# Statistics for Iwasawa Invariants of elliptic curves, II

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

We study the average behaviour of the Iwasawa invariants for Selmer groups of
elliptic curves. These results lie at the intersection of arithmetic
statistics and Iwasawa theory. We obtain unconditional lower bounds for the
density of rational elliptic curves with prescribed Iwasawa invariants.

###### Key words and phrases:

Arithmetic statistics, Iwasawa theory, Selmer groups, elliptic curves.

###### 2010 Mathematics Subject Classification:

11R18, 11F11, 11F33, 11R23 (primary); 11F85 (secondary).

##  1\. Introduction

In recent years, research in arithmetic statistics has gained considerable
popularity. This subject was initiated by H. Cohen and H. W. Lenstra in [7],
where they introduced a plausible heuristic on the distribution of class
groups of number fields. Over the years, these heuristics have been suitably
modified and these ideas have since become a powerful source of predictions
for number fields. One route that arithmetic statistics took was in the
direction of elliptic curves, which will be our primary focus.

A fundamental result in the theory of elliptic curves is the _Mordell–Weil
Theorem_. It states that given an elliptic curve $E_{/\mathbb{Q}}$, its
$\mathbb{Q}$-rational points form a finitely generated abelian group, i.e.,

$E(\mathbb{Q})\simeq\mathbb{Z}^{r}\oplus T$

where $r$ is a non-negative integer called the _(Mordell–Weil) rank_ and $T$
is a finite group, called the _torsion subgroup_. This torsion subgroup is
well-understood and its possible structure is known by the work of B. Mazur
(see [18, 19]). On the other hand, the rank remains mysterious and to-date
there is no known algorithm to compute it. The _rank distribution conjecture_
of elliptic curves claims that half of all elliptic curves have rank zero, the
remaining half have rank one, and all higher ranks constitute zero percent of
elliptic curves (even though there may exist infinitely many such elliptic
curves). Therefore, a suitably-defined _average rank_ would be $1/2$. The best
results in this direction are by M. Bharagava and A. Shankar (see [3, 4]).
They show that the average rank of elliptic curves over $\mathbb{Q}$ is
strictly less than one, and that both rank zero and rank one cases comprise
non-zero densities across all elliptic curves over $\mathbb{Q}$. Proving
results about the Mordell–Weil rank almost always involves a thorough analysis
of the Selmer group. In [1, 2], Bhargava–Shankar explicitly computed the
average sizes of certain Selmer groups to deduce asymptotic results of
Mordell–Weil ranks and made the following conjecture.

######  Conjecture 1.1.

Let $n$ be any positive integer. Then, when all elliptic curves $E$ are
ordered by height, the average size of the $n$-Selmer group, denoted by
$\mathrm{Sel}_{n}(E/\mathbb{Q})$, is $\sigma(n)$, the sum of the divisors of
$n$.

This conjecture has been verified for $n=2,3,4,5$ and was enough to deduce
powerful partial results for the rank distribution conjecture.

Another subject of active research is Iwasawa theory, which was introduced in
the late 1950’s. It started as the study of class groups over infinite towers
of number fields (see [12, 13]). In [17], Mazur initiated the study of Iwasawa
theory of Selmer groups of elliptic curves. This was motivated by the close
relationship between the group of units of number fields and the rational
points elliptic curves. Over the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$, Mazur conjectured that for an elliptic curve with good
_ordinary_ reduction at an odd prime $p$, the $p$-primary Selmer group is
cotorsion as a module over the Iwasawa algebra. This conjecture is now settled
by the work of K. Kato, see [14, Theorem 17.4].

The Iwasawa algebra (denoted by $\Lambda$) is isomorphic to the power series
ring $\mathbb{Z}_{p}\llbracket T\rrbracket$. The algebraic structure of the
Selmer group (as a $\Lambda$-module) is encoded by Iwasawa invariants, $\mu$
and $\lambda$. The $p$-adic Weierstrass Preparation Theorem asserts that the
characteristic ideal of the Pontryagin dual of the Selmer group is generated
by a unique element $f_{E}^{(p)}(T)$, which can be expressed as a power of $p$
times a distinguished polynomial. Then, the $\mu$-invariant is the power of
$p$ dividing $f_{E}^{(p)}(T)$ and the $\lambda$-invariant is its degree.

In [17], there are examples of elliptic curves over $\mathbb{Q}$ with good
ordinary reduction at $p$ such that the $\mu$-invariant associated to the
Selmer group is positive. On the other hand, there is a long standing
conjecture of R. Greenberg (see [11, Conjecture 1.11]) that if
$E_{/\mathbb{Q}}$ is an elliptic curve with good _ordinary_ reduction at $p$
and $E[p]$ is _irreducible_ , then the $\mu$-invariant is zero. This motivated
us to study Iwasawa invariants _on average_ , in [15]. The primary goal was to
study the following two separate but interrelated problems.

  1. (1)

For a fixed elliptic curve $E$, how do the Iwasawa invariants vary as $p$
varies over all odd primes $p$ at which $E$ has good reduction?

  2. (2)

For a fixed prime $p$, how do the Iwasawa invariants vary as $E$ varies over
all elliptic curves (with good reduction at $p$)?

The first question is easier to tackle and was studied by Greenberg in [11] in
the special case of elliptic curves with good _ordinary_ reduction at $p$. The
second question lies at the intersection of arithmetic statistics and Iwasawa
theory, and requires new ideas.

It is well-known that the $\lambda$-invariant associated to the Selmer group
of an elliptic curve is _at least_ equal to its Mordell–Weil rank (see Lemma
2.2). As a first step, in [15], we developed a technique to distinguish
between when the $\lambda$-invariant is _exactly equal_ to the Mordell–Weil
rank and when it is _strictly greater_ than the rank. This involved using the
_truncated Euler characteristic formula_ (see Section 3 for the precise
definition). This invariant of the elliptic curve is closely related to the
$p$-adic Birch and Swinnerton-Dyer (BSD) conjecture and captures information
about the size of the Tate–Shafarevich group, the Tamagawa number, the
anomalous primes, and the torsion points of the elliptic curve. Further, we
calculated upper bounds for the density of the set of rank 0 elliptic curves
defined over $\mathbb{Q}$ whose Euler characteristic is _not_ a unit, i.e.,
when the $\lambda$-invariant is strictly larger than the rank. Unfortunately,
these results are conditional and depend on Cohen–Lenstra type heuristics on
the variation of the Tate–Shafarevich group by C. Delaunay (see [9]). In this
paper, we refine our methods and prove Theorem 5.3. Our refined technique
allows for better control over the Euler characteristic formula. The benefit
is three-fold: given an odd prime $p$ and a non-negative integer $n$, we can
prove an _unconditional_ explicit _lower bound_ for the proportion of elliptic
curves with Mordell–Weil rank _at least 2_ or rank _at most 1_ with the
truncated Euler characteristic divisible by $p^{n}$. Since 100% of the
elliptic curves are expected to have rank at most 1, our result
(conjecturally) provides a lower bound for the proportion of elliptic curves
with rank _at most 1_ and the truncated Euler characteristic divisible by
$p^{n}$.

Another important question in the theory of elliptic curves is the _rank
boundedness conjecture_ which asks whether there is an upper bound for the
Mordell–Weil rank of elliptic curves over number fields. This question is
widely open and experts are unable to arrive at a consensus on what to expect,
see [20, Section 3] for a detailed survey. Those in favour of unboundedness
argue that this phenomenon provably occurs in other global fields, and that
the proven lower bound for this upper bound increases every few years. The
current record, is an elliptic curve over $\mathbb{Q}$ with Mordell–Weil rank
at least 28, discovered by N. Elkies. On the other hand, a recent series of
papers by B. Poonen _et. al._ provides a justified heuristic inspired by ideas
from arithmetic statistics which suggests otherwise (see for example, [22]).
Evidently, rank unboundedness is a deep problem. However, upon restricting our
focus to the cyclotomic $\mathbb{Z}_{p}$-extension of a number field, it is
possible to make somewhat precise statements that are analogous. We have
already discussed that the $\lambda$-invariant of the Selmer group of an
elliptic curve is closely related to its Mordell–Weil rank. In Theorem 6.4 and
Corollary 6.5, we prove that given any integer $n$, there is an explicit lower
bound for the density of the set of elliptic curves with good ordinary
reduction at $p$ for which $\lambda+\mu\geq n$. This lower bound depends on
$p$ (and $n$), is strictly positive, and becomes smaller as $p$ or $n$ become
larger. A key ingredient is the observation that a lower bound of the sum
$\lambda+\mu$ depends on the number of primes (distinct from $p$) for which
$p$ divides the Tamagawa number and whether $p$ is an anomalous prime (see
Lemma 6.3). In view of Greenberg’s conjecture for irreducible residual
representation $E[p]$, our results imply that $\lambda\geq n$ for a _positive_
proportion of elliptic curves that depends only on $p$ and $n$.

_Organization:_ Including this introduction, the article has 7 sections.
Section 2 is preliminary in nature: here, we introduce the Iwasawa invariants
and the main object of interest, i.e., the Selmer groups. In Section 3, we
discuss the notion of truncated Euler characteristic. Next, in Section 4, we
arrange elliptic curves (defined over $\mathbb{Q}$) by naïve height and
calculate what proportion of them have a certain Kodaira type. In Section 5,
given an odd prime $p$ and a fixed integer $n\geq 0$, we associate to each
elliptic curve with good ordinary reduction at $p$, its (truncated) Euler
characteristic formula. We analyze what proportion of elliptic curves have
Mordell–Weil rank at most 1 (conjectured to happen 100% of the time) and the
truncated Euler characteristic divisible by $p^{n}$. Finally, in Section 6 we
state and prove the main theorem of this article. Given an odd prime $p$ and a
fixed integer $n\geq 0$, we prove a lower bound for what proportion of
elliptic curves defined over $\mathbb{Q}$ with good ordinary reduction at $p$
is the sum of Iwasawa invariants $\geq n$. In Section 7, we have included some
computational tables.

