# Statistics for Iwasawa invariants of elliptic curves

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

We study the average behaviour of the Iwasawa invariants for the Selmer groups
of elliptic curves, setting out new directions in arithmetic statistics and
Iwasawa theory.

###### Key words and phrases:

Iwasawa invariants, Selmer groups of elliptic curves

###### 2010 Mathematics Subject Classification:

Primary 11R23, 11R45, 11G05

##  1\. Introduction

Iwasawa theory began as the study of class groups over infinite towers of
number fields. In [18], B. Mazur initiated the study of Iwasawa theory of
elliptic curves. The main object of study is the $p$-primary Selmer group of
an elliptic curve $E$, taken over the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$. Mazur conjectured that when $p$ is a prime of good ordinary
reduction, the $p$-primary Selmer group is cotorsion as a module over the
Iwasawa algebra, denoted by $\Lambda$. This conjecture was settled by K. Kato,
see [13, Theorem 17.4].

Note that the Iwasawa algebra $\Lambda$ 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 certain invariants which have been
extensively studied. First consider the case when $E$ has good ordinary
reduction at $p$. By the $p$-adic Weierstrass Preparation Theorem, the
characteristic ideal of the Pontryagin dual of the Selmer group is generated
by a unique element $f_{E}^{(p)}(T)$, that can be expressed as a power of $p$
times a distinguished polynomial. The $\mu$-invariant is the power of $p$
dividing $f_{E}^{(p)}(T)$ and the $\lambda$-invariant is its degree. R.
Greenberg has conjectured that when the residual representation on the
$p$-torsion subgroup of $E$ is irreducible, then the $\mu$-invariant of the
Selmer group vanishes, see [11, Conjecture 1.11]. Further, if the $p$-primary
part of the Tate-Shafarevich group, denoted by
$\Sha(E/\mathbb{Q})[p^{\infty}]$, is finite, then one can show that the
$\lambda$-invariant is at least as large as the Mordell-Weil rank of $E$ (see
Lemma 3.3). However, this $\lambda$ may indeed be strictly larger than the
rank, and one of our main objectives is to determine its behaviour on average
.

When $E$ has supersingular reduction at $p$, the Selmer group is not
$\Lambda$-cotorsion. This makes the analysis of the algebraic structure of the
torsion part of the Selmer group particularly difficult. Instead, we consider
the plus and minus Selmer groups introduced by S. Kobayashi in [15], which are
known to be $\Lambda$-cotorsion. The Iwasawa invariants $\mu^{+}$ and
$\lambda^{+}$ (resp. $\mu^{-}$ and $\lambda^{-}$) of the plus (resp. minus)
Selmer group are defined in an analogous manner. In the supersingular case as
well, there is much computational evidence towards the conjecture that the
$\mu$-invariants $\mu^{+}$ and $\mu^{-}$ vanish. Once again, under standard
hypotheses on the Tate-Shafarevich group, both $\lambda^{+}$ and $\lambda^{-}$
are known to be greater than or equal to the Mordell-Weil rank of $E$ (see
Lemma 3.3).

The main goal of this article is prove results about the variation of the
Iwasawa invariants as the pair $(E,p)$ varies such that $E$ has good reduction
(ordinary or supersingular) at $p$. More precisely, we analyze 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 $p$)?

Greenberg studied the first question when $E_{/\mathbb{Q}}$ has rank zero and
$p$ varies over the primes of good ordinary reduction (see [11, Theorems 4.1
and 5.1]). In Theorem 3.8, we generalize this result to include the case of
supersingular primes. We show that a conjecture of J. Coates and R. Sujatha on
the vanishing of $\mu$-invariants of fine Selmer groups holds for density one
primes (see Corollary 3.10). Under natural assumptions, we prove similar
results for higher rank elliptic curves (see Theorems 3.13 and 3.16). The
results in both ordinary and supersingular cases lead us to make the following
conjecture (which is proved for elliptic curves of rank zero).

######  Conjecture.

Let $E_{/\mathbb{Q}}$ be an elliptic curve of rank $r_{E}$. For $100\%$ of the
primes $p$ at which $E$ has good ordinary reduction (resp. supersingular),
$\mu=0$ and $\lambda=r_{E}$ (resp. $\mu^{+}=\mu^{-}=0$ and
$\lambda^{+}=\lambda^{-}=r_{E}$).

The second question is at the intersection of arithmetic statistics and
Iwasawa theory. The area of arithmetic statistics concerns the behaviour of
number theoretic objects in families, and offers a probabilistic model that
seeks to explain numerous phenomena in the statistical behaviour of Selmer
groups. The investigations in this paper show that there is promise in the
analysis of the average behaviour of Iwasawa invariants. The main results we
prove are Theorems 4.3, 4.4 and 4.5. Our results indicate that it is
reasonable to expect that for a fixed prime $p$, as we vary over all rank 0
elliptic curves over $\mathbb{Q}$ with good ordinary (resp. supersingular)
reduction at $p$ ordered by height, a positive proportion of them have trivial
$p$-primary Selmer group (considered over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$). In fact, the results suggest that
the proportion of elliptic curves of rank $0$ with trivial $p$-primary Selmer
group approaches $100\%$ as $p\rightarrow\infty$ (see Conjecture 4.7).

##  2\. Background and Preliminaries

### 2.1.

Let
$\Gamma\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(\mathbb{Q}^{\operatorname{cyc}}/\mathbb{Q})\simeq\mathbb{Z}_{p}$.
The Iwasawa algebra $\Lambda$ is the completed group algebra
$\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\mathrel{\mathop{\ordinarycolon}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$.
After fixing a topological generator $\gamma$ of $\Gamma$, there is an
isomorphism of rings $\Lambda\cong\mathbb{Z}_{p}\llbracket T\rrbracket$, by
sending $\gamma-1$ to the formal variable $T$.

