# Arithmetic statistics for the Fine Selmer group in Iwasawa theory

Anwesh Ray<EMAIL_ADDRESS>and  R.Sujatha<EMAIL_ADDRESS>Department of Mathematics
University of British Columbia
Vancouver BC, Canada V6T 1Z2

###### Abstract.

We study arithmetic statistics for Iwasawa invariants associated to the fine
Selmer groups associated to elliptic curves. The results also provide insight
into the vanishing of the classical Iwasawa $\mu$-invariant associated to
certain families of nonabelian number fields.

###### Key words and phrases:

Arithmetic statistics, Iwasawa theory, Fine Selmer groups, elliptic curves,
local torsion, Euler characteristic.

###### 2010 Mathematics Subject Classification:

11G05, 11R23 (primary); 11R45 (secondary).

##  1\. Introduction

Iwasawa studied growth questions of class groups in certain infinite towers of
number fields. These initial developments led to deep questions, many of which
are still unresolved. Given a number field $F$ and a prime number $p$, let
$F_{\operatorname{cyc}}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of
$F$. Letting $F_{n}\subset F_{\operatorname{cyc}}$ be such that
$[F_{n}\mathrel{\mathop{\mathchar 58\relax}}F]=p^{n}$, denote by
$\operatorname{Cl}_{p}(F_{n})$ the $p$-Sylow subgroup of the class group of
$F_{n}$. Iwasawa showed that for $n\gg 0$,

(1.1) $\\#\operatorname{Cl}_{p}(F_{n})=p^{p^{n}\mu+\lambda n+\nu},$

where $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{Z}$. Let
$K(F_{\operatorname{cyc}})\subset\bar{K}$ be the maximal abelian pro-$p$
extension of $F_{\operatorname{cyc}}$ in which all primes of
$F_{\operatorname{cyc}}$ are unramified. The classical $\mu=0$ conjecture of
Iwasawa states that $X(F_{\operatorname{cyc}})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(K(F_{\operatorname{cyc}})/F)$ is finitely
generated as a $\mathbb{Z}_{p}$-module. Equivalently, the invariant $\mu$ in
(1.1) is equal to $0$. The $\mu=0$ conjecture was proved by Ferrero and
Washington [FW79] for all abelian number fields $F/\mathbb{Q}$.

Mazur initiated the Iwasawa theory of elliptic curves and abelian varieties in
[Maz72]. Let $E$ be an elliptic curve and $p$ an odd prime number at which $E$
has good ordinary reduction. The main object of study is the $p$-primary
Selmer group over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$.
Greenberg conjectured that when the residual representation on the $p$-torsion
subgroup of $E$ is irreducible, the Iwasawa $\mu$-invariant of this Selmer
group must vanish, see [Gre99, Conjecture 1.11]. This means that the
Pontryagin dual of the Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$ is
a finitely generated $\mathbb{Z}_{p}$-module. The rank of the dual Selmer
group as a $\mathbb{Z}_{p}$-module is the $\lambda$-invariant, and is of
significant interest. It was proven by Kato [Kat04] and Rohlrich [Roh84] that
the rank of $E(\mathbb{Q}_{n})$ is bounded as $n\rightarrow\infty$. In fact,
$\operatorname{rank}E(\mathbb{Q}_{n})$ is always bounded above by the
$\lambda$-invariant. When the residual representation is reducible, there are
examples and explicit criteria under which the $\mu$-invariant is positive,
see [RS21, sections 3 and 7].

Let $E_{/\mathbb{Q}}$ be an elliptic curve with good reduction at an odd prime
$p$ (either ordinary or supersingular). The _fine Selmer group_ was
systematically studied by Coates and the second named author in [CS05]. This
is the subgroup $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$
of the $p$-primary Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$
consisting of all cohomology classes that are trivial when localized at the
primes above $p$ and the primes at which $E$ has bad reduction (cf.
Definitions 2.1 and 2.2). The fine Selmer group is closely related to class
group extensions studied by Iwasawa. Conjecture A in [CS05] predicts that the
$\mu$-invariant of the fine Selmer group is always zero. Let
$F=\mathbb{Q}(E[p])$ be the extension of $\mathbb{Q}$ that is _cut out_ by the
residual representation on $E[p]$. In other words, it is the Galois extension
of $\mathbb{Q}$ fixed by the kernel of the residual representation. By Serre’s
Open image theorem [Ser71, section 4, Theorem 3], if $E$ is a non-CM elliptic
curve, $\operatorname{Gal}(F/\mathbb{Q})$ is isomorphic to
$\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$ for all but finitely many
primes $p$. Thus, the extension $F/\mathbb{Q}$ is in most cases non-abelian
and in fact, non-solvable; and Iwasawa’s $\mu=0$ conjecture is wide open for
such extensions. It follows from results of Coates and the second named author
that Iwasawa’s $\mu=0$ conjecture for
$\operatorname{Gal}(K(F_{\operatorname{cyc}})/F)$ is closely related to the
$\mu=0$ conjecture for the fine Selmer group. In greater detail, the
$\mu$-invariant vanishes for the fine Selmer group over
$F_{\operatorname{cyc}}$ vanishes if and only if the classical Iwasawa
$\mu$-invariant for $X(F_{\operatorname{cyc}})$ vanishes (cf. Theorem 2.3).

The arithmetic statistics of elliptic curves is concerned with the study of
the average behaviour of certain interesting arithmetic invariants associated
with elliptic curves. One such invariant is the rank of the Mordell-Weil
group. A conjecture of Katz and Sarnak [KS99] supported with heuristics from
random matrix theory predicts that when ordered by height (or conductor or
discriminant), $50\%$ of all elliptic curves have rank $0$ and the other
$50\%$ have rank $1$. Certain unconditional results on rank distribution have
been proven by Bhargava and Shankar [BS13] who study the average cardinality
of certain Selmer groups over $\mathbb{Q}$. This provides ample motivation to
study the average behavior of Iwasawa invariants of elliptic curves associated
to Selmer groups over $\mathbb{Q}_{\operatorname{cyc}}$. The main difficulty
is that the methods of Bhargava and Shankar can only be applied to Selmer
groups over $\mathbb{Q}$ and for small prime numbers $p\leq 5$. Therefore, an
altogether different method is required to study Selmer groups over the
cyclotomic $\mathbb{Z}_{p}$-extension. Arithmetic statistics for the Iwasawa
theory of elliptic curves was initiated by Kundu and the first named author in
[KR21a]. In the present article, we study similar questions for the fine
Selmer group. In greater detail, we study the following questions:

  1. (1)

Fix a prime number $p$. What can be said about the proportion of elliptic
curves $E_{/\mathbb{Q}}$ with good reduction at $p$ for which the $p$-primary
fine Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$ is infinite (i.e.,
either $\mu>0$ or $\lambda>0$)?

  2. (2)

Given an elliptic curve $E_{/\mathbb{Q}}$ and a fixed positive number $Y>0$,
what can be said about the number of primes $p\leq Y$ at which $E$ has good
reduction and the fine Selmer group of $E$ over
$\mathbb{Q}_{\operatorname{cyc}}$ is infinite.

In section 4, we study the first question and prove explicit results. From a
statistical point of view it only makes sense to consider elliptic curves of
rank $\leq 1$ (since the elliptic curves of rank $>1$ are expected to
constitute a set of density $0$). Let $p$ be a prime $\geq 5$. Consider the
set $\mathfrak{B}_{p}$ (resp. $\mathfrak{D}_{p}$) consisting of elliptic
curves $E_{/\mathbb{Q}}$ that have Mordell-Weil rank $0$ (resp. $1$), good
reduction at $p$, and for which the $p$-primary fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is infinite.
Theorem 4.6 gives a precise upper bound for the upper density of
$\mathfrak{B}_{p}$ in terms of $p$. Stated simply, the bound is of the form

$\frac{2\zeta(10)+1}{p}+O\left(\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{p^{3/2}}\right).$

A similar bound is also obtained for the set $\mathfrak{D}_{p}$, see Theorem
4.8. One also considers the set $\mathfrak{F}_{p}$ of all elliptic curves
$E_{/\mathbb{Q}}$ with Mordell-Weil rank $0$ and good _ordinary_ reduction at
$p$, such that the entire Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is
infinite. The bound obtained for the set $\mathfrak{F}_{p}$ in [KR21a, section
4] is of the form

$O\left(\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{p^{1/2}}\right)$

(see Theorem 4.4 for the refined statement). The results show that that given
a prime $p$ the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ of most elliptic
curves is finite. Furthermore, we give an explicit upper bound for the
proportion of elliptic curves for which
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ may be infinite
and this proportion becomes smaller as $p$ gets larger. In fact, the quantity
decreases at the rate of $O(1/p)$. On the other hand, for elliptic curves of
rank 0, the Selmer group does
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ shows
similar behavior on average. However, $p$ gets larger, the upper bound
decreases at a much slower rate.

In section 5, we study a variant of the second question. Let $E$ be an
elliptic curve with Mordell-Weil rank zero. According to Corollary 2.10, the
set of primes $p$ at which
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is infinite is
contained in the set of primes $\Pi_{lt,E}\cup\mathfrak{S}$. Here,
$\Pi_{lt,E}$ is the set of primes $p$ at which $E(\mathbb{Q}_{p})[p]\neq 0$.
Such primes are referred to as the set of _local torsion primes_. On the other
hand, $\mathfrak{S}$ is an explicit finite set of primes (see _loc. cit._ for
details). Given a positive number $Y>0$, we denote by $\Pi_{lt,E}(Y)$ the set
of local torsion primes $p$ that are $\leq Y$. We prove results about the
expected size of $\\#\Pi_{lt,E}(Y)$ when $Y$ is fixed and $E$ ranges over all
elliptic curves. This gives us some insight into the number of primes $\leq Y$
at which the fine Selmer group may be infinite. We use a version of the Large
Sieve inequality due to Huxley, see Theorem 5.2. Fix $Y>0$ (for example
$Y=100$), let $P(Y)$ be the quantity defined in the discussion following the
proof of Theorem 5.3. The main result of section 5, Theorem 5.4, shows that
there is an absolute constant $c>0$ such that for any number $\beta>0$, the
proportion of all elliptic curves $E_{/\mathbb{Q}}$ for which

(1.2) $|\\#\Pi_{lt,E}(Y)-P(Y)|<\beta\sqrt{P(Y)}$

is $\geq(1-\frac{c}{\beta^{2}})$. Thus, the average size of $\\#\Pi_{lt,E}(Y)$
is close to $P(Y)$ for an explicit positive density set of elliptic curves.
For instance, setting $\beta=10\sqrt{c}$, we find that $\geq 99\%$ of elliptic
curves satisfy the above inequality (1.2). These results shed light
distribution of primes outside of which the fine Selmer groups is finite, when
the elliptic curve $E$ has Mordell-Weil rank zero.

### Acknowledgements

The first named author would like to thank Jeffrey Hatley, Debanjana Kundu and
Antonio Lei for helpful discussions and comments. He would also like to thank
Christian Wuthrich and Will Sawin for their suggestions in response to the
following question: https://mathoverflow.net/questions/409642/. The second
named author gratefully acknowledges support from NSERC Discovery grant
2019-03987.