##  2\. Preliminaries

Throughout, $p$ will be a fixed odd prime number. Let $E_{/\mathbb{Q}}$ be an
elliptic curve with good ordinary reduction at $p$. The main object of study
is the $p$-primary Selmer group of $E$ over the cyclotomic
$\mathbb{Z}_{p}$-extension, which we introduce.

Fix an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$ and for each prime
$\ell$, choose an embedding $\iota_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$. Let $S$ be
a finite set of prime numbers containing $p$ and the primes at which $E$ has
bad reduction and $\mathbb{Q}_{S}$ be the maximal algebraic extension of
$\mathbb{Q}$ which is unramified at all primes $\ell\notin S$. Denote by
$\operatorname{G}_{S}$ the Galois group
$\operatorname{Gal}(\mathbb{Q}_{S}/\mathbb{Q})$. For any discrete
$\operatorname{G}_{S}$-module $N$, let $H^{i}(\mathbb{Q}_{S}/\mathbb{Q},N)$
denote the cohomology group $H^{i}(\operatorname{G}_{S}(\mathbb{Q}),N)$. For
$n\geq 0$, let $\mathbb{Q}_{n}$ be the (unique) subfield of
$\mathbb{Q}(\mu_{p^{n+1}})$ of degree $p^{n}$ over $\mathbb{Q}$. Note that
$\mathbb{Q}_{n}$ is contained in $\mathbb{Q}_{n+1}$. Let $\mathbb{Q}_{\infty}$
be the union

$\mathbb{Q}_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\bigcup_{n\geq
0}\mathbb{Q}_{n}$

and set $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$. There are
isomorphisms of topological groups

$\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\xrightarrow{\sim}\varprojlim_{n}\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})\xrightarrow{\sim}\mathbb{Z}_{p}.$

The extension $\mathbb{Q}_{\infty}$ is the _cyclotomic
$\mathbb{Z}_{p}$-extension_ of $\mathbb{Q}$ and $\mathbb{Q}_{n}$ is its $n$-th
layer. Choose a topological generator $\gamma\in\Gamma$ and fix an isomorphism
$\mathbb{Z}_{p}\xrightarrow{\sim}\Gamma$ sending $a$ to $\gamma^{a}$. The
Iwasawa algebra $\Lambda$ is defined as the following inverse limit

$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})].$

Fix an isomorphism of $\Lambda$ with the ring of formal power series
$\mathbb{Z}_{p}\llbracket T\rrbracket$, upon identifying $\gamma-1$ with $T$.

### 2.1.

Denote by $E[p^{\infty}]$ the $\operatorname{G}_{S}$-module consisting of all
$p$-power torsion points in $E(\bar{\mathbb{Q}})$. For $\ell\in S$, and any
finite extension $L/\mathbb{Q}$, define

$J_{\ell}(E/L)\mathrel{\mathop{\mathchar
58\relax}}=\bigoplus_{w|\ell}H^{1}\left(L_{w},E\right)[p^{\infty}]$

where the direct sum is over all primes $w$ of $L$ lying above $\ell$. For any
field $L\subset\mathbb{Q}_{S}$, the _$p$ -primary Selmer group_ over $L$ is
defined as follows

$\mathrm{Sel}_{p^{\infty}}(E/L)\mathrel{\mathop{\mathchar
58\relax}}=\ker\left\\{H^{1}\left(\mathbb{Q}_{S}/L,E[p^{\infty}]\right)\xrightarrow{\Phi_{E,L}}\bigoplus_{\ell\in
S}J_{\ell}(E/L)\right\\}.$

By a well-known exercise involving Nakayama’s lemma, the Pontryagin dual

$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Hom}(\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p})$

is finitely generated as a $\Lambda$-module. It is a deep result of Kato that
$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\vee}$ is a torsion
$\Lambda$-module, see [14].

### 2.2.

Let $M$ be a cofinitely generated cotorsion $\Lambda$-module, we introduce
certain Iwasawa invariants associated to $M$. According to [27, Theorem
13.12], $M^{\vee}$ is pseudo-isomorphic to a finite direct sum of cyclic
$\Lambda$-modules, i.e., there is a map of $\Lambda$-modules

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

with finite kernel and cokernel. Here, $\mu_{i}>0$ and $f_{j}(T)$ 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

$f_{M}^{(p)}(T)=f_{M}(T)\mathrel{\mathop{\mathchar
58\relax}}=p^{\sum_{i}\mu_{i}}\prod_{j}f_{j}(T).$

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

$\mu_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{i=1}^{s}\mu_{i}&\textrm{ if }s>0\\\ 0&\textrm{
if }s=0.\end{cases}$

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

$\lambda_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{i=1}^{s}\deg f_{i}&\textrm{ if }s>0\\\
0&\textrm{ if }s=0.\end{cases}$

The minimal number of generators $g_{p}(M)$ is defined to be

$g_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\dim_{\mathbb{F}_{p}}\left(M^{\vee}\otimes_{\Lambda}\Lambda/\mathfrak{m}\right),$

where $\mathfrak{m}\subset\Lambda$ is its maximal ideal. Denote by
$\mu_{p}(E/\mathbb{Q}_{\infty})$ (resp. $\lambda_{p}(E/\mathbb{Q}_{\infty})$)
the $\mu$-invariant (resp. $\lambda$-invariant) of the Selmer group
$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})$. The quantity
$g_{p}(E/\mathbb{Q}_{\infty})$ will denote
$g_{p}(\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty}))$.

######  Lemma 2.1.

The following inequality holds

$\lambda_{p}(E/\mathbb{Q}_{\infty})+\mu_{p}(E/\mathbb{Q}_{\infty})\geq
g_{p}(E/\mathbb{Q}_{\infty}).$

###### Proof.

It follows from [11, Proposition 4.15] that
$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\vee}$ contains no non-zero
finite $\Lambda$-submodules. The result then follows from [16, Lemma 2.2]. ∎

######  Lemma 2.2.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, then
$\lambda_{p}(E/\mathbb{Q}_{\infty})\geq\operatorname{rank}E(\mathbb{Q})$.

###### Proof.

We denote by $r$ the $\mathbb{Z}_{p}$-corank of
$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}$. Since
$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})$ is a cotorsion
$\Lambda$-module, it follows that $r$ is finite. This is indeed a simple
consequence of the structure theorem for $\Lambda$-modules. From the short
exact sequence,

$0\rightarrow
E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow\Sh(E/\mathbb{Q})[p^{\infty}]\rightarrow
0,$

it follows that

(2.1)
$\operatorname{corank}_{\mathbb{Z}_{p}}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\geq\mathrm{rank}E(\mathbb{Q}).$

It is an exercise in the structure theory of $\Lambda$-modules that
$\lambda_{p}(E/\mathbb{Q}_{\infty})\geq r$. Hence, it suffices to show that
$r\geq\mathrm{rank}E(\mathbb{Q})$. This is indeed the case, since there is a
natural map

$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}$

with finite kernel. ∎

##  3\. The Truncated Euler Characteristic

Here, we introduce the notion of the Euler characteristic. The average size of
this Iwasawa theoretic invariant will be studied in due course. Even though
the following discussion holds for any cofinitely generated cotorsion
$\Lambda$-module $M$, we will specialize to the case that
$M=\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})$ where $p$ is a prime of
good ordinary reduction. Since the $p$-cohomological dimension of $\Gamma$ is
$1$, we have that $H^{i}(\Gamma,M)$ is always zero for $i\geq 2$.

######  Lemma 3.1.

For a cofinitely generated cotorsion $\Lambda$-module $M$,

$\mathrm{corank}_{\mathbb{Z}_{p}}M^{\Gamma}=\mathrm{corank}_{\mathbb{Z}_{p}}M_{\Gamma}.$

###### Proof.

See [15, Lemma 2.1]. ∎

When the cohomology groups $H^{0}(\Gamma,M)$ and $H^{1}(\Gamma,M)$ are finite,
the (classical) _Euler characteristic_ $\chi(\Gamma,M)$ is defined as follows

$\chi(\Gamma,M)=\frac{\left(\\#H^{0}(\Gamma,M)\right)}{\left(\\#H^{1}(\Gamma,M)\right)}.$

For ease of notation, set
$\chi(\Gamma,E[p^{\infty}])\mathrel{\mathop{\mathchar
58\relax}}=\chi(\Gamma,\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty}))$. The
next lemma gives a criterion for the classical Euler characteristic to be
well-defined.

######  Lemma 3.2.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at
$p>2$. The following are equivalent.

  1. (1)

The classical Euler characteristic $\chi(\Gamma,E[p^{\infty}])$ is well-
defined.

  2. (2)

$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}$ is finite.

  3. (3)

The Selmer group $\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$ is finite.

  4. (4)

The Mordell–Weil group $E(\mathbb{Q})$ is finite, i.e., the Mordell–Weil rank
is 0.