Let M be a cofinitely generated cotorsion $\Lambda$-module. The _Structure
Theorem of $\Lambda$-modules_ asserts that the Pontryagin dual of M, denoted
by $\rm{M}^{\vee}$, is pseudo-isomorphic to a finite direct sum of cyclic
$\Lambda$-modules. In other words, there is a map of $\Lambda$-modules

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

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

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

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

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

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

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

### 2.2.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at $p$. It
shall be assumed throughout that the prime $p$ is odd. Let $N$ denote the
conductor of $E$ and set $S$ to denote the set of primes which divide $Np$.
Let $\mathbb{Q}_{S}$ be the maximal algebraic extension of $\mathbb{Q}$ which
is unramified at the primes $v\notin S$. Set $E[p^{\infty}]$ to be the Galois
module of all $p$-power torsion points in $E({}\mkern
3.0mu\overline{\mkern-3.0mu\mathbb{Q}})$.

First, consider the case when $E$ has good (ordinary or supersingular)
reduction at $p$. Let $v$ be a prime in $S$. For any finite extension
$L/\mathbb{Q}$ contained in $\mathbb{Q}^{\operatorname{cyc}}$, write

$J_{v}(E/L)=\bigoplus_{w|v}H^{1}\left(L_{w},E\right)[p^{\infty}]$

where the direct sum is over all primes $w$ of $L$ lying above $v$. Then, the
_$p$ -primary Selmer group over $\mathbb{Q}$_ is defined as follows

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

This Selmer group fits into a short exact sequence

(2.1) $0\rightarrow
E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow\Sha(E/\mathbb{Q})[p^{\infty}]\rightarrow
0,$

see [5]. Here, $\Sha(E/\mathbb{Q})$ is the _Tate-Shafarevich group_

$\Sha(E/\mathbb{Q})\mathrel{\mathop{\ordinarycolon}}=\left\\{H^{1}({}\mkern
3.0mu\overline{\mkern-3.0mu\mathbb{Q}}/\mathbb{Q},E[p^{\infty}])\rightarrow\prod_{l}H^{1}({}\mkern
3.0mu\overline{\mkern-3.0mu\mathbb{Q}}_{l}/\mathbb{Q}_{l},E[p^{\infty}])\right\\}.$

Now, define

$J_{v}(E/\mathbb{Q}^{\operatorname{cyc}})=\varinjlim J_{v}(E/L)$

where $L$ ranges over the number fields contained in
$\mathbb{Q}^{\operatorname{cyc}}$ and the inductive limit is taken with
respect to the restriction maps. Taking direct limits, the _$p$ -primary
Selmer group over $\mathbb{Q}^{\operatorname{cyc}}$_ is defined as follows

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

As mentioned in the introduction, when $p$ is a prime of good supersingular
reduction, the Pontryagin dual of
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is not
$\Lambda$-torsion. In this case, one studies the plus and minus Selmer groups
which we describe below.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with supersingular reduction at
$p$. Denote by $\mathbb{Q}_{n}^{\operatorname{cyc}}$ the $n$-th layer in the
cyclotomic $\mathbb{Z}_{p}$-extension, with
$\mathbb{Q}_{0}^{\operatorname{cyc}}\mathrel{\mathop{\ordinarycolon}}=\mathbb{Q}$.
Set $\widehat{E}$ to be the formal group of $E$ over $\mathbb{Z}_{p}$. Let $L$
be a finite extension of $\mathbb{Q}_{p}$ with valuation ring
$\mathcal{O}_{L}$, let $\widehat{E}(L)$ denote
$\widehat{E}(\mathfrak{m}_{L})$, where $\mathfrak{m}_{L}$ is the maximal ideal
in $L$. Write $\mathfrak{p}$ for the (unique) prime above $p$ in
$\mathbb{Q}^{\operatorname{cyc}}$, and for the prime above $p$ in every finite
layer of the cyclotomic tower. Define the plus and minus norm groups as
follows

$\widehat{E}^{+}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\left\\{P\in\widehat{E}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\mid\operatorname{tr}_{n/m+1}(P)\in\widehat{E}(\mathbb{Q}_{m,\mathfrak{p}}^{\operatorname{cyc}}),\text{
for }0\leq m<n\text{ and }m\text{ even }\right\\},$
$\widehat{E}^{-}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\left\\{P\in\widehat{E}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\mid\operatorname{tr}_{n/m+1}(P)\in\widehat{E}(\mathbb{Q}_{m,\mathfrak{p}}^{\operatorname{cyc}}),\text{
for }0\leq m<n\text{ and }m\text{ odd }\right\\},$

where
$\operatorname{tr}_{n/m+1}\mathrel{\mathop{\ordinarycolon}}\widehat{E}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\rightarrow\widehat{E}(\mathbb{Q}_{m+1,\mathfrak{p}}^{\operatorname{cyc}})$
denotes the trace map with respect to the formal group law on $\widehat{E}$.
The completion $\mathbb{Q}^{\operatorname{cyc}}_{\mathfrak{p}}$ is the union
of completions $\bigcup_{n\geq
1}\mathbb{Q}^{\operatorname{cyc}}_{n,\mathfrak{p}}$, and set
$\widehat{E}^{\pm}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\bigcup_{n\geq
1}\widehat{E}^{\pm}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})$. Define

$J_{p}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\frac{H^{1}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}},E[p^{\infty}])}{\widehat{E}^{\pm}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}},$

where the inclusion

$\widehat{E}^{\pm}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\hookrightarrow
H^{1}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}},E[p^{\infty}])$

is induced via the Kummer map. For $v\in S\setminus\\{p\\}$, set
$J_{v}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})$ to be equal to
$J_{v}(E/\mathbb{Q}^{\operatorname{cyc}})$. The _plus and minus Selmer groups_
are defined as follows

$\operatorname{Sel}_{p^{\infty}}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\ker\left\\{H^{1}\left(\mathbb{Q}_{S}/\mathbb{Q}^{\operatorname{cyc}},E[p^{\infty}]\right)\rightarrow\bigoplus_{v\in
S}J_{v}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})\right\\}.$

For each choice of sign ${\ddagger}\in\\{+,-\\}$, set
$f_{E}^{(p),{\ddagger}}(T)$ to denote the characteristic element of
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})^{\vee}$,
with $\mu_{p}^{{\ddagger}}(E)$, $\lambda_{p}^{{\ddagger}}(E)$ defined
analogously.