##  2\. Iwasawa theory of the Fine Selmer group

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime at
which $E$ has good reduction. Fix an algebraic closure $\bar{\mathbb{Q}}$ of
$\mathbb{Q}$ and a finite set of primes $S$ containing $\\{p,\infty\\}$
outside of which $E$ has good reduction. Let $\mathbb{Q}_{S}$ be the maximal
extension of $\mathbb{Q}$ in $\bar{\mathbb{Q}}$ in which all primes
$\ell\notin S$ are unramified. Denote by $E[p^{n}]$ the $p^{n}$-torsion
subgroup of $E(\bar{\mathbb{Q}})$, and set $E[p^{\infty}]$ to be the the union
of all $p$-primary torsion groups $E[p^{n}]$. Given a field
$L\subset\mathbb{Q}_{S}$, set
$H^{i}(\mathbb{Q}_{S}/L,\cdot)\mathrel{\mathop{\mathchar
58\relax}}=H^{i}(\operatorname{Gal}(\mathbb{Q}_{S}/L),\cdot)$. For a number
field $L$ and a prime $\ell\in S$, set $K_{\ell}(\cdot/L)$ to be the direct
sum $\bigoplus_{v|\ell}H^{1}(L_{v},\cdot)$. Here, $v$ ranges over the primes
of $L$ above $\ell$.

######  Definition 2.1.

Let $L$ be a number field. The _fine Selmer group_
$\mathcal{R}_{p^{\infty}}(E/L)$ is defined to be the kernel of the restriction
map

$\operatorname{ker}\left\\{H^{1}\left(\mathbb{Q}_{S}/L,E[p^{\infty}]\right)\longrightarrow\bigoplus_{\ell\in
S}K_{\ell}(E[p^{\infty}]/L)\right\\}.$

We study the fine Selmer group from an Iwasawa theoretic point of view. This
involves passing up an infinite tower of number fields. Let
$\mathbb{Q}(\mu_{p^{\infty}})$ be the field generated by $\mathbb{Q}$ and the
$p$-power roots of unity $\mu_{p^{\infty}}\subset\bar{\mathbb{Q}}$. For
$n\in\mathbb{Z}_{\geq 1}$, let
$\mathbb{Q}_{n}\subset\mathbb{Q}(\mu_{p^{\infty}})$ be the extension of
$\mathbb{Q}$ contained in $\mathbb{Q}(\mu_{p^{\infty}})$ such that
$[\mathbb{Q}_{n}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=p^{n}$. The
_cyclotomic $\mathbb{Z}_{p}$-extension_ of $\mathbb{Q}$ is the union
$\mathbb{Q}_{\operatorname{cyc}}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{n\geq 1}\mathbb{Q}_{n}$, with Galois group
$\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{\operatorname{cyc}}/\mathbb{Q})$.
Fix a topological generator $\gamma\in\Gamma$. This gives rise to an
isomorphism $\mathbb{Z}_{p}\xrightarrow{\sim}\Gamma$, sending
$a\in\mathbb{Z}_{p}$ to $\gamma^{a}$. The _Iwasawa algebra_ is the completed
group algebra $\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})]$.
Fix an isomorphism of $\Lambda$ with the formal power series ring
$\mathbb{Z}_{p}\llbracket T\rrbracket$, where $T$ is the formal variable in
place of $(\gamma-1)\in\Lambda$.

######  Definition 2.2.

The fine Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$ is taken to be
the direct limit

(2.1)
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{n}\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{n})$

and is a cofinitely generated module over $\Lambda$.

In other words, the Pontryagin dual

$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\operatorname{cts}}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}}),\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$

is finitely generated over $\Lambda$. We refer to [CS00] for the definition of
the $p$-primary Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$, which we
denote by
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$. Note
that the fine Selmer group is a subgroup of
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$. It
follows from a deep result of Kato [Kat04] that
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ is
torsion over $\Lambda$. In particular it follows that
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ is
torsion over $\Lambda$.

Let $M$ be a cofinitely generated $\Lambda$-module. We introduce the Iwasawa
invariants associated to the finitely generated $\Lambda$-module
$M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$.
Associated with $M^{\vee}$ are its Iwasawa invariants. By the _Structure
Theorem_ for finitely generated $\Lambda$-modules [Was97, Theorem 13.12],
$M^{\vee}$ is pseudo-isomorphic to a finite direct sum of cyclic
$\Lambda$-modules. In other words, there is a map of $\Lambda$-modules

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

with finite kernel and cokernel. Here, $\alpha$ is the rank of $M^{\vee}$ as a
$\Lambda$-module, $\mu_{i}>0$ 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

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

where $\mu=\sum_{i}\mu_{i}$ and $g_{M}(T)$ is the distinguished polynomial
$\prod_{j}f_{j}(T)$. The quantity $\mu$ is the $\mu$-invariant, which is
denoted $\mu_{p}(M)$, and the $\lambda$-invariant $\lambda_{p}(M)$ is the
degree of $g_{M}(T)$. In this paper, we shall set
$\mu_{p}(E/\mathbb{Q}_{\operatorname{cyc}})$ (resp.
$\lambda_{p}(E/\mathbb{Q}_{\operatorname{cyc}})$) to denote the
$\mu$-invariant (resp. $\lambda$-invariant) of the classical Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$. On the
other hand, we set
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})$ (resp.
$\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})$) to
denote the $\mu$-invariant (resp. $\lambda$-invariant) of the fine Selmer
group $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$. At this
point, we note that for the fine Selmer group, there is no known analog of the
main conjecture. Therefore, the fine Selmer group is a purely algebraic object
and in many respects, its structure is more mysterious than the classical
Selmer group.

Let $E$ be an elliptic curve with good reduction at $p$ and let
$F\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{Q}(E[p])$, i.e., the field
which is fixed by the kernel of the mod-$p$ representation on $E[p]$. Let
$\mu_{p}(F_{\operatorname{cyc}}/F)$ be the classical Iwasawa $\mu$-invariant
associated with the maximal abelian pro-$p$ extension of
$F_{\operatorname{cyc}}$ which is unramified at all primes. However, this is
expected to be true for any number field $F$. The following result due to
Coates and the second named author establishes a relationship between the
vanishing of the classical $\mu$-invariant and the $\mu$-invariant of the fine
Selmer group.

######  Theorem 2.3.

Let $E$ be an elliptic curve and $p$ an odd prime number. Let $F$ be the
number field given by $\mathbb{Q}(E[p])$. Then,

$\mu_{p}(F_{\operatorname{cyc}}/F)=0\Leftrightarrow\mu_{p}^{\operatorname{fn}}(E/F_{\operatorname{cyc}})=0.$

###### Proof.

The statement follows directly from from [CS05, Theorem 3.4]. ∎

Conjecture A of [CS05] asserts that
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$ for all
elliptic curves $E_{/\mathbb{Q}}$. The above result shows that it is a special
case of the $\mu=0$ conjecture for number fields. One should note that the
above result in fact provides us with a criterion for the classical $\mu=0$
conjecture to hold for number field extensions $F=\mathbb{Q}(E[p])$, which
need not be abelian or even solvable. The vanishing of the
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})$ thus provides
evidence for the vanishing of the classical $\mu$-invariant of
$F_{\operatorname{cyc}}/F$, where $F$ is the number field $\mathbb{Q}(E[p])$.

Now consider for a moment the special case when the residual representation

$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$

on $E[p]$ is reducible, i.e.,
$\bar{\rho}=\left({\begin{array}[]{cc}\varphi_{1}&\ast\\\ 0&\varphi_{2}\\\
\end{array}}\right)$, for globally defined characters

$\varphi_{i}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathbb{F}_{p}^{\times}.$

######  Proposition 2.4.

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that
$\bar{\rho}$ is reducible, then,
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$.

###### Proof.

Let $\mathbb{Q}(\varphi_{i})$ be the field fixed by the kernel of
$\varphi_{i}$ and set $L$ to be the composite of $\mathbb{Q}(\varphi_{1})$ and
$\mathbb{Q}(\varphi_{2})$. Note that $\mathbb{Q}(E[p^{\infty}])/L$ is pro-$p$
and since $L$ an abelian number field, by the Theorem of Ferrero and
Washington [FW79], the Iwasawa $\mu=0$ conjecture holds for
$L_{\operatorname{cyc}}/L$. It follows from [CS05, Corollary 3.5] that in
fact, $\mu_{p}^{\operatorname{fn}}(E/L_{\operatorname{cyc}})=0$, and it is
easy to see that this implies that
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$. ∎

In contrast, the $\mu$-invariant of the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is known
to be positive in some cases when $\bar{\rho}$ is reducible. From a
statistical point of view, the case of interest is that when $\bar{\rho}$ is
surjective, in particular, irreducible. It follows from Serre’s Open image
theorem that when $E_{/\mathbb{Q}}$ is an elliptic curve without complex
multiplication, the residual representation on $E[p]$ is surjective for all
but finitely many primes $p$. A _Serre curve_ is an elliptic curve whose
adelic Galois representation has index-$2$ in
$\operatorname{GL}_{2}(\hat{\mathbb{Z}})$, see [Jon10] for a more precise
definition. Serre curves are of interest since they are the elliptic curves
for which the adelic Galois image is as large as possible. It is not hard to
see that given a Serre curve, the mod-$p$ Galois representation is irreducible
for all odd primes $p$. Jones proves in _loc. cit._ that $100\%$ of elliptic
curves ordered according to height are Serre curves.

Let $f(T)$ be the characteristic element of the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$, and write
$f(T)=a_{r}T^{r}+a_{r+1}T^{r+1}+\dots$, where $a_{r}\in\mathbb{Z}_{p}$ is non-
zero. Here, the number $r$ is called the order of vanishing of $f(T)$ at $T=0$
and $a_{r}$ is called the _leading term_. The class of $a_{r}$ modulo
$\mathbb{Z}_{p}^{\times}$ does not depend on the choice of topological
generator $\gamma\in\Gamma$. Let $\mathcal{M}(E/\mathbb{Q})$ be the subgroup
of elements of $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})$ that lie in the image
of the Kummer map

$E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow
H^{1}(\mathbb{Q}_{S}/\mathbb{Q},E[p^{\infty}]),$

and the _fine Tate-Shafarevich group_ $\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})$ is
the cokernel of

$\mathcal{M}(E/\mathbb{Q})\rightarrow\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}).$

Then, $\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})$ is naturally identified with a
subgroup of $\Sh(E/K)[p^{\infty}]$. In our applications, we would like to
understand when $\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})$ is non-zero. The
following result shows that the fine Tate-Shafarevich group vanishes precisely
when the $p$-primary part of the Tate-Shafarevich group does.

######  Theorem 2.5 (Wuthrich).

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Then,
$\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})\neq 0$ precisely when
$\Sh(E/\mathbb{Q})[p^{\infty}]\neq 0$. Furthermore, if
$\operatorname{rank}E(\mathbb{Q})>0$, then,
$\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})=\Sh(E/\mathbb{Q})[p^{\infty}]$.

###### Proof.

See [Wut07a, Theorems 3.4, 3.5]. ∎

When $\operatorname{rank}E(\mathbb{Q})=0$,
$\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})$ may be strictly smaller than
$\Sh(E/\mathbb{Q})[p^{\infty}]$, as numerical examples in [Wut07a] show.