###### Proof.

See [15, Lemma 3.2]. ∎

When the cohomology groups $H^{i}(\Gamma,M)$ are _not_ finite, there is a
generalization of the above notion. Since $H^{1}(\Gamma,M)$ is isomorphic to
the group of coinvariants $H_{0}(\Gamma,M)=M_{\Gamma}$, there is a natural map

$\Phi_{M}\mathrel{\mathop{\mathchar 58\relax}}M^{\Gamma}\rightarrow
M_{\Gamma}$

sending $x\in M^{\Gamma}$ to the residue class of $x$ in $M_{\Gamma}$. It
follows from Lemma 3.1 that the kernel of $\Phi_{M}$ is finite if and only if
its cokernel is finite. When the kernel and cokernel of $\Phi_{M}$ are finite,
we define the _truncated Euler characteristic_ , denoted by
$\chi_{t}(\Gamma,M)$, as the following quotient,

$\chi_{t}(\Gamma,M)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#\operatorname{ker}(\Phi_{M})}{\\#\operatorname{cok}(\Phi_{M})}.$

It is easy to check that when $\chi(\Gamma,M)$ is defined, so is
$\chi_{t}(\Gamma,M)$. In fact,

$\chi_{t}(\Gamma,M)=\chi(\Gamma,M).$

Express the characteristic element $f_{M}^{(p)}(T)$ as a polynomial,

$f_{M}^{(p)}(T)=c_{0}+c_{1}T+\dots+c_{d}T^{d}.$

Let $r_{M}$ denote the order of vanishing of $f_{M}^{(p)}(T)$ at $T=0$. For
$a,b\in\mathbb{Q}_{p}$, we write $a\sim b$ if there is a unit
$u\in\mathbb{Z}_{p}^{\times}$ such that $a=bu$.

######  Lemma 3.3.

Let $M$ be a cofinitely generated cotorsion $\Lambda$-module. Assume that the
kernel and cokernel of $\Phi_{M}$ are finite. Then,

  1. (1)

${r_{M}}=\mathrm{corank}_{\mathbb{Z}_{p}}(M^{\Gamma})=\mathrm{corank}_{\mathbb{Z}_{p}}(M_{\Gamma})$.

  2. (2)

$c_{r_{M}}\neq 0$.

  3. (3)

$c_{r_{M}}\sim\chi_{t}(\Gamma,M)$.

###### Proof.

See [29, Lemma 2.11]. ∎

It follows from Lemma 3.3 that the truncated Euler characteristic, when
defined, is always an integer. Note that $\chi(\Gamma,M)$ is defined if and
only if $r_{M}=0$. When this happens, the constant coefficient
$c_{0}\sim\chi(\Gamma,M)$. The following lemma indicates the precise
relationship between truncated Euler characteristic and the Iwasawa
invariants.

######  Lemma 3.4.

Let $M$ be a cofinitely generated and cotorsion $\Lambda$-module such that
$\Phi_{M}$ has finite kernel and cokernel. Let $r_{M}$ be the order of
vanishing of $f_{M}^{(p)}(T)$ at $T=0$. Then, the following are equivalent.

  1. (1)

$\chi_{t}(\Gamma,M)=1$,

  2. (2)

$\mu(M)=0$ and $\lambda(M)=r_{M}$.

###### Proof.

See [15, Lemma 3.4] ∎

The next result provides conditions for the truncated Euler characteristic to
be defined.

######  Lemma 3.5.

Let $M$ be a $p$-primary, discrete $\Lambda$-module and let $X$ be its
Pontryagin dual. Suppose that $X$ is a finitely generated torsion
$\Lambda$-module. Denote by $X[p^{\infty}]$ the $p^{\infty}$-torsion submodule
of $X$. Let $f_{1}(T),\dots,f_{n}(T)$ be distinguished polynomials such that
$X/X[p^{\infty}]$ is pseudo-isomorphic to
$\bigoplus_{i=1}^{n}\Lambda/(f_{i}(T))$. If $T^{2}\nmid f_{i}(T)$ for any $i$,
then the kernel and cokernel of $\Phi_{M}$ are finite and the truncated Euler
characteristic $\chi_{t}(\Gamma,M)$ is defined. In particular,
$\chi_{t}(\Gamma,M)$ is defined when $r_{M}\leq 1$.

###### Proof.

The assertion of the lemma follows from the proof of [29, Lemma 2.11]. ∎

When $E$ has good ordinary reduction at $p$, there is a $p$-adic analog of the
usual height pairing, which was studied extensively by P. Schneider in [23,
24]. This $p$-adic height pairing is conjecturally non-degenerate, and its
determinant is called the _$p$ -adic regulator_ (denoted by
$\operatorname{Reg}_{p}(E/\mathbb{Q})$). In Iwasawa theory, it is standard to
use the following normalized $p$-adic regulator, which is well-defined up to a
$p$-adic unit

$\mathcal{R}_{p}(E/\mathbb{Q})=\frac{\operatorname{Reg}_{p}(E/\mathbb{Q})}{p^{\mathrm{rank}E(\mathbb{Q})}}.$

The following result gives the formula for the (truncated) Euler
characteristic of the $p$-primary Selmer group (when it is defined). In the CM
case, this was proven by B. Perrin-Riou (see [21]) and in the general case by
P. Schneider (see [24]).

######  Theorem 3.6.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at
$p>2$. Then, the order of vanishing of the characteristic element
$f_{E}^{(p)}(T)$ at $T=0$ is at least equal to $\mathrm{rank}\ E(\mathbb{Q})$.
Furthermore, if

  1. (i)

$\mathcal{R}_{p}(E/\mathbb{Q})\neq 0$,

  2. (ii)

$\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite,

  3. (iii)

the truncated Euler characteristic $\chi_{t}(\Gamma,E[p^{\infty}])$ is
defined,

then,

  1. (1)

$\operatorname{ord}_{T=0}f_{E}^{(p)}(T)=\operatorname{rank}E(\mathbb{Q})$,

  2. (2)

we have the following $p$-adic Birch and Swinnerton-Dyer formula for the
truncated Euler characteristic

(3.1)
$\chi_{t}(\Gamma,E[p^{\infty}])\sim\mathcal{R}_{p}(E/\mathbb{Q})\times\frac{\\#\Sh(E/\mathbb{Q})[p^{\infty}]\times\prod_{\ell\neq
p}c_{\ell}^{(p)}\times\left(\\#\widetilde{E}(\mathbb{F}_{p})[p^{\infty}]\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

Here, $c_{\ell}$ is the Tamagawa number at $\ell\neq p$ and $c_{\ell}^{(p)}$
is its $p$-part.

######  Corollary 3.7.

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$
and assume that

  1. (1)

$\operatorname{rank}E(\mathbb{Q})\leq 1$,

  2. (2)

the $p$-adic regulator $\mathcal{R}_{p}(E/\mathbb{Q})$ is non-zero,

  3. (3)

$\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite.

Then, the truncated Euler characteristic $\chi_{t}(\Gamma,E[p^{\infty}])$ is
defined and given by (3.1).

###### Proof.

By Theorem 3.6, the order of vanishing of $f_{E}^{(p)}(T)$ is $\leq 1$. Hence,
by Lemma 3.5, the truncated Euler characteristic is defined. Therefore, by
Theorem 3.6 the truncated Euler characteristic up to a $p$-adic unit is given
by (3.1). ∎

##  4\. Densities for Weierstrass Models of Elliptic curves

The results in this section are essentially due to Cremona and Sadek [8],
except that we work with short Weierstrass equations and a different notion of
height. They are applied in the subsequent analysis of Iwasawa invariants on
average. Any curve $E_{/\mathbb{Q}}$ admits a unique global Weierstrass
equation $Y^{2}=X^{3}+aX+b$, for integers $a$ and $b$ such that
$\gcd(a^{3},b^{2})$ is not divisible by any $12$-th power. Set $E_{(a,b)}$ to
be the elliptic curve with minimal short Weierstrass equation above. We
introduce some terminology, which will be in place throughout this paper. The
height of $E$ will denote the _naïve height_ , namely

$H(E_{(a,b)})\mathrel{\mathop{\mathchar
58\relax}}=\max\left\\{4|a|^{3},27b^{2}\right\\}.$

### 4.1.

For any integral domain $R$, denote by $\mathcal{W}(R)$ the set of pairs

$\mathcal{W}(R)\mathrel{\mathop{\mathchar 58\relax}}=\\{(a,b)|a,b\in R\\}$

where to each pair $(a,b)$, we associate the curve

$E_{(a,b)}\mathrel{\mathop{\mathchar 58\relax}}Y^{2}=X^{3}+aX+b$

over $R$. The equation $E_{(a,b)}$ defines an elliptic curve if
$\Delta_{(a,b)}\mathrel{\mathop{\mathchar 58\relax}}=4a^{3}+27b^{2}$ is non-
zero. Let $\ell$ be a prime number and $v$ be the valuation on
$\mathbb{Z}_{\ell}$, normalized by $v(\ell)=1$. The Haar measure on
$\mathbb{Z}_{\ell}$ induces a measure $\mu$ on
$\mathcal{W}(\mathbb{Z}_{\ell})$ such that
$\mu\left(\mathcal{W}(\mathbb{Z}_{\ell})\right)=1$. For any tuple of non-
negative integers $(v_{1},v_{2})$, consider the set
$\mathcal{W}(v_{1},v_{2})\subseteq\mathcal{W}(\mathbb{Z}_{\ell})$ defined by

$\mathcal{W}(v_{1},v_{2})\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(a,b)\in\mathcal{W}(\mathbb{Z}_{\ell})\ |\ v(a)\geq
v_{1},v(b)\geq v_{2}\right\\}.$

The measure $\mu$ has the property that
$\mu\left(\mathcal{W}(v_{1},v_{2})\right)=\ell^{-v_{1}-v_{2}}$.