Let $E$ be an elliptic curve with good (ordinary or supersingular) reduction
at $p\geq 5$. In order to state results in both ordinary and supersingular
case at once, we set for the remainder of the article the following notation,

$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\begin{cases}\operatorname{Sel}_{p^{\infty}}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})&\textrm{if
}E\textrm{ has supersingular reduction at }p,\textrm{ where
}{\ddagger}=\pm,\\\
\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})&\textrm{if
}E\textrm{ has ordinary reduction at }p.\end{cases}$

### 2.3.

Next, we introduce the fine Selmer group. Let $E$ be an elliptic curve over
$\mathbb{Q}$ and $p$ be any odd prime. At each prime $v\in S$, set

$\mathcal{K}_{v}(E/\mathbb{Q}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\bigoplus_{\eta|v}H^{1}(\mathbb{Q}^{\operatorname{cyc}}_{\eta},E[p^{\infty}]).$

The _$p$ -primary fine Selmer group_ of $E$ is defined as follows

$\operatorname{Sel}^{0}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\ker\left\\{H^{1}(\mathbb{Q}_{S}/\mathbb{Q}^{\operatorname{cyc}},E[p^{\infty}])\longrightarrow\bigoplus_{v\in
S}\mathcal{K}_{v}(E/\mathbb{Q}^{\operatorname{cyc}})\right\\}.$

By the result of Kato mentioned in the introduction, the Pontryagin dual of
$\operatorname{Sel}^{0}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is
known to be a cotorsion $\Lambda$-module independent of the reduction type at
$p$. Further, it is conjectured that this fine Selmer group is a cotorsion
$\mathbb{Z}_{p}$-module (i.e. the corresponding $\mu$-invariant vanishes), see
[6, Conjecture A].

### 2.4.

In what follows, M will be a cofinitely generated cotorsion $\Lambda$-module.
Note that $H^{i}(\Gamma,\rm{M})$ is always zero for $i\geq 2$.

######  Lemma 2.1.

Let M be a cofinitely generated cotorsion $\Lambda$-module. Then,

$\operatorname{corank}_{\mathbb{Z}_{p}}\rm{M}^{\Gamma}=\operatorname{corank}_{\mathbb{Z}_{p}}\rm{M}_{\Gamma}.$

###### Proof.

Note that $H^{1}(\Gamma,\rm{M})$ may be identified with $\rm{M}_{\Gamma}$ (see
[19, Proposition 1.7.7]). It follows from [12, Theorem 1.1] that

$\operatorname{corank}_{\Lambda}\rm{M}=\operatorname{corank}_{\mathbb{Z}_{p}}H^{0}(\Gamma,\rm{M})-\operatorname{corank}_{\mathbb{Z}_{p}}H^{1}(\Gamma,\rm{M}).$

Since $\rm{M}$ is assumed to be cotorsion over $\Lambda$, the result follows.
∎

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

$\chi(\Gamma,\rm{M})=\prod_{i\geq
0}\left(\\#H^{i}(\Gamma,\rm{M})\right)^{(-1)^{i}}.$

On the other hand, when the cohomology groups $H^{i}(\Gamma,\rm{M})$ are not
finite, there is a generalized version denoted by $\chi_{t}(\Gamma,\rm{M})$.
Since $H^{1}(\Gamma,\rm{M})$ is isomorphic to the group of coinvariants
$H_{0}(\Gamma,\rm{M})=\rm{M}_{\Gamma}$, there is a natural map

$\Phi_{\rm{M}}\mathrel{\mathop{\ordinarycolon}}\rm{M}^{\Gamma}\rightarrow\rm{M}_{\Gamma}$

sending $x\in\rm{M}^{\Gamma}$ to the residue class of $x$ in
$\rm{M}_{\Gamma}$. We say that the _truncated Euler characteristic_ is defined
if the kernel and cokernel of $\Phi_{\rm{M}}$ are finite. In this case, the
$\chi_{t}(\Gamma,\rm{M})$ is defined to be the following quotient,

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

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

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

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

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

Let $r_{\rm{M}}$ denote the order of vanishing of $f_{\rm{M}}^{(p)}(T)$ 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 2.2 (S. Zerbes).

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

  1. (1)

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

  2. (2)

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

  3. (3)

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

###### Proof.

See [37, Lemma 2.11]. ∎

In particular, the classical Euler characteristic $\chi(\Gamma,\rm{M})$ is
defined if and only if $r_{\rm{M}}=0$. When this happens, the constant
coefficient $c_{0}\sim\chi(\Gamma,\rm{M})$.

We specialize the discussion on Euler characteristics to Selmer groups of
elliptic curves. Let ${\ddagger}$ be a choice of sign and recall the
definition of
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$.
When $E$ has good ordinary reduction at $p$, the choice of ${\ddagger}$ is
irrelevant. Denote by $\chi_{t}^{{\ddagger}}(\Gamma,E[p^{\infty}])$ (resp.
$\chi^{{\ddagger}}(\Gamma,E[p^{\infty}])$) the truncated (resp. classical)
Euler characteristic of the Selmer group
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$.
The invariants $\mu_{p}^{{\ddagger}}(E)$ and $\lambda_{p}^{{\ddagger}}(E)$
simply refer to $\mu_{p}(E)$ and $\lambda_{p}(E)$ when $E$ has good ordinary
reduction at $p$. When $E$ has good ordinary reduction at $p$, we shall drop
the sign ${\ddagger}$ from the notation. It follows from Lemma 2.2 that the
truncated Euler characteristic is always an integer.

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

In this section, we study the variation of the classical and the truncated
Euler characteristic as $p$ varies. Fix a pair $(E,p)$ such that

  1. (1)

$p$ is odd.

  2. (2)

$E$ is defined over $\mathbb{Q}$ and has good reduction at $p$.

We record some lemmas which are required throughout this section. Recall that
when $E$ has good ordinary reduction,
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$
simply refers to
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$.

######  Theorem 3.1.

Let ${\ddagger}\in\\{+,-\\}$ be a choice of sign. Let $E_{/\mathbb{Q}}$ be an
elliptic curve with good reduction at $p>2$. Then, there is a natural map

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

with finite kernel and cokernel.