Wuthrich gives an explicit formula for the leading term of the characteristic
element $a_{r}$, see [Wut07b, Corollary 6.2]. We let $D=D_{E,p}$ be the
cokernel of the natural map $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ to the
$p$-adic completion of $E(\mathbb{Q}_{p})$. At a prime $\ell$ at which $E$ has
bad reduction, $c_{\ell}(E)$ will denote the Tamagawa number at $\ell$. Given
two numbers $a,b$, we write $a\sim b$ if $a=ub$ for some unit
$u\in\mathbb{Z}_{p}^{\times}$. There is a $p$-adic height pairing on the
Selmer group $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})$, see [Wut07b, section
5]. The regulator
$\operatorname{Reg}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})\right)$ is the
determinant of this height pairing.

######  Theorem 2.6 (Wuthrich).

Let $E$ be an elliptic curve over $\mathbb{Q}$ with potentially good reduction
at $p$. Assume that the following conditions are satisfied:

  1. (1)

$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is a cotorsion
$\Lambda$-module.

  2. (2)

The fine Tate-Shafarevich group $\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})$ is
finite.

  3. (3)

The cyclotomic height pairing on
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is
nondegenerate.

Then,

$r=\operatorname{max}\left(0,\operatorname{rank}E(\mathbb{Q})-1\right)$

and we have the following formula for the leading term $a_{r}$

(2.2)
$a_{r}\sim\operatorname{Reg}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})\right)\times\left(\frac{\\#\operatorname{Tors}_{\mathbb{Z}_{p}}(D)\times\prod_{\ell\neq
p}c_{\ell}(E)\times\\#\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})}{\\#J}\right).$

Note that under the assumptions of the above theorem, $r=0$ and the regulator
$\operatorname{Reg}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})\right)=1$ when
$\operatorname{rank}E(\mathbb{Q})\leq 1$. As a result of the above formula, we
obtain an important consequence towards the vanishing of the Iwasawa
invariants of the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$.

######  Corollary 2.7.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ an odd prime. Assume
that $E$ has potentially good reduction at $p$ and satisfied the conditions of
Theorem 2.6. Furthermore, assume that the following conditions hold:

  1. (1)

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

  2. (2)

$p\nmid c_{\ell}(E)$ for all primes $\ell\neq p$,

  3. (3)

$\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})=0$,

  4. (4)

$\operatorname{Tors}_{\mathbb{Z}_{p}}(D)=0$.

Then, the $\mu$ and $\lambda$-invariants of the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ are $0$. In
particular, $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ has
finite cardinality.

###### Proof.

It follows from Theorem 2.6 that $r=0$. Furthermore, the regulator

$\operatorname{Reg}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})\right)=1.$

Furthermore, the conditions imply that $a_{0}=1$, hence the characteristic
element of $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is a
unit in $\Lambda$. The result follows from this. ∎

The above result has an interesting consequence from a statistical point of
view. Let us first introduce some notation. Given $x>0$, recall that $\pi(x)$
is the number of primes $p\leq x$ and that the prime number theorem states
that $\pi(x)\sim\frac{x}{\operatorname{log}(x)}$, i.e.,

$\lim_{x\rightarrow\infty}\frac{\pi(x)}{\left(x/\operatorname{log}(x)\right)}=1.$

Given a set of primes $\mathcal{S}$, set
$\mathcal{S}(x)\mathrel{\mathop{\mathchar 58\relax}}=\\{p\in\mathcal{S}\mid
p\leq x\\}$, note that $\\#\mathcal{S}(x)\leq\pi(x)$ by definition.

######  Definition 2.8.

Let $\beta\in[0,1]$ and $\mathcal{S}$ be a set of prime numbers. We say that
$\mathcal{S}$ has density $\beta$ if

$\lim_{x\rightarrow\infty}\frac{\\#\mathcal{S}(x)}{\pi(x)}=\beta.$

Note that part of the requirement is that the limit actually exists.

Thus, if the set $\mathcal{S}$ has density $1$, we say that $\mathcal{S}$
consists of $100\%$ of primes. If the complement of $\mathcal{S}$ in the set
of all primes is finite, then indeed, $\mathcal{S}$ consists of $100\%$ of
primes, however, not conversely. Given an elliptic curve $E_{/\mathbb{Q}}$,
let $\Pi_{lt,E}$ be the set of _local torsion primes_ for $E$, i.e., the set
of all primes $p$ such that $E$ has good reduction at $p$ and
$E(\mathbb{Q}_{p})$ contains a $p$-torsion point. Let $\Pi_{an,E}$ be the set
of _anomalous primes_ , i.e., primes $p$ at which $E$ has good reduction such
that $\widetilde{E}(\mathbb{F}_{p})[p]\neq 0$. Here, $\widetilde{E}$ is the
reduction of $E$ at $p$. Let $\Pi_{E}$ be the set of primes $p$ at which $E$
has good reduction such that $\operatorname{Tors}_{\mathbb{Z}_{p}}D\neq 0$.

######  Proposition 2.9.

Let $E$ be an elliptic curve over $\mathbb{Q}$, then,
$\Pi_{lt,E}\subseteq\Pi_{an,E}$. Furthermore, if
$\operatorname{rank}E(\mathbb{Q})=0$, then, $\Pi_{E}\subseteq\Pi_{lt,E}$.

###### Proof.

Let $E$ have good reduction at $p$, in other words, $E$ can be viewed as an
elliptic curve over $\operatorname{Spec}\mathbb{Z}_{p}$. Letting $E_{0}$ be
the formal group of $E$, we have a short exact sequence

$0\rightarrow E_{0}(p\mathbb{Z}_{p})\rightarrow
E(\mathbb{Q}_{p})\rightarrow\widetilde{E}(\mathbb{F}_{p})\rightarrow 0.$

Note that $E_{0}(p\mathbb{Z}_{p})\simeq\mathbb{Z}_{p}$ does not contain any
$p$-torsion, and therefore, there is an injection
$E(\mathbb{Q}_{p})[p^{\infty}]\hookrightarrow\widetilde{E}(\mathbb{F}_{p})[p^{\infty}]$.
This shows that if $p$ is a local torsion prime, then $p$ is an anomalous
prime, i.e., $\Pi_{lt,E}\subseteq\Pi_{an,E}$. On the other hand, when
$\operatorname{rank}E(\mathbb{Q})=0$ and $E(\mathbb{Q}_{p})$ has no nontrivial
$p$-torsion, it follows that $\operatorname{Tors}_{\mathbb{Z}_{p}}D=0$.
Therefore, $\Pi_{E}\subseteq\Pi_{lt,E}$, when the Mordell-Weil rank of $E$ is
$0$. ∎

######  Corollary 2.10.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ with
$\operatorname{rank}E(\mathbb{Q})=0$ for which the Tate-Shafarevich group
$\Sh(E/\mathbb{Q})$ is finite. Then, for $100\%$ of primes $p$,

$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0\text{ and
}\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0.$

The set of primes for which
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})>0$ or
$\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})>0$ is
contained in the set $\Pi_{lt,E}\cup\mathfrak{S}$, where $\mathfrak{S}$ is the
set of primes dividing $2\\#\Sh(E/\mathbb{Q})\times\prod_{\ell\neq
p}c_{\ell}(E)$.

###### Proof.

Since it is assumed that $\Sh(E/\mathbb{Q})$ is finite, it follows that
$\mbox{Ж}_{p^{\infty}}(E/\mathbb{Q})=0$ for all but finitely many primes. It
is also clear (from the fact that $c_{\ell}(E)$ is a non-zero integer) that
$\prod_{\ell\neq p}c_{\ell}(E)$ is a unit in $\mathbb{Z}_{p}$ for all but
finitely many primes $p$. It is well known that $\Pi_{\operatorname{an},E}$
has density zero, see [Mur97]. Since it is assumed that
$\operatorname{rank}E(\mathbb{Q})=0$, Proposition 2.9 asserts that
$\Pi_{E}\subseteq\Pi_{\operatorname{an},E}$, and hence $\Pi_{E}$ has density
zero. Therefore, we have shown that the conditions of Corollary 2.7 hold for
$100\%$ of primes $p$. ∎

##  3\. Statistics for local torsion primes

Let $E_{/\mathbb{Q}}$ be an elliptic curve. We recall that a prime $p$ at
which $E$ has good reduction is said to be _anomalous_ if
$p\mid\\#\widetilde{E}(\mathbb{F}_{p})$. Given an elliptic curve $E$,
$\Pi_{an,E}$ is the set of anomalous primes for $E$. The following result is
due to Greenberg, see [Gre99, Theorems 4.1 and 5.1].

######  Theorem 3.1 (Greenberg).

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication
such that $\operatorname{rank}E(\mathbb{Q})=0$ and $\Sh(E/\mathbb{Q})$ is
finite. Then, for $100\%$ of primes $p$, the $p$-primary Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ has
$\mu=0$ and $\lambda=0$. The set of primes $p$ at which either $\mu>0$ or
$\lambda>0$ is contained in $\Pi_{an,E}\cup\mathfrak{S}$, where $\mathfrak{S}$
is the finite set of primes dividing $2\\#\Sh(E/\mathbb{Q})\prod_{\ell\neq
p}c_{\ell}(E)$. Furthermore, the set of primes at which $\mu>0$ or $\lambda>0$
is infinite if and only if $\Pi_{an,E}$ is infinite.

At this point, a natural question arises. Recall that for $x>0$, the set
$\Pi_{an,E}(x)$ is the subset of $\Pi_{an,E}$ consisting of primes $p\leq x$.
There are results about the asymptotic growth of $\\#\Pi_{an,E}(x)$ as a
function in $x$. The conjecture of Lang and Trotter predicts that there is a
constant $C\geq 0$ (depending on $E$ and independent of $p$) such that

(3.1) $\\#\Pi_{an,E}(x)\sim C\frac{\sqrt{x}}{\operatorname{log}x}.$

This is only conjectural, in fact, the best known upper bound, due to V. K.
Murty (see [Mur97]) asserts that

$\\#\Pi_{an,E}(x)\ll\frac{x(\log\log x)^{2}}{(\log x)^{2}}.$

In (3.1), the constant $C$ may actually be $0$, and there are examples of
elliptic curves $E$ such that the entire set of anomalous primes $\Pi_{an,E}$
is finite. Such elliptic curves are known as _finitely anomalous curves_ , and
were studied by Ridgdill in her thesis [Rid10].

In studying the fine Selmer group, anomalous primes no longer play a role
since $\widetilde{E}(\mathbb{F}_{p})$ is not a term in the Euler
characteristic formula for the leading term of the characteristic element.
Instead we turn our attention to local torsion primes. Let $E$ be an elliptic
curve over $\mathbb{Q}$, and $p$ a prime number. Recall that

$D=D(E,p)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{cok}\left(E(\mathbb{Q})\otimes\mathbb{Z}_{p}\longrightarrow\widehat{E(\mathbb{Q}_{p})}\right),$

where $\widehat{E(\mathbb{Q}_{p})}$ is the $p$-adic completion of
$E(\mathbb{Q}_{p})$. The torsion subgroup
$\operatorname{Tors}_{\mathbb{Z}_{p}}D$ features in the numerator of (2.2).
The following conjecture on the distribution of local torsion primes is
supported by heuristics.

######  Conjecture 3.2 (David-Weston).

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex-multiplication.
Then, the set of local torsion primes is finite.

