega}}\mathcal{H}_{v}(K_{\infty},E[p^{\infty}])$

is surjective. Therefore, the restriction map

$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})\rightarrow\bigoplus_{v\in\Omega_{0}\cup
S_{p}}\mathcal{H}_{v}(K_{\infty},E[p^{\infty}])$

is also surjective. Observe that the kernel of the restriction map

$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})\rightarrow\bigoplus_{v\in
S_{p}}\mathcal{H}_{v}(K_{\infty},E[p^{\infty}])$

is $\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})$. Therefore,

$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})/\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})\simeq\bigoplus_{v\in
S_{p}}\mathcal{H}_{v}(K_{\infty},E[p^{\infty}])$

is a cofree $\Lambda$-module by Lemma 6.14. By [16, Lemma 2.6], to complete
the proof of the proposition, it suffices to show that
$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})$ has no proper finite
index $\Lambda$-submodules. Observe that
$\operatorname{Sel}^{\operatorname{str}}(E/K_{\infty})^{\vee}$ is a quotient
of $\operatorname{Sel}^{\Omega_{0}}(E/K_{\infty})^{\vee}$, and therefore is
also $\Lambda$-torsion.

Recall that $\chi$ denotes the $p$-adic cyclotomic character. Let $\omega$ be
the Teichmüller lift of the mod-$p$ cyclotomic character and
$\kappa\mathrel{\mathop{\ordinarycolon}}=\omega^{-1}\chi$. For
$s\in\mathbb{Z}$, let $A_{s}$ denote the twisted Galois module
$E[p^{\infty}]\otimes\kappa^{s}$. Since $E(K)[p]=0$ by assumption, Corollary
3.3 asserts that $H^{0}(K_{\infty},E[p^{\infty}])=0$. Further, since $A_{s}$
and $E[p^{\infty}]$ are isomorphic as $\operatorname{Gal}({}\mkern
3.0mu\overline{\mkern-3.0muK}/K_{\infty})$-modules, it follows that
$H^{0}(K_{\infty},A_{s})=0$. For a subfield $\mathcal{K}$ of $K_{\infty}$, and
$v$ a prime of $K$ which does not divide $p$, set

$\mathcal{H}_{v}(\mathcal{K},A_{s})=\prod_{\eta|v}H^{1}(\mathcal{K}_{\eta},A_{s})/(A_{s}(\mathcal{K}_{\eta})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}),$

where $\eta$ ranges over the primes of $\mathcal{K}$ above $v$. Define the
products

$\displaystyle P^{\widetilde{\Omega},\operatorname{rel}}(\mathcal{K},A_{s})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\bigoplus_{v\in\widetilde{\Omega}\setminus(\Omega_{0}\cup
S_{p})}\mathcal{H}_{v}(\mathcal{K},A_{s}),$ $\displaystyle
P^{\widetilde{\Omega},\operatorname{str}}(\mathcal{K},A_{s})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\bigoplus_{v\in\widetilde{\Omega}\setminus(\Omega_{0}\cup
S_{p})}\mathcal{H}_{v}(\mathcal{K},A_{s})\times\bigoplus_{v\in\Omega_{0}\cup
S_{p}}\left\\{\prod_{\eta|v}H^{1}(\mathcal{K}_{\eta},A_{s})\right\\}.$

Let $S_{A_{s}}^{\operatorname{rel}}(\mathcal{K})$ and
$S_{A_{s}}^{\operatorname{str}}(\mathcal{K})$ be the Selmer groups defined as
follows

$\displaystyle S_{A_{s}}^{\operatorname{rel}}(\mathcal{K})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\ker\left(H^{1}(K_{\widetilde{\Omega}}/\mathcal{K},A_{s})\rightarrow
P^{\widetilde{\Omega},\operatorname{rel}}(\mathcal{K},A_{s})\right),$
$\displaystyle S_{A_{s}}^{\operatorname{str}}(\mathcal{K})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\ker\left(H^{1}(K_{\widetilde{\Omega}}/\mathcal{K},A_{s})\rightarrow
P^{\widetilde{\Omega},\operatorname{str}}(\mathcal{K},A_{s})\right).$