###### Proof.

When $E$ has good ordinary reduction at $p$, this follows from Mazur’s control
theorem, see [18] or [11, Theorem 1.2]. In the good supersingular reduction
case, the result follows from the proof of [9, Lemma 3.9] (see also [24,
Proposition 5.1]). ∎

The following lemma gives a criterion for when the classical Euler
characteristic $\chi^{{\ddagger}}(\Gamma,E[p^{\infty}])$ is well-defined.

######  Lemma 3.2.

Let ${\ddagger}\in\\{+,-\\}$ be a choice of sign. Let $E_{/\mathbb{Q}}$ be an
elliptic curve with good reduction at $p>2$. The following are equivalent.

  1. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

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

###### Proof.

Recall that the Selmer group
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$
is a cotorsion $\Lambda$-module. By Lemma 2.1,
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})^{\Gamma}$
is finite if and only if
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})_{\Gamma}$
is finite. This shows that $\eqref{31c1}$ and $\eqref{31c2}$ are equivalent.

Theorem 3.1 asserts that there is a natural map

$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q})\rightarrow\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})^{\Gamma},$

with finite kernel and cokernel. Hence, the conditions $\eqref{31c2}$ and
$\eqref{31c3}$ are equivalent. By the work of V. Kolyvagin,
$\Sha(E/\mathbb{Q})$ is known to be finite when $E(\mathbb{Q})$ is finite (see
[16]). Thus, it follows from (2.1) that conditions $\eqref{31c3}$ and
$\eqref{31c4}$ are equivalent. ∎

######  Lemma 3.3.

Let ${\ddagger}\in\\{+,-\\}$ be a choice of sign. Let $E_{/\mathbb{Q}}$ be an
elliptic curve with good reduction at $p>2$. Assume that
$\chi_{t}^{{\ddagger}}(\Gamma,E[p^{\infty}])$ is well-defined and
$\Sha(E/\mathbb{Q})[p^{\infty}]$ is finite. Let $r_{E}$ denote the Mordell-
Weil rank of $E(\mathbb{Q})$. Then, $r_{E}$ is equal to the order of vanishing
of $f_{E}^{(p),{\ddagger}}(T)$ at $T=0$. In particular, we have that
$\lambda_{p}^{{\ddagger}}(E)\geq r_{E}$.

###### Proof.

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

$r_{E}=\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}).$

By Lemma 2.2, it suffices to show that

$r_{E}=\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})^{\Gamma}.$

Therefore the result follows from Theorem 3.1. ∎

######  Lemma 3.4.

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

  1. (1)

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

  2. (2)

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

###### Proof.

Suppose that $\chi_{t}(\Gamma,\rm{M})=1$. Write
$f_{\rm{M}}^{(p)}(T)=T^{r_{\rm{M}}}g_{\rm{M}}^{(p)}(T)$ where
$g_{\rm{M}}^{(p)}(T)\in\Lambda$ and $g_{\rm{M}}(0)\neq 0$. By Lemma 2.2,

$\mathinner{\\!\left\lvert
g_{\rm{M}}^{(p)}(0)\right\rvert}_{p}^{-1}=\chi_{t}(\Gamma,\rm{M})=1.$

In particular, $f_{\rm{M}}^{(p)}(T)$ and $g_{\rm{M}}^{(p)}(T)$ are
distinguished polynomials. Since $g_{\rm{M}}^{(p)}(0)$ is a unit, it follows
that $g_{\rm{M}}^{(p)}(T)$ is a unit. Since $g_{\rm{M}}^{(p)}(T)$ is a
distinguished polynomial, it follows that

$g_{\rm{M}}^{(p)}(T)=1\text{ and }f_{\rm{M}}^{(p)}(T)=T^{r_{\rm{M}}}.$

Therefore, $\mu(\rm{M})=0$ and $\lambda(\textrm{M})=\deg
f_{M}^{(p)}(T)=r_{M}$.

Conversely, suppose that $\mu(\rm{M})=0$ and $\lambda(\textrm{M})=r_{\rm{M}}$.
Since $\mu(\rm{M})=0$, it follows that $f_{\rm{M}}^{(p)}(T)$ and
$g_{\rm{M}}^{(p)}(T)$ are distinguished polynomials. The degree of
$f_{\rm{M}}^{(p)}(T)$ is $\lambda_{\rm{M}}=r_{\rm{M}}$. It follows that
$g_{\rm{M}}^{(p)}(T)$ is a constant polynomial and hence,
$g_{\rm{M}}^{(p)}(T)=1$. By Lemma 2.2,

$\chi_{t}(\Gamma,\textrm{M})=\mathinner{\\!\left\lvert
g_{\textrm{M}}(0)\right\rvert}_{p}^{-1}=1.$

∎

######  Proposition 3.5.

Let ${\ddagger}\in\\{+,-\\}$ be a choice of sign. Let $E_{/\mathbb{Q}}$ be an
elliptic curve of Mordell-Weil rank $r_{E}$ with good reduction at $p>2$.
Suppose that these additional conditions hold.

  1. (i)

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

  2. (ii)

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

Then the following statements are equivalent

  1. (1)

$\chi_{t}^{{\ddagger}}(\Gamma,E[p^{\infty}])=1$,

  2. (2)

$\mu_{p}^{{\ddagger}}(E)=0$ and $\lambda_{p}^{{\ddagger}}(E)=r_{E}$.

###### Proof.

Lemma 3.3 asserts that $r_{E}$ is equal to the order of vanishing of
$f_{E}^{(p),{\ddagger}}(T)$ at $T=0$. The assertion follows from Lemma 3.4. ∎

###  3.1. Elliptic curves over $\mathbb{Q}$ with rank zero

In this subsection, we study the variation of the classical Euler
characteristic as $p$ varies over primes of good reduction.

######  Corollary 3.6.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at $p>2$
for which the Mordell-Weil rank of $E$ is zero. The following are equivalent.

  1. (1)

$\chi^{{\ddagger}}(\Gamma,E[p^{\infty}])=1$,

  2. (2)

$\mu_{p}^{{\ddagger}}(E)=0$ and $\lambda_{p}^{{\ddagger}}(E)=0$.