The following heuristic is taken from [DW08, p.1]. Suppose $p$ is a local
torsion prime, i.e., $E(\mathbb{Q}_{p})[p]\neq 0$ such that $E$ has good
reduction at $p$. This in particular implies that $E(\mathbb{F}_{p})[p]\neq
0$. If $p\geq 7$, then it follows from the Weil bound that $a_{p}(E)=1$. On
the other hand, $a_{p}(E)=1$ occurs with probability $\frac{1}{4\sqrt{p}}$.
Given an elliptic curve $\widetilde{E}$ over $\mathbb{F}_{p}$ satisfying some
additional conditions, $1$ out every $p$ lifts $E$ of $\widetilde{E}$ to
$\mathbb{Z}_{p}$ has a local torsion point, see [DW08, Corollary 3.5]. The sum
$\sum_{n}\frac{1}{4\sqrt{p}\cdot p}=\frac{1}{4}\sum_{p}p^{-3/2}$ is finite.

On the other hand, Conjecture 1.2 in [Wut07b] states that the fine Selmer
group $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ associated
to an elliptic curve with $\operatorname{rank}E(\mathbb{Q})\leq 1$ is finite
for all but finitely many primes $p$. More generally, the
$\mathbb{Z}_{p}$-corank of
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is expected to
be equal to the $\mathbb{Z}_{p}$-corank of
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q})$ for all but finitely many primes $p$.
We state the conjecture slightly differently from its original formulation.

######  Conjecture 3.3 (Wuthrich).

Let $E$ be an elliptic curve over $\mathbb{Q}$, then for all but finitely many
primes $p$ of good reduction, the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ satisfies
$\mu=0$ and
$\lambda=\operatorname{max}\left(\operatorname{rank}E(\mathbb{Q})-1,0\right)$.

We specialize to the case when $\operatorname{rank}E(\mathbb{Q})=0$ to obtain
the following easy observation.

######  Proposition 3.4.

Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with
$\operatorname{rank}E(\mathbb{Q})=0$. Then, Conjecture 3.2 implies that
Conjecture 3.3.

###### Proof.

Since $\operatorname{rank}E(\mathbb{Q})=0$, the order of vanishing of
$f(T)=\operatorname{char}\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\right)$
at $T=0$ is $0$. We appeal to Wuthrich’s formula for the leading term $a_{0}$.
Note that $a_{0}$ is a unit in $\mathbb{Z}_{p}$ if and only if
$\mu\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\right)=0$
and
$\lambda\left(\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\right)=0$.

Wuthrich’s conjecture in this context states that $a_{0}=a_{0}(E,p)$ is a
$p$-adic unit for all but finitely many primes $p$. The result follows from
Corollary 2.7. ∎

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at $p$, in
other words, $E_{/\mathbb{Q}_{p}}$ admits a smooth local model
$\mathcal{E}_{/\mathbb{Z}_{p}}$. Given a finitely generated abelian group $M$,
we set $\operatorname{rank}_{p}M\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{dim}\left(M\otimes\mathbb{F}_{p}\right)$ to be the
$p$-rank of $M$.

######  Lemma 3.5.

Let $\mathcal{E}$ be an elliptic curve over $\mathbb{Z}_{p}$, then,

$\operatorname{rank}_{p}\left(\mathcal{E}(\mathbb{Z}/p^{2}\mathbb{Z})\right)=\begin{cases}1\text{
if }E(\mathbb{Q}_{p})[p]=0;\\\ 2\text{ if }E(\mathbb{Q}_{p})[p]\neq
0.\end{cases}$

###### Proof.

The assertion is a special case of [DW08, Lemma 3.1]. ∎

Given a pair $(a,b)$, we let $E_{a,b}$ be the elliptic curve defined by
$y^{2}=x^{3}+ax+b$.

######  Proposition 3.6.

Let $E_{a,b}$ be an elliptic curve associated to a pair
$(a,b)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$ and such that the j-invariant
$j(E_{a,b})\neq 0,1728$. Then, there are $p$ distinct pairs
$(A_{i},B_{i})\in\mathbb{Z}/p^{2}\times\mathbb{Z}/p^{2}$ such that
$(A_{i},B_{i})\equiv(a,b)\mod{p}$ and

$\operatorname{rank}_{p}E_{A_{i},B_{i}}(\mathbb{Z}/p^{2})=2.$

###### Proof.

See [DW08, Proposition 3.4] for a proof of the result. ∎

######  Definition 3.7.

Let $\mathfrak{S}_{p}$ be the subset of $\mathbb{F}_{p}\times\mathbb{F}_{p}$
consisting of all pairs $(a,b)$ such that

$\Delta_{a,b}\neq 0\text{ and }E_{a,b}(\mathbb{F}_{p})[p]\neq 0.$

Let $\mathfrak{S}_{p}^{j}$ be the subset of $\mathfrak{S}_{p}$ consisting of
pairs $(a,b)$ such that $j(E_{a,b})=j$. Set $\mathfrak{A}_{p}$ be the set of
pairs $(A,B)\in\mathbb{Z}/p^{2}\times\mathbb{Z}/p^{2}$ such that

$p\nmid\Delta_{A,B}\text{ and
}\operatorname{rank}_{p}\left(E_{A,B}(\mathbb{Z}/p^{2})\right)=2.$

Given $j\in\mathbb{F}_{p}$, let
$\mathfrak{A}_{p}^{j}\subseteq\mathfrak{A}_{p}$ consist of the pairs for which
$j(E_{A,B})\equiv j\mod{p}$.

We would like to estimate $\mathfrak{S}_{p}$ and $\mathfrak{A}_{p}$. Note that
by definition $\mathfrak{A}_{p}$ reduces mod-$p$ to $\mathfrak{S}_{p}$, and by
Proposition 3.6, the fibers of the reduction map
$\mathfrak{A}_{p}^{j}\rightarrow\mathfrak{S}_{p}^{j}$ have order $p$ provided
$j\neq 0,1728$. The quantity $\\#\mathfrak{S}_{p}$ can be expressed in terms
of Hurwitz class numbers associated to integral binary quadratic forms with
fixed discriminant.

We follow the notation and conventions of [Sch87, section 2]. Let
$\Delta\in\mathbb{Z}_{<0}$ with $\Delta\equiv 0,1\mod{4}$, and set

$B(\Delta)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{ax^{2}+bxy+cy^{2}\in\mathbb{Z}[x,y]\mid
a>0,b^{2}-4ac=\Delta\right\\}.$

The group $\operatorname{SL}_{2}(\mathbb{Z})$ acts on $\mathbb{Z}[x,y]$, where
a matrix $\sigma=\left({\begin{array}[]{cc}p&q\\\ r&s\\\ \end{array}}\right)$
sends $x\mapsto(px+qy)$ and $y\mapsto(rx+sy)$. Thus, if $f=ax^{2}+bxy+cy^{2}$,
the matrix $\sigma$ acts by

$f\circ\sigma=a(px+qy)^{2}+b(px+qy)(rx+sy)+c(rx+sy)^{2}.$

It can be checked that $B(\Delta)$ is stable under the action of
$\operatorname{SL}_{2}(\mathbb{Z})$ and
$B(\Delta)/\operatorname{SL}_{2}(\mathbb{Z})$ is finite, see _loc. cit._ for
additional details.

######  Definition 3.8.

The _Hurwitz class number_ $H(\Delta)$ is the order of
$B(\Delta)/\operatorname{SL}_{2}(\mathbb{Z})$.

Given an elliptic curve $E_{/\mathbb{F}_{p}}$, the Frobenius element
$\varphi\in\operatorname{End}(E)$ satisfies the equation
$\varphi^{2}-t\varphi+p=0$. Here, $t$ is an integer given by
$\\#E(\mathbb{F}_{p})=p+1-t$. According to the well-known Hurwitz bound, we
have that $|t|\leq 2\sqrt{p}$. Note that $p$ divides $t$ if and only if $E$ is
supersingular.

######  Definition 3.9.

(cf. [Sch87, Definition 4.1]) Two elliptic curves $E$ and $E^{\prime}$ over
$\mathbb{F}_{p}$ are _isogenous over $\mathbb{F}_{p}$_ if

$\\#E(\mathbb{F}_{p})=\\#E^{\prime}(\mathbb{F}_{p}).$

Let $I(t)$ be the isogeny class of elliptic curves over $\mathbb{F}_{p}$ with
$\\#E(\mathbb{F}_{p})=t$ and set $N(t)$ be the number of
$\mathbb{F}_{p}$-isomorphism classes of elliptic curves in $I(t)$.

Consider the special case when $\\#E(\mathbb{F}_{p})$ is divisible by $p$, in
other words, $t\equiv 1\mod{p}$. Note that since $|t|\leq 2\sqrt{p}$, it
follows that if $p\geq 7$, then, $E(\mathbb{F}_{p})$ contains a non-zero
$p$-torsion point precisely when $t=1$. However, if $p\leq 5$, then the two
possibilities are $t=-p+1$, in which case, $\\#E(\mathbb{F}_{p})=2p$, and
$t=1$, i.e., $\\#E(\mathbb{F}_{p})=p$.

Let $\bar{\mathfrak{S}}_{p}$ be the set of isomorphism classes of elliptic
curves $E$ over $\mathbb{F}_{p}$ such that $E(\mathbb{F}_{p})[p]\neq 0$. Thus,
we have that

$\bar{\mathfrak{S}}_{p}=\begin{cases}N(1)&\text{ for }p\geq 7,\\\
N(-p+1)+N(1)&\text{ for }p\leq 5.\end{cases}$

Waterhouse showed computed the number of elliptic curves in a given isogeny
class can be computed in terms of certain class numbers, see [Wat69]. These
results were later reformulated in terms of Hurwitz class numbers by Schoof.

######  Theorem 3.10 (Waterhouse, Schoof).

Let $t$ be an integer such that $p\nmid t$ and $t^{2}<4p$, then,
$N(t)=H(t^{2}-4p)$.

###### Proof.

The reader is referred to [Sch87, Theorem 4.6] for a proof of the result. ∎

######  Corollary 3.11.

Let $p$ be a prime and let $\bar{\mathfrak{S}}_{p}$ be as above. Then, we have
that

$\\#\bar{\mathfrak{S}}_{p}=\begin{cases}H(1-4p)&\text{ for }p\geq 7,\\\
H(p^{2}+1-6p)+H(1-4p)&\text{ for }p\leq 5.\end{cases}$

###### Proof.

From the discussion above Theorem 3.10,

$\bar{\mathfrak{S}}_{p}=\begin{cases}N(1)&\text{ for }p\geq 7,\\\
N(-p+1)+N(1)&\text{ for }p\leq 5.\end{cases}$

The result then follows from Theorem 3.10. ∎

######  Proposition 3.12.

Let $p$ be any prime and $\mathfrak{S}_{p}$ be as above. Then, we have that

$\\#\mathfrak{S}_{p}\leq\left(\frac{p-1}{2}\right)\left\\{z_{p}H(p^{2}+1-6p)+H(1-4p)\right\\}<Cp^{3/2}\operatorname{log}p(\operatorname{loglog}p)^{2},$

where $C>0$ is an effective constant (independent of $p$) and
$z_{p}=\begin{cases}1&\text{ if }p\leq 5,\\\ 0&\text{ if }p\geq 7.\end{cases}$

###### Proof.

The elliptic curves $E_{a,b}$ and $E_{a^{\prime},b^{\prime}}$ are isomorphic
over $\mathbb{F}_{p}$ if there exists $c\in\mathbb{F}_{p}^{\times}$ such that

$a^{\prime}=c^{4}a\text{ and }b^{\prime}=c^{6}b.$

Therefore, the number of elliptic curves $E_{a^{\prime},b^{\prime}}$
isomorphic to a given elliptic curve $E_{a,b}$ is at most
$\left(\frac{p-1}{2}\right)$. Therefore, we have that

$\begin{split}\\#\mathfrak{S}_{p}\leq\left(\frac{p-1}{2}\right)\\#\bar{\mathfrak{S}}_{p}<Cp^{3/2}\operatorname{log}p(\operatorname{loglog}p)^{2},\end{split}$

where the first inequality is the assertion of Corollary 3.11, and the second
follows from a well known bound on Hurwitz numbers [LJ87, Proposition 1.8]. ∎

######  Corollary 3.13.

For $\mathfrak{A}_{p}$ be as in Definition 3.7, we have that

(3.2)
$\\#\frac{\mathfrak{A}_{p}}{p^{4}}<\frac{2}{p}+Cp^{-3/2}\operatorname{log}p(\operatorname{loglog}p)^{2},$

where $C>0$ is an effective constant.