Since $\operatorname{Sel}^{\operatorname{str}}(E/K_{\infty})$ is
$\Lambda$-cotorsion, we have that
$S_{A_{s}}^{\operatorname{str}}(K_{\infty})^{\Gamma}$ is finite for all but
finitely many values of $s$. Hence, $S_{A_{s}}^{\operatorname{str}}(K)$ is
finite for all but finitely many values of $s$. As in the proof of [13,
Proposition 4.14], write $M=A_{s}$ and $M^{*}=A_{-s}$. Denote by $S_{M}(K)$
the Selmer group defined by the relaxed conditions
$S_{A_{s}}^{\operatorname{rel}}(K)$. By the discussion on [13, p. 100], we
have that $S_{M^{*}}(K)$ is the strict Selmer group
$S_{A_{-s}}^{\operatorname{str}}(K)$. Let $s$ be an integer such that
$S_{M^{*}}(K)$ is finite. Since $S_{M^{*}}(K)$ is finite and $M^{*}(K)=0$, it
follows that the map $H^{1}(K_{\widetilde{\Omega}}/K,M)\rightarrow
P^{\widetilde{\Omega},\operatorname{rel}}(K,M)$ is surjective (see [13,
Proposition 4.13]). It follows from the proof of [13, Proposition 4.14] that
$\operatorname{Sel}^{\operatorname{rel}}(E/K_{\infty})$ has no proper finite
index $\Lambda$-submodules. This completes the proof. ∎

######  Proposition 6.16.

Let $E_{1}$ and $E_{2}$ be $p$-congruent elliptic curves over $K$ such that

  1. (1)

Assumptions 6.2 and 6.12 are satisfied,

  2. (2)

$E_{1}(K)[p]=0$ (or equivalently, $E_{2}(K)[p]=0$).

Then the imprimitive $\lambda$-invariants coincide, i.e.,

$\lambda^{\Omega_{0}}(E_{1}/K_{\infty})=\lambda^{\Omega_{0}}(E_{2}/K_{\infty}).$

Equivalently,

$\lambda(E_{1}/K_{\infty})+\sum_{v\in\Omega_{0}}\sigma_{E_{1}}^{(v)}=\lambda(E_{2}/K_{\infty})+\sum_{v\in\Omega_{0}}\sigma_{E_{2}}^{(v)}.$

###### Proof.

It follows from Assumption 6.12 that
$\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})[p]$ is finite for $i=1,2$.
Corollary 6.10 asserts that

(6.3)
$\operatorname{Sel}^{\Omega_{0}}(E_{1}/K_{\infty})[p]\simeq\operatorname{Sel}^{\Omega_{0}}(E_{2}/K_{\infty})[p].$

By Proposition 6.15, we know
$\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})$ has no proper
$\Lambda$-submodules of finite index for $i=1,2$. Hence they are cofree
$\mathbb{Z}_{p}$-modules. Therefore,

$\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})\simeq(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\lambda_{i}},$

where
$\lambda_{i}\mathrel{\mathop{\ordinarycolon}}=\lambda^{\Omega_{0}}(E_{i}/K_{\infty})$.
As a result,

$\lambda^{\Omega_{0}}(E_{i}/K_{\infty})=\operatorname{dim}_{\mathbb{F}_{p}}\left(\operatorname{Sel}^{\Omega_{0}}(E_{i}/K_{\infty})[p]\right).$

Finally, (6.3) implies that

$\lambda^{\Omega_{0}}(E_{1}/K_{\infty})=\lambda^{\Omega_{0}}(E_{2}/K_{\infty}).$

∎

##  7\. Large $\lambda$-Invariant

In this section, we use results from the previous section to construct
elliptic curves with _large_ $5$-adic (anti-cyclotomic) $\lambda$-invariants.
Throughout this section, we will work in the following setting. Let
$K=\mathbb{Q}(\sqrt{-1})$, $p=5$ and $K^{\operatorname{ac}}$ denote the
anticyclotomic $\mathbb{Z}_{5}$-extension of $K$. Recall that $\Sigma$ is the
set of primes that are finitely decomposed in $K^{\operatorname{ac}}$. Note
that a prime $v\nmid 5$ lies in $\Sigma$ if and only if it is a split prime in
$K$. If $v\in\Sigma\setminus\\{5\\}$ and $v^{\prime}\in
v(K^{\operatorname{ac}})$, then $K^{\operatorname{ac}}_{v^{\prime}}$ is an
unramified $\mathbb{Z}_{5}$-extension of $v$. Therefore,
$K^{\operatorname{ac}}_{v^{\prime}}=K_{v}^{\operatorname{cyc}}$.