  3. (3)

$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$
is finite.

  4. (4)

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

###### Proof.

Since $E$ is assumed to have rank zero, the Tate-Shafarevich group
$\Sha(E/\mathbb{Q})$ is finite. It follows from Lemma 3.2 that the Euler
characteristic $\chi^{{\ddagger}}(\Gamma,E[p^{\infty}])$ is well-defined, and
by Proposition 3.5 that $\eqref{35p1}$ and $\eqref{35p2}$ are equivalent. From
the Structure Theorem of finitely generated $\Lambda$-modules, it is clear
that $\eqref{35p2}$ and $\eqref{35p3}$ are equivalent conditions. It is known
that
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$
contains no proper finite index submodules (see [11, Proposition 4.14] for the
case when $p$ is ordinary and [9, Theorem 3.14] for when $p$ is
supersingular). Hence, $\eqref{35p3}$ and $\eqref{35p4}$ are equivalent. ∎

Let $E_{/\mathbb{Q}}$ be an elliptic curve. Denote by $S^{\operatorname{bad}}$
the finite set of primes at which $E$ has bad reduction. Denote by
$S^{\operatorname{good}}$ the primes at which $E$ has good reduction (i.e.,
either ordinary reduction or supersingular reduction); write
$S^{\operatorname{good}}=S^{\operatorname{ord}}\cup S^{\operatorname{ss}}$.
When $E$ has good reduction at a fixed prime $p$, we set $\widetilde{E}$ to
denote the reduced curve over $\mathbb{F}_{p}$.

The following result was initially proved by Greenberg for good ordinary
primes.

######  Theorem 3.7.

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $E(\mathbb{Q})$ is
finite. Let $\Sigma\subset S^{\operatorname{ord}}$ be the set of primes at
which $p$ divides $\\#\widetilde{E}(\mathbb{F}_{p})$. Let
$\Sigma^{\prime}\subset S^{\operatorname{ord}}$ be the finite set of primes
$p$ such that either

  1. (1)

$p=2$.

  2. (2)

$p$ divides $\\#\Sha(E/\mathbb{Q})$.

  3. (3)

$p$ divides the Tamagawa product $\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)$.

Then for all primes $p\in
S^{\operatorname{ord}}\setminus(\Sigma\cup\Sigma^{\prime})$, we have that
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})=0$.

Observe that $\Sigma^{\prime}$ is always a finite set. However, the set of
primes $\Sigma$, referred to as the set of _anomalous primes_ , is known to be
a set of Dirichlet density zero. The Lang-Trotter Conjecture predicts that the
proportion of anomalous primes $p<X$ is $C\cdot\frac{\sqrt{X}}{\log X}$ for
some constant $C$. It has recently been shown that all but a density zero set
of elliptic curves have infinitely many anomalous primes [1, Corollary 4.3].
However, in special cases one can in fact show that $C=0$ [25, §1.1]. By the
Hasse inequality, a prime $p\geq 7$ is anomalous for an elliptic curve defined
over $\mathbb{Q}$ if and only if
$a_{p}=p+1-\\#\widetilde{E}(\mathbb{F}_{p})=1$. When $p\geq 7$ and an elliptic
curve has 2-torsion, $a_{p}$ is even and hence $C=0$. For more examples where
the set $\Sigma$ is finite, and detailed discussion on this subject, we refer
the reader to [18, 20, 23].

###### Proof of Theorem 3.7.

Since the Mordell-Weil rank of $E$ is assumed to be zero, the Euler
characteristic $\chi(\Gamma,E[p^{\infty}])$ is well-defined. The following
formula for the Euler characteristic is well known (see [5, Theorem 3.3])

(3.1)
$\chi(\Gamma,E[p^{\infty}])\sim\frac{\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

Since the Euler characteristic is an integer, the result follows immediately
from Corollary 3.6. ∎

We prove a similar result for primes of supersingular reduction.

######  Theorem 3.8.

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $E(\mathbb{Q})$ is
finite. Let $\Upsilon\subset S^{\operatorname{ss}}$ be the finite set of
primes $p$ such that either

  1. (1)

$p=2$.

  2. (2)

$p$ divides $\\#\Sha(E/\mathbb{Q})$.

  3. (3)

$p$ divides the Tamagawa product $\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)$.

Then for all primes $p\in S^{\operatorname{ss}}\setminus\Upsilon$, we have
that

$\operatorname{Sel}_{p^{\infty}}^{+}(E/\mathbb{Q}^{\operatorname{cyc}})=0\text{
and
}\operatorname{Sel}_{p^{\infty}}^{-}(E/\mathbb{Q}^{\operatorname{cyc}})=0.$

###### Proof.

Since $E(\mathbb{Q})$ is assumed to be finite, the Euler characteristic
$\chi^{\pm}(\Gamma,E[p^{\infty}])$ is well-defined. Since $E$ has
supersingular reduction at $p$, $E[p]$ is irreducible as a
$\operatorname{G}_{\mathbb{Q}}$-module; hence, $E(\mathbb{Q})[p]$ is the
trivial group. Finiteness of $E(\mathbb{Q})$ implies that
$E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}=0$. Therefore

$\\#\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q})=\\#\Sha(E/\mathbb{Q})[p^{\infty}].$

By [17, Theorem 5.15], we know that

(3.2)
$\chi^{\pm}(\Gamma,E[p^{\infty}])=\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right).$

The result follows from Corollary 3.6. ∎

######  Remark 3.9.

It is clear that the $\mu$ and $\lambda$ invariants do not change for each of
the layers of the cyclotomic $\mathbb{Z}_{p}$-extension. Let $E_{/\mathbb{Q}}$
be a rank 0 elliptic curve and $p\in
S^{\operatorname{ord}}\setminus(\Sigma\cup\Sigma^{\prime})$ or $p\in
S^{\operatorname{ss}}\setminus\Upsilon$. Since
$\lambda_{p}^{{\ddagger}}(E)\geq r_{E}$, the Mordell-Weil rank of the elliptic
curve must remain zero at each layer in the cyclotomic
$\mathbb{Z}_{p}$-extension.