Let $\operatorname{T}$ denote a Kodaira type at $\ell$, the choices are as
$\operatorname{T}=\operatorname{I}_{0},\operatorname{I}_{n},\operatorname{I}_{n}^{*}$.
Note that after a finite base-change of $\mathbb{Q}_{\ell}$, one of these
types are attained. Here $\operatorname{I}_{0}$ is the Kodaira type for good
reduction. For each Kodaira type $\operatorname{T}$, let
$\mathcal{W}_{\operatorname{T}}(\mathbb{Z}_{\ell})$ be the subset of
$\mathcal{W}(\mathbb{Z}_{\ell})$ consisting of $(a,b)$ such that $E_{(a,b)}$
has Kodaira type $\operatorname{T}$ at $\ell$. Denote by
$\mathcal{W}_{\operatorname{I}_{\geq n}}(\mathbb{Z}_{\ell})$ the union

$\mathcal{W}_{\operatorname{I}_{\geq
n}}(\mathbb{Z}_{\ell})\mathrel{\mathop{\mathchar 58\relax}}=\bigcup_{m\geq
n}\mathcal{W}_{\operatorname{I}_{m}}(\mathbb{Z}_{\ell}).$

The local density of elliptic curves of type $\operatorname{T}$ is the measure

$\rho_{T}(\ell)\mathrel{\mathop{\mathchar
58\relax}}=\mu\left(\mathcal{W}_{\operatorname{T}}(\mathbb{Z}_{\ell})\right).$

The pair $(a,b)$ is minimal if the associated Weierstrass equation is minimal,
i.e., if either $v(a)<4$ or $v(b)<6$. Let
$\mathcal{W}_{M}(\mathbb{Z}_{\ell})\subset\mathcal{W}(\mathbb{Z}_{\ell})$
consist of models which are minimal in this sense. A global model over
$\mathbb{Z}$ is _minimal_ if and only if it localizes to the subset
$\mathcal{W}_{M}(\mathbb{Z}_{\ell})$ at every prime $\ell$. Let
$\rho^{M}(\ell)$ be the local density of the Weierstrass equations that are
minimal, i.e.,

$\rho^{M}(\ell)\mathrel{\mathop{\mathchar
58\relax}}=\mu\left(\mathcal{W}_{M}(\mathbb{Z}_{\ell})\right).$

We write $\rho^{M}_{\operatorname{T}}(\ell)$ for the local density of minimal
Weierstrass equations of type $\operatorname{T}$, i.e.,

$\rho_{\operatorname{T}}^{M}(\ell)\mathrel{\mathop{\mathchar
58\relax}}=\mu\left(\mathcal{W}_{M}^{T}(\mathbb{Z}_{\ell})\right)=\mu\left(\mathcal{W}^{\operatorname{T}}(\mathbb{Z}_{\ell})\cap\mathcal{W}_{M}(\mathbb{Z}_{\ell})\right).$

Computation of local densities will be related to global densities, and we
begin by computing $\rho^{M}(\ell)$.

######  Proposition 4.1.

Let $\ell$ be any prime and $\rho^{M}(\ell)$ be defined as above. Then

$\rho^{M}(\ell)=1-\ell^{-10}.$

###### Proof.

Note that $(a,b)\in\mathcal{W}_{M}(\mathbb{Z}_{\ell})$ if it does not reduce
to $(0,0)\in\mathbb{Z}/\ell^{4}\times\mathbb{Z}/\ell^{6}$. Hence, out of a
total of $\ell^{10}$ residue classes, there are $\ell^{10}-1$ which lift to
$\mathcal{W}_{M}(\mathbb{Z}_{\ell})$. Therefore,

$\rho^{M}(\ell)=\frac{\ell^{10}-1}{\ell^{10}}=1-\ell^{-10}.$

∎

Next, we compute the density corresponding to the Kodaira types
$\operatorname{T}=\operatorname{I}_{0}$ (i.e., good reduction) and
$\operatorname{I}_{n}$ for $n\geq 1$. We begin by proving the following lemma.

######  Lemma 4.2.

Let $\ell\geq 5$ be a prime, $n\geq 1$ and
$(a,b)\in\mathcal{W}_{M}(\mathbb{Z}_{\ell})$ with $\Delta_{(a,b)}\neq 0$.
Then, the elliptic curve $E_{(a,b)}$ has Kodaira type $\operatorname{I}_{n}$
if and only if

  1. (1)

$(a,b)\not\equiv(0,0)\mod{\ell}$,

  2. (2)

$v(\Delta_{(a,b)})=n$.

###### Proof.

Throughout this proof, $K$ will denote a finite extension of
$\mathbb{Q}_{\ell}$ in which the Kodaira type is realized and $\pi$ is the
uniformizer of $\mathcal{O}_{K}$. We show that the result follows from Tate’s
algorithm (see [26, §0-Summary]). According to this algorithm, we first
require that $v(\Delta_{(a,b)})>0$, i.e., the reduced curve $\tilde{E}$ has a
singular point. We make a change of variables so that the singular point of
$\tilde{E}$ is $(0,0)$. If ${(a,b)}\equiv(0,0)\mod{\ell}$, then this condition
is already satisfied for $\tilde{E}$. In this case, $\pi|b_{2}$ (see [25, p.
364] for the definition) and one can conclude from the algorithm that the
Kodaira type is not $\operatorname{I}_{n}$. Hence, it is required that
${(a,b)}\not\equiv(0,0)\mod{\ell}$. It suffices to observe that if
additionally, $v(\Delta_{(a,b)})=n$, then the Kodaira type is
$\operatorname{I}_{n}$. Upon extending the field $\mathbb{Q}_{\ell}$ by a
finite extension, it is possible to make $(0,0)$ the singular point after a
transformation replacing $x$ by $x+c$, where $c$ is an $\ell$-adic unit. For
the new equation $\pi\nmid b_{2}$, and as a result, the Kodaira type is
$\operatorname{I}_{n}$, according to the discussion in loc. cit. ∎

######  Proposition 4.3.

Let $\ell\geq 5$ be a prime and $\rho_{\operatorname{T}}^{M}(\ell)$ be as
defined previously. Then, for $\operatorname{T}=\operatorname{I}_{n}$, the
following equalities hold.

  1. (1)

$\rho_{\operatorname{I}_{0}}^{M}(\ell)=1-\ell^{-1}$.

  2. (2)

$\rho_{\operatorname{I}_{n}}^{M}(\ell)=\frac{(\ell-1)^{2}}{\ell^{n+2}}$ for
$n\geq 1$.

  3. (3)

$\rho_{\operatorname{I}_{\geq n}}^{M}(\ell)=\frac{(\ell-1)}{\ell^{n+1}}$.

###### Proof.

For $(1)$ it suffices to show that the number of solutions
$(a,b)\in\mathbb{Z}/\ell\times\mathbb{Z}/\ell$ to the equation
$\Delta_{(a,b)}=0$ is $\ell$. When $\ell\neq 2,3$, we see that
$27b^{2}=-4a^{3}$ if and only if $-3a$ is a quadratic residue modulo $\ell$.
As $a$ ranges over $\mathbb{Z}/\ell\mathbb{Z}$, we find that $-3a$ also ranges
over $\mathbb{Z}/\ell\mathbb{Z}$. If $a=0$, there is one corresponding
solution, namely, $(0,0)$ and if $-3a$ is a quadratic residue, then there are
exactly $2$-solutions. Note that there are an equal number of quadratic
residues as there are non-residues. When $\ell=2$ (resp. $\ell=3$), any
solution is of the form $(a,0)$ (resp. $(0,b)$). Hence, in all, there are a
total of $\ell$-solutions, and this completes the proof of part (1).

Lemma 4.2 asserts that $\mathcal{W}_{\operatorname{I}_{n}}(\mathbb{Z}_{\ell})$
consists of all minimal pairs $(a,b)\not\equiv(0,0)\mod{\ell}$ for which
$v(\Delta_{(a,b)})=n$. This is determined by the residue of $(a,b)$ in
$\mathbb{Z}/\ell^{n+1}\mathbb{Z}\times\mathbb{Z}/\ell^{n+1}\mathbb{Z}$. In
all, there are $\ell-1$ solutions of $\Delta_{(a,b)}=0$ for
$(0,0)\neq(\bar{a},\bar{b})\in\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/\ell\mathbb{Z}$.
For any integer $m\geq 1$, each solution lifts to $\ell^{m-1}$ solutions of
$\Delta_{(a,b)}=0$ in
$\mathbb{Z}/\ell^{m}\mathbb{Z}\times\mathbb{Z}/\ell^{m}\mathbb{Z}$. Hence, the
number of pairs in
$\mathbb{Z}/\ell^{n+1}\mathbb{Z}\times\mathbb{Z}/\ell^{n+1}\mathbb{Z}$ is
equal to
$\ell^{2}\left(\ell^{n-1}(\ell-1)\right)-\ell^{n}(\ell-1)=\ell^{n}(\ell-1)^{2}$.
Therefore, we have that

$\rho_{\operatorname{I}_{n}}^{M}(\ell)=\frac{\ell^{n}(\ell-1)^{2}}{\ell^{2(n+1)}}=\frac{(\ell-1)^{2}}{\ell^{n+2}}.$

In the same way

$\rho_{\operatorname{I}_{\geq
n}}^{M}(\ell)=\frac{\ell^{n-1}(\ell-1)}{\ell^{2n}}=\frac{(\ell-1)}{\ell^{n+1}}.$

∎

### 4.2.

We now discuss global Weierstrass equations satisfying local constraints. Let
$S$ be a collection of prime numbers (possibly all prime numbers). For each
prime $\ell\in S$, let $\mathcal{U}_{\ell}$ be a subset of
$\mathcal{W}(\mathbb{Z}_{\ell})$ which is defined by a congruence condition.
In other words, for some integer $N>0$ (depending on $\ell$),
$\mathcal{U}_{\ell}$ is the set that lifts a certain collection of residue
classes $\\{x_{1},\dots,x_{t}\\}$ in
$\mathcal{W}(\mathbb{Z}/\ell^{N}\mathbb{Z})$. Then, the measure
$\mu\left(\mathcal{U}_{\ell}\right)$ is equal to $t/\ell^{2N}$. Let $\Phi$ be
the data consisting of the set of primes $S$ and the choice of
$\mathcal{U}_{\ell}$ for each prime $\ell\in S$. We refer to $\Phi$ as a
_local datum over $S$ defined by congruence conditions._ Let