######  Lemma 7.1.

Let $E_{1}$ and $E_{2}$ be $5$-congruent elliptic curves defined over $K$. Let
$v\nmid 5$ be a prime such that $v\in\Sigma$. Then the following assertions
hold

  1. (1)

If $E_{1}$ has good reduction at $v$ and $E_{2}$ has bad reduction at $v$,
then

$\operatorname{corank}_{\mathbb{Z}_{p}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{1}[5^{\infty}]\right)-\operatorname{corank}_{\mathbb{Z}_{p}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{2}[5^{\infty}]\right)\geq
0.$

  2. (2)

Furthermore, if $\operatorname{Frob}_{v}$ acts trivially on $E_{1}[5]$, then

$\operatorname{corank}_{\mathbb{Z}_{p}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{1}[5^{\infty}]\right)-\operatorname{corank}_{\mathbb{Z}_{p}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{2}[5^{\infty}]\right)>0.$

Furthermore, $E_{2}$ has split multiplicative reduction at $v$.

###### Proof.

For part (1), and the first assertion of part (2), we refer to the proof of
[22, Lemma 3.1] for the argument. By [2, Lemma 2.3], $E_{2}$ cannot have
additive reduction at $v$. Moreover, by [2, Lemma 2.11], if $E_{2}$ has non-
split multiplicative reduction at $v$, then,

$\operatorname{corank}_{\mathbb{Z}_{5}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{1}[5^{\infty}]\right)=\operatorname{corank}_{\mathbb{Z}_{5}}\mathcal{H}_{v}\left(K^{\operatorname{ac}},E_{2}[5^{\infty}]\right).$

Hence, the strict inequality forces $E$ to have split multiplicative reduction
at $v$. ∎

Consider the elliptic curve $E\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}-x$.
This is the elliptic curve 32a2 (Cremona label). This is a rank 0 elliptic
curve with good ordinary reduction at $p=5$ and CM by $\mathcal{O}_{K}$. Rubin
has proved that for all primes $q$, the $q$-part of the Selmer group is
trivial (see for example [22, p. 188]). Over $K$, the elliptic curve $E$ has
Mordell-Weil rank equal to $0$ and $\Sha(E/K)[5^{\infty}]=0$. Since
$E(K)[5]=0$, it follows that $\operatorname{Sel}(E/K)=0$.

######  Lemma 7.2.

The 5-primary Selmer group $\operatorname{Sel}(E/K^{\operatorname{ac}})$ is
equal to $0$.

###### Proof.

To prove the lemma we will show that the natural map

$\operatorname{Sel}(E/K)\rightarrow\operatorname{Sel}(E/K^{\operatorname{ac}})^{\Gamma}$

is surjective. It suffices to show that the local map

$\gamma_{v}\mathrel{\mathop{\ordinarycolon}}J_{v}(E/K)\rightarrow
J_{v}(E/K^{\operatorname{ac}})$

is injective at all primes. For $v\nmid 5$, the kernel of $\gamma_{v}$ has
order at most $c_{v}^{(5)}$, where $c_{v}$ is the Tamagawa number and
$c_{v}^{(5)}$ its $5$-primary part, see [15, Proposition 4.1]. For the given
elliptic curve $E$, one checks that $c_{v}^{(5)}=0$ for all primes $v$. Next,
we consider the kernel of $\gamma_{v}$ for $v|5$, the map $\gamma_{v}$ is
injective if $\widetilde{E}(\mathbb{F}_{v})[5]=0$, see the proof of [9,
Proposition 3.5]. Since $5$ splits in $K$, $\mathbb{F}_{v}=\mathbb{F}_{5}$.
The abelian group
$\widetilde{E}(\mathbb{F}_{5})\simeq\mathbb{Z}/2\times\mathbb{Z}/4$.
Therefore, the map $\gamma_{v}$ is injective. Therefore,
$\operatorname{Sel}(E/K^{\operatorname{ac}})^{\Gamma}=0$. The assertion
follows from an application of Nakayama’s Lemma. ∎

In [31, Theorem 5.3], Rubin and Silverberg have shown that for every rational
number $t$, there exist polynomials $a(t)$, $b(t)$, and $c(t)$ such that

$E_{t}\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}+a(t)x^{2}+b(t)x+c(t)$

is an elliptic curve defined over $\mathbb{Q}$, with good reduction at $p=5$,
discriminant

$\Delta(E_{t})=4^{3}\left(5t^{4}-2t^{2}+1\right)^{5}\left(25t^{8}-100t^{6}-210t^{4}-20t^{2}+1\right)^{5},$

and the additional property that $E_{t}[5]\simeq E[5]$.