The following result gives evidence for the conjecture of Coates and Sujatha
on the vanishing of the $\mu$-invariant of fine Selmer groups. In fact, more
is true.

######  Corollary 3.10.

Let $E$ be a rank 0 elliptic curve defined over $\mathbb{Q}$. Then, the
$p$-primary fine Selmer group
$\operatorname{Sel}^{0}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is
trivial for density one primes.

###### Proof.

Given any elliptic curve, it has good reduction at all but finitely many
primes. When $p$ is a prime of good (ordinary or supersingular) reduction,
note that the $p$-primary fine Selmer group is a subgroup of
$\operatorname{Sel}^{{\ddagger}}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$.
The result is immediate from Theorems 3.7 and 3.8. ∎

###  3.2. Elliptic curves over $\mathbb{Q}$ with positive rank

Next, we consider the case when the elliptic curve $E$ has positive rank.

When $E$ has good ordinary reduction at $p$, there is a notion of the $p$-adic
height pairing, which is the $p$-adic analog of the usual height pairing, and
was studied extensively in [27, 28]. This pairing is conjectured to be non-
degenerate, and the $p$-adic regulator $R_{p}(E/\mathbb{Q})$ is defined to be
the determinant of this pairing.

######  Theorem 3.11.

[B. Perrin-Riou [22], P. Schneider [28, Theorem 2’]] Let $E_{/\mathbb{Q}}$ be
an elliptic curve of Mordell-Weil rank $r_{E}$ with good ordinary reduction at
$p$. Assume that

  1. (1)

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

  2. (2)

the $p$-adic height pairing is non-degenerate, i.e., the $p$-adic regulator is
non-zero.

Then, the order of vanishing of $f_{E}^{(p)}(T)$ at $T=0$ is equal to
$r_{E}\mathrel{\mathop{\ordinarycolon}}=\operatorname{rank}E(\mathbb{Q})$.
Express $f_{E}^{(p)}(T)$ as $T^{r_{E}}g_{E}^{(p)}(T)$. Then,
$g_{E}^{(p)}(0)\neq 0$ and

$g_{E}^{(p)}(0)\sim\frac{\left(\frac{R_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)\times\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

The above theorem implies that the right hand side of the equation is an
integer since the left hand side is. This result does not assume that the
truncated Euler characteristic is well-defined. The statement is proven in
greater generality, in fact, for abelian varieties over number fields.

######  Corollary 3.12.

Let $E_{/\mathbb{Q}}$ be an elliptic curve of Mordell-Weil rank $r_{E}$ with
good ordinary reduction at $p$. Assume that

  1. (1)

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

  2. (2)

the $p$-adic Height pairing is non-degenerate.

  3. (3)

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

Then, we have that

$\chi_{t}(\Gamma,E[p^{\infty}])\sim\frac{\left(\frac{R_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)\times\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

###### Proof.

The assertion a direct consequence of Theorem 3.11 and Lemma 2.2. ∎

To analyze Iwasawa invariants on average, we shall apply Theorem 3.11. Let
$E_{/\mathbb{Q}}$ be an elliptic curve. As before, we denote by
$S^{\operatorname{ord}}$ the set of primes $p$ at which $E$ has good ordinary
reduction. Let $\Sigma$ be the set of anomalous primes and $\Sigma^{\prime}$
the finite set of primes for which

  1. (1)

$p=2$,

  2. (2)

$p$ divides $\\#\Sha(E/\mathbb{Q})$.

  3. (3)

$p$ divides the Tamagawa product $\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)$.

Denote by $v_{p}$ the $p$-adic valuation on $\mathbb{Q}_{p}$ normalized by
$v_{p}(p)=1$. Let $\Pi\subset S^{\operatorname{ord}}$ be the set of primes $p$
at which $v_{p}(R_{p}(E/\mathbb{Q}))\geq r_{E}$. In other words, it is the set
of primes for which $p$ divides
$\left(\frac{R_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)$. When $r_{E}=0$, the
$p$-adic regulator $R_{p}(E/\mathbb{Q})=1$, therefore the set $\Pi$ is empty.
On the other hand, the set of primes $\Pi$ need not be empty when $r_{E}\geq
1$. The following result generalizes Theorem 3.7 and is an easy consequence of
Theorem 3.11.

######  Theorem 3.13.

Let $E_{/\mathbb{Q}}$ be an elliptic curve such that $r_{E}\geq 1$. Then for
all primes $p\in
S^{\operatorname{ord}}\setminus(\Sigma\cup\Sigma^{\prime}\cup\Pi)$, we have
that $f_{E}^{(p)}(T)=T^{r_{E}}$. In particular, $\mu_{p}(E)=0$ and
$\lambda_{p}(E)=r_{E}$.

###### Proof.

Express the characteristic element $f_{E}^{(p)}(T)$ as
$T^{r_{E}}g_{E}^{(p)}(T)$. Theorem 3.11 asserts that $g_{E}^{(p)}(0)\neq 0$
and

$g_{E}^{(p)}(0)\sim\frac{\left(\frac{R_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)\times\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\left(\\#E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

Thus, $g_{E}^{(p)}(0)$ is a $p$-adic unit for $p\in
S^{\operatorname{ord}}\setminus(\Sigma\cup\Sigma^{\prime}\cup\Pi)$. Therefore,
$g_{E}^{(p)}(T)$ is a unit in $\Lambda$. Since $f_{E}^{(p)}(T)$ is a product
of a distinguished polynomial with a power of $p$, it follows that
$g_{E}^{(p)}(T)=1$ and $f_{E}^{(p)}(T)=T^{r_{E}}$. ∎

While it is known that the set $\Sigma^{\prime}$ is finite and $\Sigma$ is a
density zero set of primes, the same is not known for $\Pi$. For $N\geq 1$,
let $\Pi^{\leq N}$ be the set of primes $p\in\Pi$ for which $5\leq p\leq N$.
Calculations in [4] suggest that the set $\Pi$ is likely to have Dirichlet
density zero, at least for elliptic curves of rank $1$ (see also [36]). We
compute $\Pi^{\leq 1000}$ for the first 10 elliptic curves $E/\mathbb{Q}$ of
rank $2$ (ordered by conductor). The calculations in the following table were
done on sage.

| Cremona Label | $\Pi^{\leq 1000}$ |  | Cremona Label | $\Pi^{\leq 1000}$
---|---|---|---|---|---
$1.$ | $389a$ | $\emptyset$ | $6.$ | $643a$ | $\emptyset$
$2.$ | $433a$ | $\\{13\\}$ | $7.$ | $655a$ | $\\{7,31\\}$
$3.$ | $446d$ | $\\{7\\}$ | $8.$ | $664a$ | $\\{59\\}$
$4.$ | $563a$ | $\emptyset$ | $9.$ | $681c$ | $\emptyset$
$5.$ | $571b$ | $\emptyset$ | $10.$ | $707a$ | {29}.