$\iota_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{W}(\mathbb{Z})\rightarrow\mathcal{W}(\mathbb{Z}_{\ell})$

denote the natural inclusion map. Let $\mathcal{W}_{M}(\mathbb{Z})$ be the set
of minimal Weierstrass models,

$\mathcal{W}_{M}(\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(a,b)\in\mathcal{W}(\mathbb{Z})\mid\iota_{\ell}\left((a,b)\right)\in\mathcal{W}_{M}(\mathbb{Z}_{\ell})\text{
for all }\ell\right\\}.$

######  Definition 4.4.

Let $\Phi=(S,\\{\mathcal{U}_{\ell}\\}_{\ell\in S})$ be as above, and let
$\mathcal{W}_{\Phi}$ consist of the set of
$(a,b)\in\mathbb{Z}\times\mathbb{Z}$ such that
$\iota_{\ell}\left((a,b)\right)\in\mathcal{U}_{\ell}$ for each prime $\ell\in
S$. For $x>0$, let $\mathcal{W}_{\Phi}(x)$ be the subset of
$\mathcal{W}_{\Phi}$ consisting of pairs $(a,b)$ with $H(E_{(a,b)})\leq x$.
The density of $\mathcal{W}_{\Phi}$ is defined with respect to the naïve
height, as follows

$\mathfrak{d}_{\Phi}\mathrel{\mathop{\mathchar
58\relax}}=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{W}_{\Phi}(x)}{\\#\mathcal{W}(x)}.$

######  Theorem 4.5.

Let $S$ be a collection of primes and
$\Phi=(S,\\{\mathcal{U}_{\ell}\\}_{\ell\in S})$ be a local datum over $S$
defined by congruence conditions. Then, the density $\mathfrak{d}_{\Phi}$ is
as follows

$\mathfrak{d}_{\Phi}=\prod_{\ell\in S}\mu\left(\mathcal{U}_{\ell}\right),$

where the limit is taken with respect to the ascending order for $S$.

###### Proof.

The proof of this result follows from [8, Proposition 3.2]. The only
difference is in the notion of height used, here we work with the naïve
height. Their proof however applies to this case with very minor adaptations.
∎

The following is an application of the above result.

######  Corollary 4.6.

The following assertions are true.

  1. (1)

The density of the set of minimal Weierstrass equations is given by

$\rho^{M}=\prod_{\ell}(1-\ell^{-10})=\frac{1}{\zeta(10)},$

as $\ell$ runs over all prime numbers.

  2. (2)

Let $\Sigma$ be a finite set of primes $\geq 5$. Let $\mathcal{W}_{\Sigma,n}$
be the set of minimal Weierstrass equations for which the Kodaira-type of the
associated elliptic curve $E_{(a,b)}$ is $\operatorname{I}_{n}$ for the primes
$\ell\in\Sigma$. Then,

$\mathfrak{d}\left(\mathcal{W}_{\Sigma,n}\right)=\prod_{\ell\in\Sigma}\frac{(\ell-1)^{2}}{\ell^{n+2}}\times\prod_{\ell\notin\Sigma}(1-\ell^{-10}).$

###### Proof.

The proof of (1) is immediate from Proposition 4.1. Part (2) is an immediate
consequence of Theorem 4.5 along with Propositions 4.1 and 4.3(2). ∎

Let $\mathscr{E}$ denote the set of _all_ elliptic curves $E_{/\mathbb{Q}}$
ordered by naïve height. For $x>0$ and any subset $\mathscr{E}^{\prime}$ of
$\mathscr{E}$, define

$\mathscr{E}^{\prime}(x)\mathrel{\mathop{\mathchar
58\relax}}=\\{E\in\mathscr{E}|H(E)\leq x\\}.$

The upper density of $\mathscr{E}^{\prime}$ is defined as follows

$\mathfrak{d}\left(\mathscr{E}^{\prime}\right)\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{x\rightarrow\infty}\frac{\mathscr{E}^{\prime}(x)}{\mathscr{E}(x)}.$

We have the following result.

######  Lemma 4.7.

With the above notation,

$\lim_{x\rightarrow\infty}\frac{\\#\mathscr{E}(x)}{\\#\mathcal{W}(x)}=\rho^{M}=\frac{1}{\zeta(10)}.$

###### Proof.

Note that $\mathscr{E}$ and $\mathcal{W}_{M}(\mathbb{Z})$ differ by the set of
global minimal Weierstrass models with $\Delta_{(a,b)}=0$, thus defining
singular curves. The set with $\Delta_{(a,b)}=0$ is of density $0$ in the set
of all Weierstrass equations $\mathcal{W}$. Therefore, by Corollary 4.6,

$\lim_{x\rightarrow\infty}\frac{\\#\mathscr{E}(x)}{\\#\mathcal{W}(x)}=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{W}_{M}(x)}{\\#\mathcal{W}(x)}=\rho^{M}=\frac{1}{\zeta(10)}.$

∎

##  5\. The Euler characteristic on Average

We fix a prime number $p$ and let $E$ vary over all elliptic curves defined
over the rationals with good ordinary reduction at $p$, ordered by height.
Associated to each such elliptic curve $E_{/\mathbb{Q}}$ is its $p$-adic
truncated Euler characteristic, $\chi_{t}(\Gamma,E[p^{\infty}])$. Recall from
Lemma 3.5 that this generalization of the usual Euler characteristic is always
well-defined if $\mathrm{rank}\ E(\mathbb{Q})\leq 1$. When $\mathrm{rank}\
E(\mathbb{Q})=0$, then the usual Euler characteristic
$\chi(\Gamma,E[p^{\infty}])$ is well-defined and in that case,
$\chi_{t}(\Gamma,E[p^{\infty}])=\chi(\Gamma,E[p^{\infty}])$.

Throughout this section, assume that $\mathcal{R}_{p}(E/\mathbb{Q})\neq 0$.
This is a well-known conjecture. For evidence towards this conjecture, we
refer the reader to [5, 28]. It follows from Corollary 3.7 that
$\chi_{t}(\Gamma,E[p^{\infty}])$ is defined and given (up to a $p$-adic unit)
as follows

$\chi_{t}(\Gamma,E[p^{\infty}])\sim\frac{\mathcal{R}_{p}(E/\mathbb{Q})\times\\#\Sh(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{\ell}c_{\ell}^{(p)}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})[p^{\infty}]\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

Henceforth, we use the following shorthand notation

  1. (1)

$r_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\mathinner{\\!\left\lvert\mathcal{R}_{p}(E/\mathbb{Q})\right\rvert}_{p}^{-1}$,

  2. (2)

$s_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\\#\Sh(E/\mathbb{Q})[p^{\infty}]$,

  3. (3)

$\tau_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\ell}c_{\ell}^{(p)}(E)$,

  4. (4)

$\alpha_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\\#\left(\widetilde{E}(\mathbb{F}_{p})[p]\right)$.

The above formula can be rewritten as

(5.1) $\chi_{t}(\Gamma,E[p^{\infty}])\sim\frac{r_{p}(E)\times
s_{p}(E)\times\tau_{p}(E)\times\alpha_{p}(E)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

### 5.1.

We study the following question.

######  Question 5.1.

Given an integer $n>0$, what is the proportion of elliptic curves
$E_{/\mathbb{Q}}$ with good ordinary reduction at $p$ for which
$\chi_{t}(\Gamma,E[p^{\infty}])\geq p^{n}$. More precisely, let
$\mathscr{E}_{p^{n}|\chi}$ be the set of elliptic curves with good ordinary
reduction at $p$ for which either one of the following conditions is satisfied

  1. (i)

$\mathrm{rank}\ E(\mathbb{Q})\geq 2$,

  2. (ii)

$\mathrm{rank}\ E(\mathbb{Q})\leq 1$ and
$p^{n}|\chi_{t}(\Gamma,E[p^{\infty}])$.

Then, what is the density of the set $\mathscr{E}_{p^{n}|\chi}$?

It is conjectured that the set of elliptic curves $E$ with $\mathrm{rank}\
E(\mathbb{Q})\geq 2$ is a density zero set. Therefore, the density of
$\mathscr{E}_{p^{n}|\chi}$ _should coincide_ with the density of elliptic
curves $E_{/\mathbb{Q}}$ for which the truncated Euler characteristic
$\chi_{t}(\Gamma,E[p^{\infty}])$ is well-defined and divisible by $p^{n}$. In
the above formula (5.1), the quantities $r_{p}(E)$ and $s_{p}(E)$ are fine
invariants for which it appears to be difficult to prove unconditional results
on their average behavior.