Let $f(t)=(5t^{4}-2t^{2}+1)(25t^{8}-100t^{6}-210t^{4}-20t^{2}+1)$. We can
check that $\gcd(f(t),f^{\prime}(t))=1$. Let $\mathcal{A}$ be the set of prime
numbers $\ell>5$ such that $f(t)$ and $f^{\prime}(t)$ are prime to each other
modulo $\ell$. Let $\mathcal{A}^{\prime}$ be the subset of $\mathcal{A}$
consisting of those primes which split completely in $K(E[5])$. By the
Chebotarev density theorem, the set $\mathcal{A}^{\prime}$ is infinite with
positive density. We remark that primes in $\mathcal{A}^{\prime}$ must split
completely in $K$. Hence, these primes are finitely decomposed in
$K^{\operatorname{ac}}/K$.

Given a positive integer $k$, choose
$\ell_{1},\ldots,\ell_{k}\in\mathcal{A}^{\prime}$. Then, for all $1\leq i\leq
k$ and $v|\ell_{i}$ in $K$, the Frobenius, $\operatorname{Frob}_{v}$ acts
trivially on $E[5]$. By arguments similar to [22, p. 189], there exists $t$
for which $E_{t}$ satisfies the following properties

  1. (1)

$E[5]\simeq E_{t}[5]$.

  2. (2)

$E_{t}$ has bad reduction at each $\ell_{i}$.

Note that primes above $\ell_{i}$ split in $K$; hence $E_{t}$ has bad
reduction at (both) primes above each $\ell_{i}$. By Lemma 7.1, the elliptic
curve $E_{t}$ must have split multiplicative reduction at $v\mid\ell_{i}$ for
$1\leq i\leq k$. Since $E[5]\simeq E_{t}[5]$ is unramified at $v|\ell_{i}$, it
follows that $v$ does not divide ${}\mkern 3.0mu\overline{\mkern-3.0muN}_{t}$.
Thus, $v|(N_{t}/{}\mkern 3.0mu\overline{\mkern-3.0muN}_{t})$. Hence, we deduce
that $v\in\Omega(E_{t})\subseteq\Omega_{0}$.

######  Lemma 7.3.

With notation as above, suppose that $\Omega(E_{t})\subseteq\Sigma$. Then, the
5-primary Selmer group $\operatorname{Sel}(E_{t}/K)$ is $\Lambda$-cotorsion
with $\mu=0$.

###### Proof.

Recall that $E(K)[5]=0$. The elliptic curve $E$, has additive reduction at the
only prime of bad reduction, which is 2. Therefore, $\Omega(E)$ is empty. By
assumption, $\Omega(E_{t})\subseteq\Sigma$. Thus, we can apply Theorem 6.11.
The claim follows from Lemma 7.2. ∎

Using Proposition 6.16 and (6.2), we can compare $\lambda$-invariants of
$\operatorname{Sel}(E/K^{\operatorname{ac}})$ and
$\operatorname{Sel}(E_{t}/K^{\operatorname{ac}})$. We obtain that,

$\lambda(E_{t}/K^{\operatorname{ac}})=\lambda(E/K^{\operatorname{ac}})+\sum_{v\in\Omega_{0}}\left(\sigma_{E}^{(v)}-\sigma_{E_{t}}^{(v)}\right)\geq\lambda(E/K^{\operatorname{ac}})+\sum_{v\in\Omega(E_{t})}\left(\sigma_{E}^{(v)}-\sigma_{E_{t}}^{(v)}\right)$

where $\sigma_{E}^{(v)}$ (resp. $\sigma_{E}^{(v)}$) denotes the
$\mathbb{Z}_{p}$-corank of $\mathcal{H}_{v}(E/K_{\infty})$ (resp.
$\mathcal{H}_{v}(E_{t}/K_{\infty})$). It follows from our discussion that for
any given integer $k$, we can construct an elliptic curve $E_{t}$ with bad
reduction at $\ell_{1},\ldots,\ell_{k}$ such that

$\lambda(E_{t}/K^{\operatorname{ac}})\geq\lambda(E/K^{\operatorname{ac}})+\sum_{i=1}^{k}\sum_{v|\ell_{i}}\left(\sigma_{E}^{(v)}-\sigma_{E_{t}}^{(v)}\right)\geq\lambda(E/K^{\operatorname{ac}})+2k\geq
2k.$

We note that we have used the fact that the rational primes $\ell_{i}$ split
in $K$.