###### Proof.

We subdivide $\mathfrak{A}_{p}$ into two disjoint sets
$\mathfrak{A}_{p}^{(1)}$ and $\mathfrak{A}_{p}^{(2)}$. Let
$\mathfrak{A}_{p}^{(1)}$ be the pairs $(A,B)\in\mathfrak{A}_{p}$ such that the
$j$-invariant is either $0$ or $1728$ modulo $p$. In other words,
$\mathfrak{A}_{p}^{(1)}$ consists of all pairs $(A,B)\in\mathfrak{A}_{p}$ such
that $p\mid AB$. Clearly, the number of pairs $(A,B)$ such that $p\mid A$ is
$p^{3}$. Thus we have the following obvious upper bound:

(3.3) $\\#\mathfrak{A}_{p}^{(1)}<2p^{3}.$

Let $\mathfrak{A}_{p}^{(2)}$ be the complement of $\mathfrak{A}_{p}^{(1)}$ in
$\mathfrak{A}_{p}$. For any pair $(A,B)\in\mathfrak{A}_{p}^{(2)}$, we have by
construction that $j(E_{A,B})\not\equiv 0,1728\mod{p}$. Proposition 3.6
asserts that given a pair $(a,b)\in\mathfrak{S}_{p}$ with $j$-invariant not
equal to $0$ or $1728$, there are $p$ lifts $(A,B)\in\mathfrak{A}_{p}$ of
$(a,b)$. Therefore, we have the following upper bound:
$\\#\mathfrak{A}_{p}^{(2)}\leq p\\#\mathfrak{S}_{p}$. Invoking Proposition
3.12, we obtain the following bound:

(3.4)
$\\#\mathfrak{A}_{p}^{(2)}<Cp^{5/2}\operatorname{log}p(\operatorname{loglog}p)^{2}.$

Putting together (3.3) and (3.4), we obtain

$\frac{\\#\mathfrak{A}_{p}}{p^{4}}=\frac{\\#\mathfrak{A}_{p}^{(1)}}{p^{4}}+\frac{\\#\mathfrak{A}_{p}^{(2)}}{p^{4}}<\frac{2}{p}+Cp^{-3/2}\operatorname{log}p(\operatorname{loglog}p)^{2}.$

∎

######  Remark 3.14.

In the proof of the above result, we have invoked the obvious upper bound for
$\\#\mathfrak{A}_{p}^{(1)}$, which contributes to the term $\frac{2}{p}$ in
the estimate (3.2). Given a pair $(A,B)$ such that $p\mid AB$, note that the
pair $(A,B)$ is contained in $\mathfrak{A}_{p}^{(1)}$ precisely when
$(\bar{A},\bar{B})=(A,B)\mod{p}$ is contained in $\mathfrak{S}_{p}$. Let
$\mathfrak{S}_{p}^{\star}$ be the subset of $\mathfrak{S}_{p}$ consisting of
all pairs $(a,b)$ such that either $a$ or $b$ is $0$. In the above proof, we
have used the obvious bound $\\#\mathfrak{S}_{p}^{\star}<2p$ to obtain that

$\\#\mathfrak{A}_{p}^{(1)}\leq p^{2}\\#\mathfrak{S}_{p}^{\star}<2p^{3}.$

The main term $2/p$ in the estimate $\eqref{bound on Up}$ can be replaced by
$\\#\mathfrak{S}_{p}^{\star}/p^{2}$. Thus, a better understanding of
$\\#\mathfrak{S}_{p}^{\star}$ would potentially give a better estimate.

##  4\. Results for a fixed prime and varying elliptic curve

Let $E_{/\mathbb{Q}}$ be an elliptic curve. Note that $E$ is defined by a
Weierstrass equation $E\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+ax+b$,
where the integers $a$ and $b$ such that for all primes $\ell$, $\ell^{6}\nmid
b$ whenever $\ell^{4}\mid a$. Such a Weierstrass equation is unique and
referred to as the _minimal Weierstrass equation_. The _height_ of $E$ is
defined to be the quantity $H(E)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\left\\{|a|^{3},b^{2}\right\\}$.

Let $\mathcal{E}$ be the set of isomorphism classes of elliptic curves defined
over $\mathbb{Q}$. Given a subset $\mathcal{S}$ of $\mathcal{E}$, and $x>0$,
we set

$\mathcal{S}_{<x}\mathrel{\mathop{\mathchar 58\relax}}=\\{E\in\mathcal{S}\mid
H(E)<x\\},$

and $\mathfrak{d}(\mathcal{S},x)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#\mathcal{S}_{<x}}{\\#\mathcal{E}_{<x}}$. We say that
$\mathcal{S}$ has upper (resp. lower) density $\delta$ if
$\limsup_{x\rightarrow\infty}\mathfrak{d}(\mathcal{S},x)=\delta$ (resp.
$\liminf_{x\rightarrow\infty}\mathfrak{d}(\mathcal{S},x)=\delta$). If

$\lim_{x\rightarrow\infty}\mathfrak{d}(\mathcal{S},x)=\delta,$

then, we say that $\mathcal{S}$ has density $\delta$. Part of the requirement
here is that the above limit exists. We let $\bar{\mathfrak{d}}(\mathcal{S})$
(resp. $\underline{\mathfrak{d}}(\mathcal{S})$) be the upper (resp. lower)
density of $\mathcal{S}$. The density is denoted $\mathfrak{d}(\mathcal{S})$.
Note that when $\mathfrak{d}(\mathcal{S})$ is defined, so are
$\underline{\mathfrak{d}}(\mathcal{S})$ and $\bar{\mathfrak{d}}(\mathcal{S})$,
and

$\mathfrak{d}(\mathcal{S})=\underline{\mathfrak{d}}(\mathcal{S})=\bar{\mathfrak{d}}(\mathcal{S}).$

The conjecture on rank distribution by Katz and Sarnak [KS99] predicts that
when ordered by height, $50\%$ of elliptic curves have rank $0$, the other
$50\%$ of rank $1$ and $0\%$ of rank $>1$. In other words, let
$\mathcal{E}^{0},\mathcal{E}^{1}$ and $\mathcal{E}^{>1}$ be the subset of
elliptic curves of rank $0,1$ and $>1$ respectively. Then, the conjecture
predicts that
$\mathfrak{d}(\mathcal{E}^{0})=\mathfrak{d}(\mathcal{E}^{1})=\frac{1}{2}$, and
$\mathfrak{d}(\mathcal{E}^{>1})=0$.

Certain unconditional results have been proven by Bhargava and Shankar who
study the average size of the the $p$-Selmer group
$\operatorname{Sel}_{p}(E/\mathbb{Q})$. This is the $p$-torsion subgroup of
the $p$-primary Selmer group $\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q})$.
For instance, it the preprint [BS13], it is shown that the average size of the
$5$-Selmer group is $6$ as $E$ varies over $\mathcal{E}$. This result implies
that $\underline{\mathfrak{d}}(\mathcal{E}^{0})\geq\frac{1}{5}$ and
$\bar{\mathfrak{d}}(\mathcal{E}^{>1})\leq\frac{1}{5}$.

In this section, we fix a prime number $p\geq 5$ and study the following
question.

######  Question 4.1.

As $E$ varies over all elliptic curves defined over $\mathbb{Q}$, what can be
said about the proportion of elliptic curves $E$ are the following equivalent
conditions satisfied:

  1. (1)

$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ has finite
cardinality.

  2. (2)

The $\mu$ and $\lambda$-invariants of
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ are both equal
to $0$.

  3. (3)

The leading term $f(0)$ of the characteristic element $f(T)$ of
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is a unit in
$\mathbb{Z}_{p}$.

Note that when the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is
finite, it is in fact equal to $0$, see [KR21a, Corollary 3.6]. This is only
possible when the Mordell-Weil rank of $E$ is $0$ and $E$ has ordinary
reduction at $p$ (for further details, the reader may refer to [KR21a, section
4]). Since the fine Selmer group is a subgroup of the Selmer group, it follows
that if
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$, then
so is $\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$.
However, it is indeed possible for the fine Selmer group
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ to be finite,
while the entire Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is
infinite. As the following example shows, this is indeed possible even in the
special case when the elliptic curve $E$ has Mordell-Weil rank $0$ and good
ordinary reduction at $p$.

######  Example 4.2.

We refer to [Wut07b, section 9.1], consider the elliptic curve

$E\mathrel{\mathop{\mathchar 58\relax}}y^{2}+y=x^{3}-x^{2}-10x-20,$

and consider the prime $p=5$. The $\mu$-invariant of the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is $1$,
in fact,

$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}=\Lambda/p.$

On the other hand,
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is finite and
non-zero. In particular, the $\mu$ and $\lambda$-invariants of the fine Selmer
group are zero, however,
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\neq 0$.

More generally, if
$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is finite and
non-zero at a prime of good ordinary reduction, then the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ must be
infinite, since otherwise it would be zero.

###  4.1. Statistics for the Selmer group

Before we proceed to prove results for the fine Selmer group, let us briefly
recall the statistical results for the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ proved by
Kundu and the first named author in [KR21a, section 4]. Some of these results
may be marginally refined using [Ray21, Lemma 6.4]. In this subsection, we
summarize the results from these sources. Let $\mathfrak{F}_{p}$ be the the
set of elliptic curves $E\in\mathcal{E}$, such that

  1. (1)

$\operatorname{rank}E(\mathbb{Q})=0$,

  2. (2)

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

  3. (3)

$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})\neq 0$.

Let $\mathcal{S}_{p}$ be the set of elliptic curves $E_{/\mathbb{Q}}$ such
that $\Sh(E/\mathbb{Q})[p^{\infty}]\neq 0$. A conjecture of Delaunay [Del07]
predicts that $\bar{\mathfrak{d}}(\mathcal{S}_{p})=1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)=\frac{1}{p}+\frac{1}{p^{3}}-\frac{1}{p^{4}}+\dots$.
The heuristics supporting this conjecture have been further refined by Poonen
and Rains, see [PR12].

######  Conjecture 4.3 (Del-p).

The proportion of all elliptic curves such that $E$ for which
$p\mid\\#\Sh(E/\mathbb{Q})$ is given by

$\left(1-\prod_{i\geq 1}\left(1-\frac{1}{p^{2i-1}}\right)\right).$

In preparation of the next result, recall the definition of the set
$\mathfrak{S}_{p}$ from Definition 3.7.