First, we look at the quantity $\\#E(\mathbb{Q})[p^{\infty}]$ which appears in
the denominator of (5.1). In [10], W. Duke has shown that almost all elliptic
curves over $\mathbb{Q}$ have trivial torsion. In particular,
$\\#E(\mathbb{Q})[p^{\infty}]=1$ for $100\%$ of elliptic curves. Thus on
average, this term does not contribute to $\chi_{t}(\Gamma,E[p^{\infty}])$. To
prove lower bounds for the density of $\mathscr{E}_{p^{n}|\chi}$, we analyze
the quantity

$\xi_{p}(E)\mathrel{\mathop{\mathchar
58\relax}}=\tau_{p}(E)\times\alpha_{p}(E)^{2}.$

Given a set of primes $\Sigma=\\{\ell_{1},\dots,\ell_{m}\\}$ not containing
$p$, $\ell_{i}\geq 5$, and an integer $k\geq 1$, we define two sets of
elliptic curves. Let $\mathscr{E}_{\Sigma,k}$ (resp.
$\mathscr{E}_{\Sigma,k}^{\prime}$) consist of $E_{/\mathbb{Q}}\in\mathscr{E}$
such that the following conditions are satisfied

  1. (i)

$E$ has good ordinary reduction at $p$,

  2. (ii)

$p$ divides $c_{\ell}(E)$ if $\ell\in\Sigma$, and the Kodaira-type of $E$ at
$\ell$ is $\operatorname{I}_{np}$ for some integer $n\leq k$,

  3. (iii)

for $\ell\notin\Sigma\cup\\{2,3,p\\}$, the Kodaira- type of $E$ at $\ell$ is
_not_ of the form $\operatorname{I}_{m}$ for any $m\geq p$,

  4. (iv)

$\alpha_{p}(E)=1$ (resp. $p|\alpha_{p}(E)$).

It follows from the definition that $\mathscr{E}_{\Sigma,k}$ and
$\mathscr{E}_{\Sigma,k}^{\prime}$ are all mutually disjoint. Associated with
the sets $\mathscr{E}_{\Sigma,k}$ and $\mathscr{E}_{\Sigma,k}^{\prime}$ are
sets of minimal Weierstrass equations, determined by congruence data
$\Phi_{\Sigma,k}=\\{\mathcal{U}_{\ell,k}\mid\ell=p\text{ or }\ell\geq 5\\}$
and $\Phi_{\Sigma,k}^{\prime}=\\{\mathcal{U}_{\ell,k}^{\prime}\mid\ell=p\text{
or }\ell\geq 5\\}$ respectively, which we now define. For each prime
$\ell\notin\Sigma\cup\\{2,3,p\\}$ define
$\mathcal{U}_{\ell,k}=\mathcal{U}_{\ell,k}^{\prime}$ to be the complement of
$\mathcal{W}_{\operatorname{I}_{\geq p}}(\mathbb{Z}_{\ell})$ in the minimal
condition $\mathcal{W}_{M}(\mathbb{Z}_{\ell})$. For $\ell\in\Sigma$, set
$\mathcal{U}_{\ell,k}=\mathcal{U}_{\ell,k}^{\prime}=\bigsqcup_{j=1}^{k}\mathcal{W}_{\operatorname{I}_{jp}}(\mathbb{Z}_{\ell})$.
Let $\mathfrak{S}_{p}\subset\mathcal{W}(\mathbb{Z}/p\mathbb{Z})$ consisting of
all pairs $(a,b)$ for which

  1. (i)

$\Delta_{(a,b)}\neq 0$,

  2. (ii)

$\\#{E_{(a,b)}}(\mathbb{F}_{p})\not\equiv 0,1\mod{p}$.

The criterion $\\#{E_{(a,b)}}\not\equiv 1\mod p$ ensures that $E_{(a,b)}$ has
good ordinary reduction at $p$. The condition
$\\#{E_{(a,b)}}(\mathbb{F}_{p})\not\equiv 0\mod p$ is equivalent to saying
that $p$ is not an anomalous prime (in the sense of [17]). Let
$\mathfrak{S}_{p}^{\prime}\subset\mathcal{W}(\mathbb{Z}/p\mathbb{Z})$ be the
subset for which the following conditions are satisfied

  1. (i)

$\Delta_{(a,b)}\neq 0$,

  2. (ii)

$\\#{E_{(a,b)}}(\mathbb{F}_{p})\equiv 0\mod{p}$.

Data for $\mathfrak{S}_{p}$ and $\mathfrak{S}_{p}^{\prime}$ is compiled in
Tables 1 and 2, respectively. Consider the reduction map,

$\pi_{p}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{W}_{M}(\mathbb{Z}_{p})\rightarrow\mathcal{W}(\mathbb{Z}/p\mathbb{Z})$

and define $\mathcal{U}_{p,k}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{p}^{-1}\left(\mathfrak{S}_{p}\right)$ and
$\mathcal{U}_{p,k}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{p}^{-1}\left(\mathfrak{S}_{p}^{\prime}\right)$. The congruence
data for the sets $\mathscr{E}_{\Sigma,k}$ and
$\mathscr{E}_{\Sigma,k}^{\prime}$ are defined by $\mathcal{U}_{\ell,k}$ and
$\mathcal{U}_{\ell,k}^{\prime}$, respectively as $\ell$ ranges over all
primes. Set $\mathscr{E}_{\Sigma}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{k\geq 1}\mathscr{E}_{\Sigma,k}$ and
$\mathscr{E}_{\Sigma}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{k\geq 1}\mathscr{E}_{\Sigma,k}^{\prime}$.

######  Lemma 5.2.

Let $\Sigma$ be a finite set of primes not containing $\\{2,3,p\\}$. For any
integer $k$,

$\begin{split}&\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)>\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{(\ell^{10}-1)\ell^{jp-8}}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
&\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}^{\prime}\right)>\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{(\ell^{10}-1)\ell^{jp-8}}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}^{\prime}}{p^{10}-1}\right).\end{split}$

In particular,

$\begin{split}&\mathfrak{d}\left(\mathscr{E}_{\Sigma}\right)>\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
&\mathfrak{d}\left(\mathscr{E}_{\Sigma}^{\prime}\right)>\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}^{\prime}}{p^{10}-1}\right).\end{split}$

###### Proof.

We compute the upper density of $\mathscr{E}_{\Sigma,k}$ and
$\mathscr{E}_{\Sigma}$. The computation for $\mathscr{E}_{\Sigma,k}^{\prime}$
and $\mathscr{E}_{\Sigma}^{\prime}$ is analogous. Recall that
$\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)$ denotes the limit

$\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{\Sigma,k}(x)}{\\#\mathscr{E}(x)}.$

It follows from Lemma 4.7 that

(5.2)
$\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)=\zeta(10)\times\limsup_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{\Sigma,k}(x)}{\\#\mathcal{W}(x)}.$

In this setting, Theorem 4.5 asserts that

$\lim_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{\Sigma,k}(x)}{\\#\mathcal{W}(x)}=\prod_{\ell}\mu\left(\mathcal{U}_{\ell,k}\right).$

Using Propositions 4.1 and 4.3(3) for the case when
$\ell\not\in\Sigma\cup\\{2,3,p\\}$ and Proposition 4.3(2) when
$\ell\in\Sigma$, we obtain

(5.3)
$\mu\left(\mathcal{U}_{\ell,k}\right)=\rho^{M}(\ell)=\begin{cases}1-\ell^{-10}-\frac{\ell-1}{\ell^{p+1}}&\textrm{when
}\ell\notin\Sigma\cup\\{2,3,p\\}\\\
\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{\ell^{jp+2}}&\textrm{when
}\ell\in\Sigma.\end{cases}$

Recall that $\mathcal{U}_{p,k}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{p}^{-1}\left(\mathfrak{S}_{p}\right)$, and hence,

(5.4) $\mu\left(\mathcal{U}_{p,k}\right)=\frac{\\#\mathfrak{S}_{p}}{p^{2}}.$

From (5.2), (5.3) and (5.4), we obtain the following expression

$\begin{split}\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)=&\zeta(10)\times\prod_{\ell\notin\Sigma\cup\\{2,3,p\\}}\left(1-\ell^{-10}-\frac{\ell-1}{\ell^{p+1}}\right)\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{\ell^{jp+2}}\right)\times\left(\frac{\\#\mathfrak{S}_{p}}{p^{2}}\right)\\\
>&\prod_{\ell\notin\Sigma\cup\\{2,3,p\\}}\left(\frac{1-\ell^{-10}-\frac{\ell-1}{\ell^{p+1}}}{1-\ell^{-10}}\right)\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{(1-\ell^{-10})\ell^{jp+2}}\right)\times\left(\frac{\\#\mathfrak{S}_{p}}{(1-p^{-10})p^{2}}\right)\\\
=&\prod_{\ell\notin\Sigma\cup\\{2,3,p\\}}\left(1-\frac{(\ell-1)}{(\ell^{10}-1)\ell^{p-9}}\right)\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{k}\frac{(\ell-1)^{2}}{(\ell^{10}-1)\ell^{jp-8}}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right).\\\
\end{split}$

Now notice that

$\prod_{\ell\notin\Sigma\cup\\{2,3,p\\}}\left(1-\frac{(\ell-1)}{(\ell^{10}-1)\ell^{p-9}}\right)>\prod_{\ell\notin\Sigma\cup\\{2,3,p\\}}\left(1-\frac{1}{\ell^{p}}\right)>\frac{1}{\zeta(p)},$

from which we obtain the result for $\mathfrak{d}(\mathscr{E}_{\Sigma,k})$. In
order to deduce the result for $\mathscr{E}_{\Sigma}$, it suffices to note
that for all $k\geq 1$, the set $\mathscr{E}_{\Sigma,k}$ is contained in
$\mathscr{E}_{\Sigma}$, and hence,

$\begin{split}\mathfrak{d}\left(\mathscr{E}_{\Sigma}\right)&\geq\lim_{k\rightarrow\infty}\mathfrak{d}\left(\mathscr{E}_{\Sigma,k}\right)\\\
&>\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\sum_{j=1}^{\infty}\frac{(\ell-1)^{2}}{(\ell^{10}-1)\ell^{jp-8}}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
&=\frac{1}{\zeta(p)}\times\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right).\end{split}$

∎

For a given integer $n$, let $\mathcal{S}_{n}$ be the set of all sets
$\Sigma=\\{\ell_{1},\dots,\ell_{n}\\}$ of primes not containing $\\{2,3,p\\}$
such that $\\#\Sigma=n$. If $n\leq 0$, the set $\mathcal{S}_{n}$ is taken to
be empty. The notation $\Sigma\in\mathcal{S}_{n}$ means that $\Sigma$ is a set
in the collection $\mathcal{S}_{n}$. The sum over $\mathcal{S}_{0}$ is taken
to be equal to $1$, and that over $\mathcal{S}_{n}$ for $n<0$ is taken to be
$0$. From our computations, we obtain the following lower bound for the
density $\mathscr{E}_{p^{n}|\chi}$,

$\mathfrak{d}\left(\mathscr{E}_{p^{n}|\chi}\right)\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{p^{n}|\chi}(x)}{\\#\mathscr{E}(x)}.$

######  Theorem 5.3.

With notation as above,

$\begin{split}\mathfrak{d}\left(\mathscr{E}_{p^{n}|\chi}\right)>&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
+&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n-2}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}^{\prime}}{p^{10}-1}\right).\end{split}$

We remind the reader that data for $\mathfrak{S}_{p}$ and
$\mathfrak{S}_{p}^{\prime}$ is recorded in Section 7.