### Example

We work out an explicit example when $p=5$. The number field
$\mathbb{Q}(E[5])$ is defined by the polynomial

$\begin{split}&x^{32}-40x^{31}+12000x^{29}-10760x^{28}\\\
&-266800x^{27}+11468000x^{26}-1096600000x^{25}\\\
&-5566913400x^{24}+142092440000x^{23}+4369939320000x^{22}\\\
&-378135400000x^{21}-257156918536000x^{20}-6566363103320000x^{19}\\\
&-68617519071000000x^{18}+71545316608000000x^{17}\\\
&+10219246387427710000x^{16}+120765118794306400000x^{15}\\\
&+1192541074750776000000x^{14}+11957464808846300000000x^{13}\\\
&+82847553195104300000000x^{12}+1265538538816402500000000x^{11}\\\
&+17074052628101627500000000x^{10}+172311923953883400000000000x^{9}\\\
&+1455441332493456906250000000x^{8}+9447284167114381875000000000x^{7}\\\
&+49318124991568150000000000000x^{6}+129093097413215156250000000000x^{5}\\\
&-2350246524757890625000000000x^{4}-682311431690296875000000000000x^{3}\\\
&-475786411839253906250000000000x^{2}+246274993230703125000000000000x\\\
&+1450426905571313476562500000000.\end{split}$

Since the extension is rather large it becomes difficult to find primes which
split in it via direct calculation. We appeal to known results about the
splitting of primes in fields of the form $\mathbb{Q}(E[N])$. The conductor of
$E$ is $2^{5}=32$ and hence all prime numbers $\ell\neq 2$ are unramified in
$\mathbb{Q}(E[5])$. If $\ell$ is a prime number, different from 2 and 5, which
splits in $\mathbb{Q}(E[5])$, then indeed it is easy to see that the mod-$5$
characteristic polynomial of $\operatorname{Frob}_{\ell}$ must equal
$(X-1)^{2}=X^{2}-2X+1$. This is equivalent to requiring that

  1. (1)

$\ell\equiv 1\mod{5}$, and

  2. (2)

$a_{\ell}(E)\equiv 2\mod{5}$, where
$a_{\ell}(E)\mathrel{\mathop{\ordinarycolon}}=\ell+1-\\#E(\mathbb{F}_{\ell})$.

This is however not a sufficient condition for $\ell$ to split in
$\mathbb{Q}(E[5])$. Let $\Phi_{5}(X,Y)$ be the modular function (see [11,
section 11] for a precise definition). Define

$\Phi_{5}^{E}(X)\mathrel{\mathop{\ordinarycolon}}=\Phi_{5}(X,j(E))=\Phi_{5}(X,1728).$

A prime $\ell$ splits in $\mathbb{Q}(E[5])$ if in addition to the above two
conditions,

  1. (3)

$\Phi_{5}^{E}(X)$ splits in $\mathbb{F}_{\ell}[X]$ into distinct linear
factors.

We refer the reader to [1, Proposition 5.6.3] for a proof of the claim. It is
not difficult to calculate $\Phi_{5}^{E}(X)$, the reader is referred to [11,
section 13]. Implementing the algorithms in [6, 7], the modular polynomials
may be calculated, and in particular