######  Theorem 4.4.

(Kundu $\&$ R.) Let $p\geq 5$ be a prime number and $\mathfrak{F}_{p}$ be
defined as above. Assume $\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite for all
elliptic curves $E_{/\mathbb{Q}}$ and that the conjecture (Del-p) is
satisfied. Then, there is an effective constant $C_{1}>0$ such that the
following bounds are satisfied:

(4.1)
$\begin{split}\bar{\mathfrak{d}}(\mathfrak{F}_{p})\leq&\zeta(10)\frac{\\#\mathfrak{S}_{p}}{p^{2}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1),\\\
<&C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{\sqrt{p}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1).\end{split}$

Moreover, the effective constant $C_{1}$ can be taken to be $\zeta(10)C$,
where $C$ is the constant from Proposition 3.12.

###### Proof.

The result is essentially a rephrasing of [KR21a, Theorem 4.3], however,
thanks to the minor update in the statement taken from [Ray21], and for the
convenience of the reader, we summarize the proof here. Let $E$ be an elliptic
curve with good ordinary reduction at $p$, and $f(T)$ be the characteristic
element of
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$.
Express $f(T)$ as a power series. Suppose that $E\in\mathfrak{F}_{p}$, then,
in particular, $E$ has good ordinary reduction at $p$ and
$\operatorname{rank}E(\mathbb{Q})=0$. On the other hand, it is assumed that
$\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite, and therefore the Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q})$ is finite. It follows that the
constant term $f(0)$ is non-zero $p$-adic integer. By the well known Euler
characteristic formula,

$f(0)\sim\frac{\\#\Sh(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\\#\left(E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

In particular, if the three terms $\\#\Sh(E/\mathbb{Q})[p^{\infty}]$,
$\left(\prod_{l}c_{l}(E)\right)$ and $\\#\widetilde{E}(\mathbb{F}_{p})$ are
units in $\mathbb{Z}_{p}$, then $f(0)$ will be a unit in $\mathbb{Z}_{p}$. We
study the average behaviour of the following quantities for fixed $p$ and
varying $E\in\mathcal{E}$,

  1. (1)

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

  2. (2)

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

  3. (3)

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

Following [KR21a, Definition 4.1], let $\mathcal{E}_{1},\mathcal{E}_{2}$ and
$\mathcal{E}_{3}$ to be the set of _all_ elliptic curves $E_{/\mathbb{Q}}$
such that $p$ divides $s_{p}(E)$, $\tau_{p}(E)$ and $\delta_{p}(E)$
respectively. Note that if $f(0)$ is a $p$-adic unit, then
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$. Thus
if $E\in\mathfrak{F}_{p}$, then $f(0)$ is not a unit in $\mathbb{Z}_{p}$,
hence, $E\in\mathcal{E}_{1}\cup\mathcal{E}_{2}\cup\mathcal{E}_{3}$. From the
containment $\mathfrak{F}_{p}\subseteq\bigcup_{i=1}^{3}\mathcal{E}_{i}$, we
obtain the following bound:

$\bar{\mathfrak{d}}(\mathfrak{F}_{p})\leq\sum_{i=1}^{3}\bar{\mathfrak{d}}(\mathcal{E}_{i}).$

Since we assume that (Del-p) is satisfied, we have that

$\bar{\mathfrak{d}}(\mathcal{E}_{1})\leq\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right).$

It follows from [KR21a, Theorem 4.2] or [HKR21, Corollary 8,8] that

$\bar{\mathfrak{d}}(\mathcal{E}_{2})\leq\zeta(p)-1.$

Next, it follows from [KR21a, Theorem 4.14] and Proposition 3.12 (or [Ray21,
Lemma 6.4]) that

$\bar{\mathfrak{d}}(\mathcal{E}_{3})\leq\zeta(10)\frac{\\#\mathfrak{S}_{p}}{p^{2}}<C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{\sqrt{p}},$

where $C_{1}=\zeta(10)C$. Putting everything together, the result follows. ∎

######  Remark 4.5.

The result shows that when $p$ is suitably large the density of the set
$\mathfrak{F}_{p}$ is quite small.

  * •

Note that $\sqrt{p}$ grows faster than
$\operatorname{log}p(\operatorname{loglog}p)^{2}$ as $p\rightarrow\infty$,
hence, the term
$C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{\sqrt{p}}$ goes
to $0$ as $p\rightarrow\infty$. However, the exact values of
$\\#\frac{\mathfrak{S}_{p}}{p^{2}}$ can be calculated for small enough primes,
and for this, we refer to Table 1 (or, [KR21b, Table 2]).

  * •

The term $\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)=\frac{1}{p}+\frac{1}{p^{3}}-\frac{1}{p^{4}}+\dots$
grows like $\frac{1}{p}$ and hence is asymptotically smaller than the term
$C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{\sqrt{p}}$ as a
function of $p$.

  * •

It is easy to see that

$\zeta(p)-1=2^{-p}+\sum_{n\geq
3}n^{-p}<2^{-p}+\int_{2}^{\infty}x^{-p}dx=2^{-p}\left(\frac{p+1}{p-1}\right).$

Thus, Theorem 4.4 shows that $\bar{\mathfrak{d}}(\mathfrak{F}_{p})\rightarrow
0$ as $p\rightarrow\infty$, and in fact, as a function of $p$ is
$O\left(\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{\sqrt{p}}\right)$,
thus indicating that given a prime $p$, the most elliptic curves
$E_{/\mathbb{Q}}$ of rank $0$ and good ordinary reduction at $p$ have
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$. In
fact, the above remark shows that the contribution of the Tamagawa factors is
very little in comparison to that of anomalous primes.

###  4.2. Statistics for the Fine Selmer group in the rank-zero case

We now turn our attention to the fine Selmer group. In this subsection, we
study the case when $\operatorname{rank}E(\mathbb{Q})=0$. Let $p\geq 5$ be a
prime number and let $\mathfrak{B}_{p}$ be the set of elliptic curves
$E_{/\mathbb{Q}}$ such that the following conditions are satisfied:

  1. (1)

$E$ has good reduction at $p$,

  2. (2)

$\operatorname{rank}E(\mathbb{Q})=0$,

  3. (3)

$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is infinite
(equivalently, either
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})>0$ or
$\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})>0$).

Thus, we no longer impose the good ordinary condition at $p$.

######  Theorem 4.6.

Let $p\geq 5$ be a prime and $\mathfrak{B}_{p}$ be defined as above. Assume
that $\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite for all elliptic curves
$E_{/\mathbb{Q}}$ and that the conjecture (Del-p) is satisfied. Then, we have
the following bound on the upper density of $\mathfrak{B}_{p}$,

(4.2)
$\begin{split}\bar{\mathfrak{d}}(\mathfrak{B}_{p})\leq&\zeta(10)\frac{\\#\mathfrak{A}_{p}}{p^{4}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1),\\\
\leq&\frac{2\zeta(10)}{p}+C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{p^{3/2}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1).\\\ \end{split}$

Here $C_{1}$ is explicit constant given by $C_{1}=\zeta(10)C$ from Corollary
3.13.

###### Proof.

Consider the following quantities:

  1. (1)

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

  2. (2)

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

  3. (3)

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

Recall from the proof of Theorem 4.4 that $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ consist of all elliptic curves $E_{/\mathbb{Q}}$ such that
$p$ divides $s_{p}(E)$ and $\tau_{p}(E)$ respectively. Let $\mathcal{E}_{4}$
be the set of all elliptic curves $E_{/\mathbb{Q}}$ such that $p$ divides
$\beta_{p}(E)$, i.e., all elliptic curves for which $p$ is a local torsion
prime over $\mathbb{Q}_{p}$. It follows from Corollary 2.7 that
$\mathfrak{B}_{p}$ is contained in the union
$\mathcal{E}_{1}\cup\mathcal{E}_{2}\cup\mathcal{E}_{4}$. Hence,

(4.3)
$\bar{\mathfrak{d}}(\mathfrak{B}_{p})\leq\bar{\mathfrak{d}}(\mathcal{E}_{1})+\bar{\mathfrak{d}}(\mathcal{E}_{2})+\bar{\mathfrak{d}}(\mathcal{E}_{4}).$

From the proof of Theorem 4.4, we have that

$\bar{\mathfrak{d}}(\mathcal{E}_{1})+\bar{\mathfrak{d}}(\mathcal{E}_{2})\leq\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1).$

We prove an upper bound for $\bar{\mathfrak{d}}(\mathcal{E}_{4})$. We recall
that $\mathfrak{A}_{p}$ is the set of pairs
$(A,B)\in\mathbb{Z}/p^{2}\times\mathbb{Z}/p^{2}$ such that
$p\nmid\Delta_{A,B}\text{ and
}\operatorname{rank}_{p}\left(E_{A,B}(\mathbb{Z}/p^{2})\right)=2$. It follows
from Lemma 3.5 that an elliptic curve $E_{A,B}$ is contained in
$\mathcal{E}_{4}$ if and only the reduction of $(A,B)$ modulo $p^{2}$ lies in
$\mathfrak{A}_{p}$. Thus it would seem that $\mathfrak{d}(\mathcal{E}_{4})$ is
simply equal to
$\frac{\\#\mathfrak{A}_{p}}{\\#\left(\mathbb{Z}/p^{2}\times\mathbb{Z}/p^{2}\right)}=\frac{\\#\mathfrak{A}_{p}}{p^{4}}$,
but this is not quite the case, since not all pairs
$(A,B)\in\mathbb{Z}\times\mathbb{Z}$ are minimal. It is well known that the
proportion of all pairs $(A,B)$ that are minimal is $\frac{1}{\zeta(10)}$, see
for instance [CS20]. In greater detail, let $\mathcal{W}$ be the set of all
pairs of integers $(A,B)$, and $\mathcal{W}_{<x}$ be the subset of pairs such
that the height is $<x$. Note that since each elliptic curve $E_{/\mathbb{Q}}$
admits a unique Weierstrass equation, we may identify $\mathcal{E}$ with a
subset of $\mathcal{W}$. It is easy to see that

$\lim_{x\rightarrow\infty}\frac{\\#\mathcal{E}_{4,<x}}{\\#\mathcal{W}_{<x}}=\frac{\\#\mathfrak{A}_{p}}{p^{4}}$

and since

$\lim_{x\rightarrow\infty}\frac{\\#\mathcal{E}_{<x}}{\\#\mathcal{W}_{<x}}=\frac{1}{\zeta(10)},$

it follows that

$\mathfrak{d}(\mathcal{E}_{4})=\lim_{x\rightarrow
x}\frac{\\#\mathcal{E}_{4,<x}}{\\#\mathcal{E}_{<x}}=\zeta(10)\frac{\\#\mathfrak{A}_{p}}{p^{4}}.$

Appealing to Corollary 3.13, we have that

$\mathfrak{d}(\mathcal{E}_{4})<\frac{2\zeta(10)}{p}+C_{1}p^{-3/2}\operatorname{log}p(\operatorname{loglog}p)^{2}.$

The result now follows from (4.3). ∎

Asymptotically in $p$, the bound (4.2) is

$\frac{2\zeta(10)+1}{p}+O\left(p^{-3/2}\operatorname{log}p(\operatorname{loglog}p)^{2}\right),$

which is significantly smaller than the asymptotic estimate
$O\left(p^{-1/2}\operatorname{log}p(\operatorname{loglog}p)^{2}\right)$ from
Theorem 4.4. This tells us that the fine Selmer group is finite a lot more
often than the classical Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$,
since we have saved on an entire factor of
$p^{1/2}\operatorname{log}p(\operatorname{loglog}p)^{2}$ in our estimates.