###### Proof.

Recall that $\xi_{p}(E)=\tau_{p}(E)\times\alpha_{p}(E)^{2}$. Let
$\mathscr{E}^{\prime}\subset\mathscr{E}$ consist of those elliptic curves for
which $p^{n}|\xi_{p}(E)$. It follows from (5.1) that if
$E\in\mathscr{E}^{\prime}$ and $E(\mathbb{Q})[p]=0$, then
$p^{n}|\chi_{t}(\Gamma,E[p^{\infty}])$. Recall that by the aforementioned
theorem of Duke, $E(\mathbb{Q})[p]=0$ for $100\%$ of elliptic curves. Hence
$\mathfrak{d}\left(\mathscr{E}_{p^{n}|\chi}\right)\geq\mathfrak{d}\left(\mathscr{E}^{\prime}\right)$.

Let $\mathcal{E}_{1}$ (resp. $\mathcal{E}_{2}$) be the set of elliptic curves
$E_{/\mathbb{Q}}$ for which $\alpha_{p}(E)=1$ and $p^{n}$ divides
$\tau_{p}(E)$ (resp. $p|\alpha_{p}(E)$ and $p^{n-2}|\tau_{p}(E)$). The two
sets are disjoint and

$\mathscr{E}^{\prime}\supseteq\mathcal{E}_{1}\cup\mathcal{E}_{2}.$

For $\Sigma\in\mathcal{S}_{n}$, the set $\mathscr{E}_{\Sigma}$ is contained in
$\mathcal{E}_{1}$. On the other hand, for $\Sigma\in\mathcal{S}_{n-2}$, the
set $\mathscr{E}_{\Sigma}^{\prime}$ is contained in $\mathcal{E}_{2}$. Note
that, if $n\leq 2$, the statement is vacuous since $\mathcal{S}_{n-2}$ is the
empty set. Since the sets $\mathscr{E}_{\Sigma}$ and
$\mathscr{E}_{\Sigma}^{\prime}$ are all mutually disjoint, we obtain a lower
bound

$\mathfrak{d}\left(\mathscr{E}_{p^{n}|\chi}\right)\geq\sum_{\Sigma\in\mathcal{S}_{n}}\mathfrak{d}\left(\mathscr{E}_{\Sigma}\right)+\sum_{\Sigma\in\mathcal{S}_{n-2}}\mathfrak{d}\left(\mathscr{E}_{\Sigma}^{\prime}\right).$

The result now follows from Lemma 5.2. ∎

##  6\. Iwasawa Invariants on Average

In this section, $p$ will be a fixed odd prime number and $E$ will vary over a
certain set of elliptic curves. For any positive integer $n$, we show that the
set of elliptic curves $E$ for which
$\mu_{p}(E/\mathbb{Q}_{\infty})+\lambda_{p}(E/\mathbb{Q}_{\infty})\geq n$ has
positive density, and further, there is an explicit lower bound for this
density expressible in terms of $p$ and $n$ alone. It is a conjecture of
Greenberg (see [11, Conjecture 1.11]) that when $E$ has good ordinary
reduction at $p$ and the residual representation $E[p]$ is irreducible as a
module over $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, then
$\mu_{p}(E/\mathbb{Q}_{\infty})=0$. Therefore, one expects that
$\mu_{p}(E/\mathbb{Q}_{\infty})=0$ for $100\%$ of the elliptic curves.
Assuming Greenberg’s conjecture, the results in this section imply that
$\lambda_{p}(E/\mathbb{Q}_{\infty})\geq n$ for a certain explicitly defined
proportion of elliptic curves. We now make these notions more precise. For an
elliptic curve $E_{/\mathbb{Q}}$ with good ordinary reduction at $p$, recall
that $g_{p}(E/\mathbb{Q}_{\infty})$ denotes the minimum number of generators
for $\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\vee}$. Recall that
Lemma 2.1 asserts that

(6.1) $\mu_{p}(E/\mathbb{Q}_{\infty})+\lambda_{p}(E/\mathbb{Q}_{\infty})\geq
g_{p}(E/\mathbb{Q}_{\infty}).$

Let $n$ be a positive integer and set $\mathscr{E}_{g\geq n}$ (resp.
$\mathscr{E}_{\mu+\lambda\geq n}$) to be the set of elliptic curves
$E_{/\mathbb{Q}}$ ordered by naïve height for which the following conditions
are satisfied

  1. (i)

$E$ has good ordinary reduction at $p$,

  2. (ii)

$g_{p}(E/\mathbb{Q}_{\infty})\geq n$ (resp.
$\mu_{p}(E/\mathbb{Q}_{\infty})+\lambda_{p}(E/\mathbb{Q}_{\infty})\geq n$).

Here, we analyze lower bounds for the density
$\mathfrak{d}\left(\mathscr{E}_{g\geq n}\right)$. It follows from (6.1) that

$\mathfrak{d}\left(\mathscr{E}_{\mu+\lambda\geq
n}\right)\geq\mathfrak{d}\left(\mathscr{E}_{g\geq n}\right).$

We prove an explicit lower bound for $\mathfrak{d}\left(\mathscr{E}_{g\geq
n}\right)$, thus obtaining one for
$\mathfrak{d}\left(\mathscr{E}_{\mu+\lambda\geq n}\right)$ as well.

### 6.1.

We first prove an algebraic criterion for the invariant $g$ to be larger than
a given number $n$. As always, let $E_{/\mathbb{Q}}$ be an elliptic curve with
good ordinary reduction at a fixed odd prime $p$. Let $E(\mathbb{Q})^{*}$ be
the inverse limit $E(\mathbb{Q})^{*}\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}E(\mathbb{Q})/p^{n}$, and set

$\widehat{E(\mathbb{Q})^{*}}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Hom}\left(E(\mathbb{Q})^{*},\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$

its Pontryagin dual.

######  Lemma 6.1.

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ be a prime for which
$\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite. Then, the map $\Phi_{E,\mathbb{Q}}$
fits into an exact sequence

$H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]\right)\xrightarrow{\Phi_{E,\mathbb{Q}}}\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q})\rightarrow\widehat{E(\mathbb{Q})^{*}}.$

###### Proof.

The compact Selmer group $\mathfrak{S}_{p^{\infty}}(E/\mathbb{Q})$ is defined
as the inverse limit

$\mathfrak{S}_{p^{\infty}}(E/\mathbb{Q})\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathrm{Sel}_{p^{n}}(E/\mathbb{Q}).$

By the Cassels-Poitou-Tate exact sequence, we know that the map
$\Phi_{E,\mathbb{Q}}$ fits into a long exact sequence (see for example, [6, p.
9])

$0\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow
H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]\right)\xrightarrow{\Phi_{E,\mathbb{Q}}}\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q})\rightarrow\widehat{\mathfrak{S}_{p^{\infty}}(E/\mathbb{Q})}\rightarrow\dots.$

When $\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite, it follows that
$\mathfrak{S}_{p^{\infty}}(E/\mathbb{Q})$ is identified with
$E(\mathbb{Q})^{*}$ (see [6, proof of Proposition 1.9]). ∎

######  Definition 6.2.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good ordinary reduction at the
prime $p$. Define $\mathfrak{a}_{E}^{(p)}$ to be the number of primes
$\ell\neq p$ for which $p$ divides the Tamagawa number $c_{\ell}(E)$. Set

$\mathfrak{b}_{E}^{(p)}\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}1&\textrm{when }p|\\#\widetilde{E}(\mathbb{F}_{p})\\\
0&\textrm{otherwise.}\end{cases}$

Finally, write $\mathfrak{c}_{E}^{(p)}\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{a}_{E}^{(p)}+\mathfrak{b}_{E}^{(p)}$.

We have the following characterization of the minimal number of generators
$g_{p}(E/\mathbb{Q}_{\infty})$.

######  Lemma 6.3.

Let $E_{/\mathbb{Q}}$ be an elliptic curve for which the following conditions
are satisfied.

  1. (1)

$E$ has good ordinary reduction at $p$,

  2. (2)

$E(\mathbb{Q})[p]=0$,

  3. (3)

$\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite.