######  Remark 3.14.

For elliptic curves $E_{/\mathbb{Q}}$ with $r_{E}=1$, the above theorem
asserts that for primes outside a set (possibly of Dirichlet density zero),
the characteristic element $f_{E}^{(p)}(T)=T$. Under the hypotheses that the
$p$-adic height pairing on $E(\mathbb{Q}_{n}^{\operatorname{cyc}})$ is non-
degenerate for all $n$ and that
$\\#\Sha(E/\mathbb{Q}_{n}^{\operatorname{cyc}})[p^{\infty}]$ is finite for all
$n$, we know that $f_{\operatorname{Sel}^{0}}^{(p)}(T)=1$ (see [35, pp.
104-105]). Thus, in this case, the fine Selmer group
$\operatorname{Sel}^{0}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is
not only cofinitely generated as a $\mathbb{Z}_{p}$-module, but in fact it is
finite. On the other hand, when $r_{E}>1$ even though the fine Selmer group
should be cofinitely generated as a $\mathbb{Z}_{p}$-module, it is not
expected to be finite.

Next, we consider the supersingular case. Let $E_{/\mathbb{Q}}$ be an elliptic
curve with good supersingular reduction at $p\geq 3$. Note that $p$ divides
$a_{p}\mathrel{\mathop{\ordinarycolon}}=p+1-\\#\widetilde{E}(\mathbb{F}_{p})$.
Hasse’s bound states that $\mathinner{\\!\left\lvert
a_{p}\right\rvert}<2\sqrt{p}$; hence if $p\geq 5$, it forces that $a_{p}=0$.
However, when $p=3$, it is indeed possible for $a_{p}\neq 0$. For simplicity,
we shall assume that $a_{p}=0$. In this setting, the Main conjectures were
formulated in [15]. When $a_{p}=0$, the Main conjecture has been proved in a
preprint of X. Wan (see [33]). Let ${\ddagger}\in\\{+,-\\}$ be a choice of
sign. Perrin Riou [21] formulated a $p$-adic L-function in the supersingular
case, which is closely related to the plus and minus $p$-adic $L$-function
defined by R. Pollack. The $p$-adic Birch and Swinnerton-Dyer conjecture
formulated by D. Bernardi and Perrin-Riou in [2] predicts a formula for the
leading term of the $p$-adic L-function. This conjecture is reformulated in
terms of Pollack’s $p$-adic $L$-functions in [32].

Let $\log_{p}$ be a branch of the $p$-adic logarithm, $\chi$ the $p$-adic
cyclotomic character, and $r_{E}$ be the Mordell-Weil rank of $E$. Let
${\ddagger}\in\\{+,-\\}$, and denote by $R^{{\ddagger}}_{p}(E/\mathbb{Q})$ the
signed $p$-adic regulator (defined up to $p$-adic unit). The convention in
loc. cit. is to choose a generator $\gamma$ of the cyclotomic $\mathbb{Z}_{p}$
extension, and divide the regulator (defined w.r.t. the choice of $\gamma$) by
$\log_{p}(\chi(\gamma))^{r_{E}}$. We are however, not interested in the exact
value of the regulator, but only the value up to a $p$-adic unit. Therefore,
we simply work with the fraction
$\left(\frac{R^{{\ddagger}}_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)$. Lemma 3.3
asserts that the order of vanishing of $f_{E}^{(p),{\ddagger}}(T)$ at $T=0$ is
equal to $r_{E}$. Express $f_{E}^{(p),{\ddagger}}(T)$ as a product
$T^{r_{E}}g_{E}^{(p),{\ddagger}}(T)$. The following conjecture is equivalent
to the $p$-adic Birch and Swinnerton-Dyer conjecture.

######  Conjecture 3.15.

Let $E$ be an elliptic curve with good supersingular reduction at the prime
$p$ and ${\ddagger}\in\\{+,-\\}$. Then,

$g_{E}^{(p),{\ddagger}}(0)\sim\left(\frac{R_{p}^{{\ddagger}}(E/\mathbb{Q})}{p^{r_{E}}}\right)\times\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l\in
S^{\operatorname{bad}}}c_{l}(E)\right).$

Let $\Pi^{{\ddagger}}\subset S^{\operatorname{ss}}$ be the set of primes $p$
at which $v_{p}(R_{p}^{{\ddagger}}(E/\mathbb{Q}))\geq r_{E}$. In other words,
it is the set of primes for which $p$ divides
$\left(\frac{R_{p}^{{\ddagger}}(E/\mathbb{Q})}{p^{r_{E}}}\right)$. The
following result is proved using the same strategy as that of Theorem 3.13, so
we skip the proof.

######  Theorem 3.16.

Assume that Conjecture 3.15 holds. Let $E_{/\mathbb{Q}}$ be an elliptic curve
with Mordell-Weil rank $r_{E}\geq 1$. Let $p$ be a prime at which $E$ has good
supersingular reduction. Then for all primes $p\in
S^{\operatorname{ss}}\setminus(\Sigma\cup\Sigma^{\prime}\cup\Pi^{{\ddagger}})$,
we have that $f_{E}^{(p),{\ddagger}}(T)=T^{r_{E}}$. In particular,
$\mu_{p}^{{\ddagger}}(E)=0$ and $\lambda_{p}^{{\ddagger}}(E)=r_{E}$.