###  4.3. Statistics for the Fine Selmer group in the rank-one case

We shift our attention to the case when our elliptic curves $E_{/\mathbb{Q}}$
have Mordell-Weil rank $1$. Let $\widehat{E(\mathbb{Q}_{p})}$ be the $p$-adic
completion of $E(\mathbb{Q}_{p})$ given by
$\widehat{E(\mathbb{Q}_{p})}\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\frac{E(\mathbb{Q}_{p})}{p^{n}E(\mathbb{Q}_{p})}$.
Recall that $D$ is the cokernel of the map
$E(\mathbb{Q})\otimes\mathbb{Z}_{p}\rightarrow\widehat{E(\mathbb{Q}_{p})}$.
The group $\widehat{E(\mathbb{Q}_{p})}$ decomposes into a direct sum
$\mathbb{Z}_{p}\oplus T$, where $T$ is the torsion subgroup of
$\widehat{E(\mathbb{Q}_{p})}$ (see [Hir19, Lemma 1, (iv)]). Note that $T$ is
non-zero when $p$ is a local torsion prime. The summand $\mathbb{Z}_{p}$
arises from the formal group of $E$.

Assume that $\operatorname{rank}E(\mathbb{Q})=1$. Write
$E(\mathbb{Q})\otimes\mathbb{Z}_{p}=\mathbb{Z}_{p}\oplus T^{\prime}$, where
$T^{\prime}$ is a finite group. The map
$E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}\rightarrow\widehat{E(\mathbb{Q}_{p})}$
gives rise to a map

$\mathbb{Z}_{p}\oplus T^{\prime}\rightarrow\mathbb{Z}_{p}\oplus T.$

The map on the first factors is denoted by $\phi_{E}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$. This is the composite of
the maps

$\mathbb{Z}_{p}\hookrightarrow\mathbb{Z}_{p}\oplus
T^{\prime}\rightarrow\mathbb{Z}_{p}\oplus T\rightarrow\mathbb{Z}_{p},$

where the first map is the inclusion into the first factor and the last map is
the projection on the first factor.

######  Lemma 4.7.

Let $E$ be an elliptic curve and $p$ a prime as above and suppose that
$\operatorname{Tors}_{\mathbb{Z}_{p}}(D)\neq 0$. Then, at least one of the
following hold:

  1. (1)

$p$ is not a local torsion prime of $E$,

  2. (2)

$\phi_{E}$ is not an isomorphism.

###### Proof.

Suppose that $p$ is not a local torsion prime. Then, it is clear $D$ is a
quotient of $\operatorname{cok}\phi_{E}$. Hence, if $D\neq 0$, it follows that
$\operatorname{cok}\phi_{E}$ is non-zero. This implies that $\phi_{E}$ is not
an isomorphism. ∎

We let $\mathcal{E}_{5}$ be the set of elliptic curves $E_{/\mathbb{Q}}$ such
that

  1. (1)

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

  2. (2)

the map $\phi_{E}$ is _not_ an isomorphism.

The upper density $\bar{\mathfrak{d}}(\mathcal{E}_{5})$ shall play a role in
our results. We are unable to prove any satisfactory bound on
$\bar{\mathfrak{d}}(\mathcal{E}_{5})$ which is why we resort to heuristics.
The probablity that an elliptic curve has rank $1$ is expected to be
$\frac{1}{2}$. This follows from the rank distribution conjecture. The map
$\phi_{E}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is multiplication by an
element $\alpha\in\mathbb{Z}_{p}$, and is an isomorphism precisely when
$\alpha$ is a unit. Thus, the probablity that $\phi_{E}$ is _not_ an
isomorphism can be expected to be $\frac{1}{p}$. Thus, the heuristic indicates
that $\bar{\mathfrak{d}}(\mathcal{E}_{5})=\frac{1}{2p}$. We state our results
in terms of the the density $\bar{\mathfrak{d}}(\mathcal{E}_{5})$, and do not
explicitly assume that this heuristic is correct. Let $p\geq 5$ be a prime
number and let $\mathfrak{D}_{p}$ be the set of elliptic curves
$E_{/\mathbb{Q}}$ such that the following conditions are satisfied:

  1. (1)

$E$ has good reduction at $p$,

  2. (2)

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

  3. (3)

$\mathcal{R}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$ is infinite.

######  Theorem 4.8.

Let $p\geq 5$ be a prime and $\mathfrak{D}_{p}$ be defined as above. Assume
that $\Sh(E/\mathbb{Q})[p^{\infty}]$ is finite for all elliptic curves
$E_{/\mathbb{Q}}$ and that the conjecture (Del-p) is satisfied. Then, we have
the following bound on the upper density of $\mathfrak{D}_{p}$,

(4.4)
$\begin{split}\bar{\mathfrak{d}}(\mathfrak{D}_{p})\leq&\bar{\mathfrak{d}}(\mathcal{E}_{5})+\zeta(10)\frac{\\#\mathfrak{A}_{p}}{p^{4}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1),\\\
\leq&\bar{\mathfrak{d}}(\mathcal{E}_{5})+\frac{2\zeta(10)}{p}+C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{p^{3/2}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1).\\\ \end{split}$

Here $C_{1}$ is explicit constant given by $C_{1}=\zeta(10)C$ from Corollary
3.13.

###### Proof.

Let $E$ be an elliptic curve in $\mathfrak{D}_{p}$. Consider the following
quantities:

  1. (1)

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

  2. (2)

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

  3. (3)

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

Recall that $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ and $\mathcal{E}_{4}$ consist
of all elliptic curves $E_{/\mathbb{Q}}$ such that $p$ divides $s_{p}(E)$,
$\tau_{p}(E)$ and $\beta_{p}(E)$ respectively. It follows from Corollary 2.7
that $E$ is either contained in the union $\mathcal{E}_{1}\cup\mathcal{E}_{2}$
or $\operatorname{Tors}_{\mathbb{Z}_{p}}(D)\neq 0$. It follows from Lemma 4.7
that if $\operatorname{Tors}_{\mathbb{Z}_{p}}(D)\neq 0$, then
$E\in\mathcal{E}_{4}\cup\mathcal{E}_{5}$. Therefore, combining the above
observations, we find that $\mathfrak{D}_{p}$ is contained in the union
$\mathcal{E}_{1}\cup\mathcal{E}_{2}\cup\mathcal{E}_{4}\cup\mathcal{E}_{5}$,
and hence,

$\bar{\mathfrak{d}}(\mathfrak{D}_{p})\leq\bar{\mathfrak{d}}(\mathcal{E}_{1})+\bar{\mathfrak{d}}(\mathcal{E}_{2})+\bar{\mathfrak{d}}(\mathcal{E}_{4})+\bar{\mathfrak{d}}(\mathcal{E}_{5}).$

From the proof of Theorem 4.6,

$\begin{split}\bar{\mathfrak{d}}(\mathcal{E}_{1})+\bar{\mathfrak{d}}(\mathcal{E}_{2})+\bar{\mathfrak{d}}(\mathcal{E}_{4})\leq&\zeta(10)\frac{\\#\mathfrak{A}_{p}}{p^{4}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1)\\\
<&C_{1}\frac{\operatorname{log}p(\operatorname{loglog}p)^{2}}{p^{3/2}}+\left(1-\prod_{i\geq
1}\left(1-\frac{1}{p^{2i-1}}\right)\right)+(\zeta(p)-1),\end{split}$

and the result follows from this. ∎

##  5\. Sieve Methods and the distribution of Local torsion primes

The distribution of local torsion primes is closely connected to the behavior
of Iwasawa invariants of elliptic curves. Let $E_{/\mathbb{Q}}$ be an elliptic
curve with Mordell-Weil rank zero for which the Tate-Shafarevich group
$\Sh(E/\mathbb{Q})$ is finite. Then Corollary 2.10 asserts that
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$ and
$\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$ for all
primes $p\notin\Pi_{lt,E}\cup\mathfrak{S}$. Here, $\mathfrak{S}$ is the set of
primes that divide $2\\#\Sh(E/\mathbb{Q})\times\prod_{\ell\neq p}c_{\ell}(E)$.
In particular, if the set $\Pi_{lt,E}$ is finite, then
$\mu_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$ and
$\lambda_{p}^{\operatorname{fn}}(E/\mathbb{Q}_{\operatorname{cyc}})=0$ for all
but finitely many primes $p$.

In this section we shall fix a number $Y>0$, and prove results about the
expectation for $\\#\Pi_{lt,E}(Y)$, as $E$ ranges over all elliptic curves
over $\mathbb{Q}$. We shall prove our results for elliptic curves on average.
For this purpose, the shall employ the use of a certain refined version of the
Large sieve inequality due to Bombieri and Davenport [BD69]. The version that
we use has previously been used in various other contexts, see [Duk97, Jon10].
The version we require is slightly more general than the sieve used in the
aforementioned works. We follow the notation and conventions in Jones’s thesis
[Jon05, p.10]. Let $k\in\mathbb{Z}_{\geq 1}$ and $B\subset\mathbb{R}^{k}$ be
the box defined by $B\mathrel{\mathop{\mathchar
58\relax}}=\prod_{j=1}^{k}[M_{j},M_{j}+N_{j}]$. For each point $n\in
B\cap\mathbb{Z}^{k}$ let $c(n)$ be a complex number, and we set

$S(x)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{n\in B}c(n)e(n\cdot x),$

where $e(x)\mathrel{\mathop{\mathchar 58\relax}}=e^{2\pi ix}$.

######  Definition 5.1.

Let $A\subset\mathbb{R}^{k}$ be a finite subset and let
$\delta=(\delta_{1},\dots,\delta_{k})$ be a tuple of positive numbers. We say
that $A$ is _$\delta$ -well spaced_ if for all
$\alpha=(\alpha_{1},\dots,\alpha_{k})\in A$ and
$\beta=(\beta_{1},\dots,\beta_{k})\in A$ such that $\alpha\neq\beta$,

$\operatorname{max}\left\\{\frac{|\alpha_{i}-\beta_{i}|}{\delta_{i}}\mid
i=1,\dots,k\right\\}\geq 1.$

In other words, for each pair $(\alpha,\beta)$ such that $\alpha\neq\beta$,
there is a coordinate $i$ such that $|\alpha_{i}-\beta_{i}|\geq\delta_{i}$.

The following generalization is due to Huxley, see [Hux68, Theorem 1].