Then, $g_{p}(E/\mathbb{Q}_{\infty})\geq\mathfrak{c}_{E}^{(p)}$. In particular,

(6.2)
$\lambda_{p}(E/\mathbb{Q}_{\infty})+\mu_{p}(E/\mathbb{Q}_{\infty})\geq\mathfrak{c}_{E}^{(p)}.$

###### Proof.

Since it is assumed that $E(\mathbb{Q})[p]=0$, it follows that
$\widehat{E(\mathbb{Q})^{*}}\simeq(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\mathrm{rank}\
E(\mathbb{Q})}$. Lemma 6.1 asserts that there is an exact sequence

$H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]\right)\xrightarrow{\Phi_{E,\mathbb{Q}}}\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q})\rightarrow\widehat{E(\mathbb{Q})^{*}}.$

Consider the diagram

(6.3)
${0}$${\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}$${H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]\right)}$${\operatorname{im}\Phi_{E,\mathbb{Q}}}$${0}$${0}$${\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}}$${H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]\right)^{\Gamma}}$${\left(\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q}_{\infty})\right)^{\Gamma}.}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h^{\prime}}$

Since $E(\mathbb{Q})[p]=0$, it follows from standard arguments that the map
$g$ is an isomorphism (see [6, p. 34]). As a result, the snake lemma applied
to (6.3) implies that there is a short exact sequence

(6.4)
$0\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}\rightarrow\ker
h^{\prime}\rightarrow 0.$

Next we have the diagram

(6.5) ${0}$${\operatorname{im}\Phi_{E,\mathbb{Q}}}$${\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q})}$${\widehat{E(\mathbb{Q})^{*}}}$${0}$${\left(\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q}_{\infty})\right)^{\Gamma}}$${\left(\bigoplus_{\ell\in
S}J_{\ell}(E/\mathbb{Q}_{\infty})\right)^{\Gamma}}$${0.}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\scriptstyle{j}$

This yields

(6.6) $0\rightarrow\ker h^{\prime}\rightarrow\ker
h\rightarrow\widehat{E(\mathbb{Q})^{*}}.$

Using the sequences (6.4) and (6.6), we estimate

$g_{p}(E/\mathbb{Q}_{\infty})=\dim_{\mathbb{F}_{p}}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}[p].$

From the short exact sequence (6.4), we obtain the following exact sequence

$0\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})[p]\rightarrow\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}[p]\rightarrow\left(\ker
h^{\prime}\right)[p]\rightarrow\frac{\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}{p\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}.$

Since it is assumed that $\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite, it follows
that

$\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})\simeq\left(\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\mathrm{rank}\
E(\mathbb{Q})}\oplus D,$

where $D$ is a finite group. It follows that

(6.7)
$\dim_{\mathbb{F}_{p}}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})[p]=\dim_{\mathbb{F}_{p}}\left(\frac{\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}{p\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}\right)+\mathrm{rank}\
E(\mathbb{Q}).$

Now,

$\displaystyle g_{p}(E/\mathbb{Q}_{\infty})$
$\displaystyle=\dim_{\mathbb{F}_{p}}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\Gamma}[p]$
$\displaystyle\geq\dim_{\mathbb{F}_{p}}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})[p]+\dim_{\mathbb{F}_{p}}\left(\ker
h^{\prime}\right)[p]-\dim_{\mathbb{F}_{p}}\left(\frac{\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}{p\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})}\right)$
$\displaystyle\geq\dim_{\mathbb{F}_{p}}\left(\ker
h^{\prime}\right)[p]+\mathrm{rank}\ E(\mathbb{Q}),$
$\displaystyle\geq\dim_{\mathbb{F}_{p}}\left(\ker h\right)[p],$

where the final inequality is a consequence of (6.6). It follows from [6,
Lemma 3.4 and Proposition 3.5] that

$\dim_{\mathbb{F}_{p}}\left(\ker h\right)[p]\geq\mathfrak{c}_{E}^{(p)},$

and this completes the proof. ∎

### 6.2.

Given a positive integer $n$, we prove an explicit lower bound for the density
of elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$, for
which $g_{p}(E/\mathbb{Q}_{\infty})\geq n$.

######  Theorem 6.4.

With notation as above,

$\begin{split}\mathfrak{d}\left(\mathscr{E}_{g\geq
n}\right)>&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
+&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n-1}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}^{\prime}}{p^{10}-1}\right).\end{split}$

###### Proof.

Recall from the definition that if $E\in\mathscr{E}_{\Sigma}$, then there are
at least $\\#\Sigma$ many primes (distinct from $2,3,p$) for which the
Tamagawa factors $c_{\ell}$ are divisible by $p$.

First, for $\Sigma\in\mathcal{S}_{n}$ and $E\in\mathscr{E}_{\Sigma}$, it
follows that $\mathfrak{c}_{E}^{(p)}\geq n$. Next, for
$\Sigma\in\mathcal{S}_{n-1}$ and $E\in\mathscr{E}_{\Sigma}^{\prime}$, there
are at least $n-1$ of the Tamagawa numbers $c_{\ell}$ are divisible by $p$ and
also, $p|\alpha_{p}(E)$. Once again, $\mathfrak{c}_{E}^{(p)}\geq n$. Since the
sets $\mathscr{E}_{\Sigma}$ and $\mathscr{E}_{\Sigma}^{\prime}$ are all
mutually disjoint, we have that

$\mathfrak{d}\left(\mathscr{E}_{g\geq
n}\right)\geq\sum_{\Sigma\in\mathcal{S}_{n}}\mathfrak{d}\left(\mathscr{E}_{\Sigma}\right)+\sum_{\Sigma\in\mathcal{S}_{n-1}}\mathfrak{d}\left(\mathscr{E}_{\Sigma}^{\prime}\right).$

The result now follows from Lemma 5.2. ∎

######  Corollary 6.5.

Let $n>0$ be an integer and $p$ a prime number. The density
$\mathfrak{d}(\mathscr{E}_{\mu+\lambda\geq n})$ of the set of elliptic curves
$E_{/\mathbb{Q}}$ for which

  1. (1)

$E$ has good ordinary reduction at $p$,

  2. (2)

$\mu_{p}(E/\mathbb{Q}_{\infty})+\lambda_{p}(E/\mathbb{Q}_{\infty})\geq n$

is positive. In particular,

$\begin{split}\mathfrak{d}(\mathscr{E}_{\mu+\lambda\geq
n})>&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}}{p^{10}-1}\right)\\\
+&\frac{1}{\zeta(p)}\times\sum_{\Sigma\in\mathcal{S}_{n-1}}\left(\prod_{\ell\in\Sigma}\left(\frac{\ell^{8}(\ell-1)^{2}}{(\ell^{10}-1)(\ell^{p}-1)}\right)\right)\times\left(\frac{p^{8}\\#\mathfrak{S}_{p}^{\prime}}{p^{10}-1}\right).\end{split}$

###### Proof.

Since $\mu_{p}(E/\mathbb{Q}_{\infty})+\lambda_{p}(E/\mathbb{Q}_{\infty})\geq
g_{p}(E/\mathbb{Q}_{\infty})$, the result follows from Theorem 6.4. ∎

##  7\. Tables

Table 1. Data for $\mathfrak{S}_{p}$ for primes $7\leq p<150$. $p$ | $\\#\mathfrak{S}_{p}/p^{2}$ | $p$ | $\\#\mathfrak{S}_{p}/p^{2}$
---|---|---|---
7 | 0.653061224489796 | 71 | 0.867883356476890
11 | 0.702479338842975 | 73 | 0.932257459185588
13 | 0.781065088757396 | 79 | 0.887357795225124
17 | 0.802768166089965 | 83 | 0.898679053563652
19 | 0.789473684210526 | 89 | 0.899886377982578
23 | 0.790170132325142 | 97 | 0.943777234562653
29 | 0.832342449464923 | 101 | 0.911675325948436
31 | 0.842872008324662 | 103 | 0.913375435950608
37 | 0.915997078159240 | 107 | 0.925845051969604
41 | 0.868530636525877 | 109 | 0.945374968437000
43 | 0.874526771227691 | 113 | 0.929751742501370
47 | 0.853779990946129 | 127 | 0.936139872279745
53 | 0.897828408686365 | 131 | 0.897674960666628
59 | 0.866417696064349 | 137 | 0.952847780915339
61 | 0.900295619457135 | 139 | 0.935665855804565
67 | 0.940966807752283 | 149 | 0.933291293184992
Table 2. Data for $\mathfrak{S}_{p}^{\prime}$ for primes $7\leq p<150$. $p$ | $\\#\mathfrak{S}_{p}^{\prime}/p^{2}$ | $p$ | $\\#\mathfrak{S}_{p}^{\prime}/p^{2}$
---|---|---|---
7 | 0.0816326530612245 | 71 | 0.0208292005554453
11 | 0.0413223140495868 | 73 | 0.0270219553387127
13 | 0.0710059171597633 | 79 | 0.0374939913475405
17 | 0.0276816608996540 | 83 | 0.0178545507330527
19 | 0.0581717451523546 | 89 | 0.0222194167403106
23 | 0.0415879017013233 | 97 | 0.0255074928260176
29 | 0.0332936979785969 | 101 | 0.00980296049406921
31 | 0.0312174817898023 | 103 | 0.0288434348194929
37 | 0.0306793279766253 | 107 | 0.00925845051969604
41 | 0.0118976799524093 | 109 | 0.0181802878545577
43 | 0.0567874526771228 | 113 | 0.0263137285613595
47 | 0.0208239022181983 | 127 | 0.0169260338520677
53 | 0.0277678889284443 | 131 | 0.0189382903094225
59 | 0.0166618787704683 | 137 | 0.0108689860940913
61 | 0.0349368449341575 | 139 | 0.0142849748977796
67 | 0.0147026063711294 | 149 | 0.0133327327597856

## Acknowledgements

DK acknowledges the support of the PIMS Postdoctoral Fellowship. AR would like
to thank Lawrence Washington and Tom Weston.

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