$\begin{split}&\Phi_{5}(X,Y)\\\
=&141359947154721358697753474691071362751004672000\\\
&+53274330803424425450420160273356509151232000X\\\
&-264073457076620596259715790247978782949376XY\\\
&+6692500042627997708487149415015068467200X^{2}\\\
&+36554736583949629295706472332656640000X^{2}Y\\\
&+5110941777552418083110765199360000X^{2}Y^{2}\\\
&+280244777828439527804321565297868800X^{3}\\\
&-192457934618928299655108231168000X^{3}Y\\\
&+26898488858380731577417728000X^{3}Y^{2}-441206965512914835246100X^{3}Y^{3}\\\
&+1284733132841424456253440X^{4}+128541798906828816384000X^{4}Y\\\
&+383083609779811215375X^{4}Y^{2}+107878928185336800X^{4}Y^{3}\\\
&+1665999364600X^{4}Y^{4}+1963211489280X^{5}-246683410950X^{5}Y\\\
&+2028551200X^{5}Y^{2}-4550940X^{5}Y^{3}+3720X^{5}Y^{4}-X^{5}Y^{5}+X^{6}.\end{split}$

Recall that primes in the set $\mathcal{A}^{\prime}$ also split in
$K=\mathbb{Q}(\sqrt{-1})$, i.e. they satisfy

  1. (4)

$\ell\equiv 1\mod 4$.

Using SageMath [34], we find primes ($\leq 10^{5}$) satisfying conditions
$(1)-(4)$. There are two primes, namely $\ell_{1}=63241$ and $\ell_{2}=63901$.
The factorization of the modular polynomial in the corresponding polynomial
rings is as follows

$\displaystyle\Phi_{5}^{E}(X)$
$\displaystyle=(x+9130)(x+26600)(x+28822)(x+31643)(x+37410)(x+60303)\textrm{
in }\mathbb{F}_{63241}(X),$ $\displaystyle\Phi_{5}^{E}(X)$
$\displaystyle=(x+15646)(x+16523)(x+16743)(x+31583)(x+36229)(x+58255)\textrm{
in }\mathbb{F}_{63901}(X).$

There exists $t=1059545078$, such that

  1. (i)

$E[5]\simeq E_{t}[5]$.

  2. (ii)

$E_{t}$ has bad reduction at the primes above $\ell_{1}$ and $\ell_{2}$.

For this value of $t$, we find that

$\displaystyle f(t)=$ $\displaystyle 13\times 401\times 63241\times
63901\times 21068381440942021\times 23007701426021875081$ $\displaystyle\times
24504438741475825204304998173516406719475833143478257969366221.$

Each of the prime factors of $f(1059545078)$ are of the form $1\pmod{4}$.
Recall that these primes factors are the primes of bad reduction of
$E_{1059545078}$. Therefore, we see that _all_ the primes of bad reduction of
$E_{1059545078}$ split completely in $K$, and hence are finitely decomposed in
$K^{\operatorname{ac}}$. This guarantees that
$\operatorname{Sel}(E_{1059545078}/K^{\operatorname{ac}})$ is a cofinitely
generated $\Lambda$-module with $\mu(E_{1059545078}/K^{\operatorname{ac}})=0$.
Our earlier discussion shows that

$\lambda(E_{1059545078}/K^{\operatorname{ac}})\geq\lambda(E/K^{\operatorname{ac}})+4\geq
4$

since we know that $\operatorname{Sel}(E/K^{\operatorname{ac}})=0$.

######  Remark 7.4.

It was possible to find primes up to $10^{7}$ satisfying properties $(1)-(4)$,
and there are a total of 24 such primes: $\ell_{1}=63241$, $\ell_{2}=63901$,
$\ell_{3}=514561$, $\ell_{4}=1311341$, $\ell_{5}=2399081$, $\ell_{6}=2502301$,
$\ell_{7}=2620301$, $\ell_{8}=2790461$, $\ell_{9}=3325121$,
$\ell_{10}=3436501$, $\ell_{11}=4046401$, $\ell_{12}=4050281$,
$\ell_{13}=4559101$, $\ell_{14}=4800421$, $\ell_{15}=5403361$,
$\ell_{16}=5609321$, $\ell_{17}=6660221$, $\ell_{18}=7601861$,
$\ell_{19}=7959521$, $\ell_{20}=8942501$, $\ell_{21}=8959921$,
$\ell_{22}=9181901$, $\ell_{23}=9187081$, and $\ell_{24}=9437321$. However,
the next part of the computation (i.e., finding $t$ satisfying $(i)$ and
$(ii)$) becomes difficult (for us) with $k\geq 3$. This is due to the product
$\prod_{i=1}^{k}\ell_{i}$ being large when $k\geq 3$.

##  8\. Acknowledgments

DK acknowledges the support of the PIMS Postdoctoral Fellowship.

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