######  Theorem 5.2 (Huxley).

Let $A$ be a $\delta$-well spaced finite set of vectors in $\mathbb{R}^{k}$
and let $B=\prod_{j=1}^{k}[M_{j},M_{j}+N_{j}]$. Then, we have the following
bound

$\sum_{\alpha\in
A}|S(\alpha)|^{2}\leq\left(\prod_{j=1}^{k}(N_{j}^{1/2}+\delta_{j}^{-1/2})^{2}\right)\sum_{n\in
B}|c(n)|^{2}.$

Fix a number $N\in\mathbb{Z}_{\geq 1}$ and for each prime $p$, we fix a subset
$\Omega(p)$ of $\left(\mathbb{Z}/p^{N}\mathbb{Z}\right)^{k}$ for each prime
$p$. For each integral vector $m\in\mathbb{Z}^{k}$ we define

$P(x;m)\mathrel{\mathop{\mathchar 58\relax}}=\\#\left\\{p\leq x\mid
m\mod{p^{N}}\in\Omega(p)\right\\}$

and

$P(x)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{p\leq x}\\#\Omega(p)p^{-Nk}.$

######  Theorem 5.3.

For each prime $p$, let $\Omega(p)$ be a subset of
$\left(\mathbb{Z}/p^{N}\mathbb{Z}\right)^{k}$. Let

$B=\prod_{j=1}^{r}[M_{j},M_{j}+N_{j}]$

be a box in $\mathbb{R}^{k}$ such that $N_{j}\geq x^{2N}$ for all $j$. Let
$V(B)$ be the volume of the box $V(B)=\prod_{j}N_{j}$, then

$\sum_{m\in
B\cap\mathbb{Z}^{k}}\left(P(x;m)-P(x)\right)^{2}\ll_{k,N}V(B)P(x).$

###### Proof.

Let $A=\bigcup_{p\leq x}A_{p}$, where

$A_{p}=\left\\{\alpha_{1}(p),\alpha_{2}(p),\dots,\alpha_{p^{Nk}-1}(p)\right\\}$

is a complete set of representatives
$\frac{1}{p^{N}}\mathbb{Z}^{k}/\mathbb{Z}^{k}$. The set is $\delta$ well-
spaced for $\delta_{i}=\frac{1}{x^{2N}}$ for all $i$. We apply Theorem 5.2 to
$A$ obtain the following result, the reader is referred to the proof of
[Jon05, Lemma 6], in which the $N=1$ is treated. The proof of this result
follows verbatim, hence we omit the details. ∎

We next apply the above result to studying the average distribution of local
torsion primes. Let $k,N=2$ and for each prime $p$, let
$\Omega(p)\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{A}_{p}\subset\mathbb{Z}/p^{2}\times\mathbb{Z}/p^{2}$.
Thus, for $Y>0$, we have that $P(Y)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{p\leq Y}\frac{\\#\mathfrak{A}_{p}}{p^{4}}$. Let $C,D>Y^{4}$,
and consider the set $\mathcal{S}_{C,D}$ of all pairs
$m=(a,b)\in\mathbb{Z}\times\mathbb{Z}$ with $|a|<C$ and $|b|<D$. Fix a number
$\beta>0$. Let $\mathcal{S}_{C,D}^{\beta}$ be the subset of
$m=(a,b)\in\mathcal{S}_{C,D}$ such that

$|\\#\Pi_{lt,E_{a,b}}(Y)-P(Y)|<\beta\sqrt{P(Y)},$

and set $\mathcal{Q}_{C,D}^{\beta}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{S}_{C,D}\backslash\mathcal{S}_{C,D}^{\beta}$.

######  Theorem 5.4.

Let $Y>0$ be a fixed number. Then, there is an absolute constant $c>0$ such
that for any number $\beta>0$ the proportion of elliptic curves $E$ for which

(5.1) $|\\#\Pi_{lt,E}(Y)-P(Y)|<\beta\sqrt{P(Y)}$

is $\geq(1-\frac{c}{\beta^{2}})$. Furthermore, if $C,D>Y^{4}$, we have that
$\frac{\\#\mathcal{Q}_{C,D}^{\beta}}{\\#\mathcal{S}_{C,D}}<\frac{c}{\beta^{2}}$
for any $\beta>0$.

######  Remark 5.5.

Setting $\beta=10\sqrt{c}$, we find that $\geq 99\%$ of elliptic curves
satisfy (5.1).

###### Proof.

Let $C$ and $D$ be integers such that $C,D>Y^{4}$. We are to show that there
is an absolute constant $c>0$ such that
$\frac{\\#\mathcal{Q}_{C,D}^{\beta}}{\\#\mathcal{S}_{C,D}}<\frac{c}{\beta^{2}}$
for any $\beta>0$. This implies that the proportion of elliptic curves
$E_{/\mathbb{Q}}$ (ordered by height) for which

$|\\#\Pi_{lt,E}(Y)-P(Y)|\geq\beta\sqrt{P(Y)}$

is $\leq c/\beta^{2}$.

Let $E_{a,b}$ be an elliptic curve associated with a minimal pair
$m=(a,b)\in\mathbb{Z}\times\mathbb{Z}$. Note that by definition,
$P(Y;m)=\\#\Pi_{lt,E}(Y)$.

According to the square sieve inequality of Theorem 5.3,

(5.2)
$\sum_{m\in\mathcal{S}_{C,D}}\left(P(m;Y)-P(Y)\right)^{2}<c\\#\mathcal{S}_{C,D}P(Y),$

where $c>0$ is an absolute constant. On the other hand,

(5.3)
$\begin{split}\sum_{m\in\mathcal{S}_{C,D}}\left(P(m;Y)-P(Y)\right)^{2}\geq&\sum_{m\in\mathcal{Q}_{C,D}}\left(P(m;Y)-P(Y)\right)^{2},\\\
=&\sum_{m\in\mathcal{Q}_{C,D}}\left(\\#\Pi_{lt,E_{m}}(Y)-P(Y)\right)^{2},\\\
\geq&\sum_{m\in\mathcal{Q}_{C,D}}\beta^{2}P(Y)=\\#\mathcal{Q}_{C,D}\beta^{2}P(Y).\end{split}$

Combining the inequalities from (5.2) and (5.3), we have shown that
$\frac{\\#\mathcal{Q}_{C,D}^{\beta}}{\\#\mathcal{S}_{C,D}}<\frac{c}{\beta^{2}}$,
and this completes the proof. ∎

The above result thus proves an explicit result for the expectation for
$\\#\Pi_{lt,E}(Y)$, which is shown to be close to $P(Y)=\sum_{p\leq
Y}\frac{\\#\mathfrak{A}_{p}}{p^{4}}$.

##  6\. Numerical data

Table 1. Data for $\mathfrak{S}_{p}/p^{2}$ 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

## References

  * [BD69] E. Bombieri and H. Davenport.  On the large sieve method.  In Number Theory and Analysis, pages 9–22. Springer, 1969.
  * [BS13] M. Bhargava and A. Shankar.  The average size of the 5-selmer group of elliptic curves is 6, and the average rank is less than 1.  arXiv preprint arXiv:1312.7859, 2013.
  * [CS00] J. Coates and R. Sujatha.  Galois cohomology of elliptic curves tata inst. fund. res.  Lecture Notes, Narosa Publishing House, 2000.
  * [CS05] J. Coates and R. Sujatha.  Fine selmer groups of elliptic curves over p-adic lie extensions.  Mathematische Annalen, 331(4):809–839, 2005.
  * [CS20] J. E. Cremona and M. Sadek.  Local and global densities for Weierstrass models of elliptic curves.  arXiv preprint arXiv:2003.08454, 2020.
  * [Del07] C. Delaunay.  Heuristics on class groups and on tate-shafarevich groups: the magic of the cohen-lenstra heuristics.  Ranks of elliptic curves and random matrix theory, 341:323–340, 2007.
  * [Duk97] W. Duke.  Elliptic curves with no exceptional primes.  Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 325(8):813–818, 1997.
  * [DW08] C. David and T. Weston.  Local torsion on elliptic curves and the deformation theory of galois representations.  Mathematical Research Letters, 15(3):599–611, 2008.
  * [FW79] B. Ferrero and L. C. Washington.  The iwasawa invariant $\mu$p vanishes for abelian number fields.  Annals of Mathematics, pages 377–395, 1979.
  * [Gre99] R. Greenberg.  Arithmetic Theory of Elliptic Curves: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (CIME) held in Cetaro, Italy, July 12-19, 1997.  Springer, 1999.
  * [Hir19] T. Hiranouchi.  Local torsion primes and the class numbers associated to an elliptic curve over $\mathbb{Q}$.  Hiroshima Mathematical Journal, 49(1):117–127, 2019.
  * [HKR21] J. Hatley, D. Kundu, and A. Ray.  Statistics for anticyclotomic iwasawa invariants of elliptic curves.  arXiv preprint arXiv:2106.01517, 2021.
  * [Hux68] M. N. Huxley.  The large sieve inequality for algebraic number fields.  Mathematika, 15(2):178–187, 1968.
  * [Jon05] N. Jones.  Almost all elliptic curves are Serre curves.  University of California Los Angeles, 2005.
  * [Jon10] N. Jones.  Almost all elliptic curves are serre curves.  Transactions of the American Mathematical Society, 362(3):1547–1570, 2010.
  * [Kat04] K. Kato.  p-adic hodge theory and values of zeta functions of modular forms.  Asterisque, 295:117–290, 2004.
  * [KR21a] D. Kundu and A. Ray.  Statistics for Iwasawa invariants of elliptic curves.  Trans. American Math. Soc., 374:7945–7965, 2021.  DOI: https://doi.org/10.1090/tran/8478.
  * [KR21b] D. Kundu and A. Ray.  Statistics for iwasawa invariants of elliptic curves, $\rm{II}$.  arXiv preprint arXiv:2106.12095, 2021.
  * [KS99] N. Katz and P. Sarnak.  Random matrices, Frobenius eigenvalues, and monodromy, volume 45.  American Mathematical Soc., 1999.
  * [LJ87] H. W. Lenstra Jr.  Factoring integers with elliptic curves.  Ann. Math., pages 649–673, 1987.
  * [Maz72] B. Mazur.  Rational points of abelian varieties with values in towers of number fields.  Invent. Math., 18(3-4):183–266, 1972.
  * [Mur97] V. K. Murty.  Modular forms and the chebotarev density theorem ii.  London Mathematical Society Lecture Note Series, pages 287–308, 1997.
  * [PR12] B. Poonen and E. Rains.  Random maximal isotropic subspaces and selmer groups.  Journal of the American Mathematical Society, 25(1):245–269, 2012\.
  * [Ray21] A. Ray.  Arithmetic statistics and diophantine stability for elliptic curves.  arXiv preprint arXiv:2109.00830, 2021.
  * [Rid10] P. C. Ridgdill.  On the frequency of finitely anomalous elliptic curves.  University of Massachusetts Amherst, 2010.
  * [Roh84] D. E. Rohrlich.  Onl-functions of elliptic curves and cyclotomic towers.  Inventiones mathematicae, 75(3):409–423, 1984.
  * [RS21] A. Ray and R. Sujatha.  On the $\mu$-invariants of residually reducible galois representations.  arXiv preprint arXiv:2104.10797, 2021.
  * [Sch87] R. Schoof.  Nonsingular plane cubic curves over finite fields.  Journal of combinatorial theory, Series A, 46(2):183–211, 1987\.
  * [Ser71] J.P. Serre.  Galois properties of finite order points of elliptic curves.  Invent. Math., 15(4):259–331, 1971.
  * [Was97] L. C. Washington.  Introduction to cyclotomic fields, volume 83.  Springer Science & Business Media, 1997.
  * [Wat69] W. C. Waterhouse.  Abelian varieties over finite fields.  In Annales scientifiques de l’École normale supérieure, volume 2, pages 521–560, 1969.
  * [Wut07a] C. Wuthrich.  The fine tate–shafarevich group.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 142, pages 1–12. Cambridge University Press, 2007.
  * [Wut07b] C. Wuthrich.  Iwasawa theory of the fine selmer group.  Journal of Algebraic Geometry, 16(1):83–108, 2007.