It seems reasonable to make the following conjecture.

######  Conjecture 3.17.

Let $E_{/\mathbb{Q}}$ be an elliptic curve of rank $r_{E}$ with good (ordinary
or supersingular) reduction at $p$. For $100\%$ of the primes,
$\mu_{p}^{{\ddagger}}(E)=0$ and $\lambda_{p}^{{\ddagger}}(E)=r_{E}$.

######  Remark 3.18.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with $r_{E}\geq 1$. If $p$ is a
prime at which the $p$-adic regulator has valuation equal to $p^{r_{E}}$, the
same argument as Remark 3.9 shows that the rank of the elliptic curve remains
unchanged in every layer of the cyclotomic $\mathbb{Z}_{p}$-extension.

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

In this section, we fix a prime $p\geq 5$ and study the variation of Iwasawa
invariants as $E$ ranges over all elliptic curves of rank zero with good
reduction at $p$. Recall that any elliptic curve $E$ over $\mathbb{Q}$ admits
a unique Weierstrass equation

(4.1) $E\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}+Ax+B$

where $A,B$ are integers and $\gcd(A^{3},B^{2})$ is not divisible by any
twelfth power. Since $p\geq 5$, such an equation is minimal. We order elliptic
curves by height and expect that similar results shall hold when they are
ordered by conductor or discriminant. Recall that the height of $E$ satisfying
the minimal equation $\eqref{weier}$ is given by
$H(E)\mathrel{\mathop{\ordinarycolon}}=\max\left(\mathinner{\\!\left\lvert
A\right\rvert}^{3},\mathinner{\\!\left\lvert B\right\rvert}^{2}\right)$.

Let $\mathcal{E}$ be set of isomorphism classes of all elliptic curves over
$\mathbb{Q}$. Let $\mathcal{J}$ be the set of elliptic curves $E$ over
$\mathbb{Q}$ satisfying the following two properties

  1. (1)

$E$ has rank zero,

  2. (2)

$E$ has good reduction (ordinary or supersingular) at $p$.

The set $\mathcal{J}$ is the (disjoint) union of two sets
$\mathcal{J}^{\operatorname{ord}}$ and $\mathcal{J}^{\operatorname{ss}}$,
consisting of rank 0 elliptic curves with ordinary and supersingular reduction
at $p$, respectively. For $X>0$, write $\mathcal{E}(X)$ for the set of
isomorphism classes of elliptic curves over $\mathbb{Q}$ of height $<X$. If
$\mathcal{S}$ is a subset of $\mathcal{E}$, set
$\mathcal{S}(X)=\mathcal{S}\cap\mathcal{E}(X)$. It is conjectured that when
ordered by height, discriminant or conductor, half of the elliptic curves over
$\mathbb{Q}$ have rank $0$ (see for example [10, Conjecture B] or [14, p.
15]). If $E\in\mathcal{J}$ has good ordinary reduction at $p$, then the Euler
characteristic formula $\eqref{ecf1}$ states that

(4.2)
$\chi(\Gamma,E[p^{\infty}])\sim\frac{\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l}c_{l}(E)\right)\times\left(\\#\widetilde{E}(\mathbb{F}_{p})\right)^{2}}{\\#\left(E(\mathbb{Q})[p^{\infty}]\right)^{2}}.$

Note that in the above equation, $\\#E(\mathbb{Q})[p^{\infty}]=1$ if $p\geq
11$. On the other hand, if $E\in\mathcal{J}$ has good supersingular reduction
at $p$, then by $\eqref{ecf2}$, we have that

(4.3)
$\chi^{\pm}(\Gamma,E[p^{\infty}])\sim\\#\Sha(E/\mathbb{Q})[p^{\infty}]\times\left(\prod_{l}c_{l}(E)\right).$

Denote by $c_{l}^{(p)}(E)$ the $p$-part of $c_{l}(E)$, given by
$c_{l}^{(p)}(E)\mathrel{\mathop{\ordinarycolon}}=p^{v_{p}(c_{l}(E))}$. The key
observation in this section is that to analyze the variation of the Euler
characteristic (and hence $\mu$ and $\lambda$-invariants) of elliptic curves,
it suffices to study the average behaviour of the following quantities for
fixed $p$ and varying $E\in\mathcal{J}$,

  1. (1)

$s_{p}(E)\mathrel{\mathop{\ordinarycolon}}=\\#\Sha(E/\mathbb{Q})[p^{\infty}]$,

  2. (2)

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

  3. (3)

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

######  Definition 4.1.

Let $\mathcal{E}_{1}(X)$, $\mathcal{E}_{2}(X)$, and $\mathcal{E}_{3}(X)$ be
the subset of elliptic curves in $\mathcal{E}(X)$ for which $p$ divides
$s_{p}(E)$, $\tau_{p}(E)$ and $\delta_{p}(E)$ respectively.

Note that no assumptions are made on the rank of elliptic curves in
$\mathcal{E}(X)$ or $\mathcal{E}_{i}(X)$. On the other hand, for elliptic
curves $E\in\mathcal{J}$, the rank is zero.

The primary goal is to obtain upper bounds for

$\mathfrak{d}_{p}^{(i)}\mathrel{\mathop{\ordinarycolon}}=\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{E}_{i}(X)}{\\#\mathcal{E}(X)}$

for $i=2,3$ (with no constraints on the rank of the elliptic curves).

In [8], C. Delauney gave heuristics for the average number of elliptic curves
with $s_{p}(E)\neq 1$. These heuristics are stated in terms of elliptic curves
ordered by conductor. However, they indicate that $\mathfrak{d}_{p}^{(1)}$
goes to $0$ as $p\rightarrow\infty$ rather fast. Since there is still not much
known about this particular question, we are unable to make further
clarifications about the behaviour of $\mathfrak{d}_{p}^{(1)}$. However, we
expect that the analysis of this part of the formula is the most difficult.

Let $\kappa=(a,b)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$ be such that the
discriminant $\Delta(\kappa)\mathrel{\mathop{\ordinarycolon}}=4a^{3}+27b^{2}$
is nonzero. The elliptic curve
$E_{\kappa}\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}+ax+b$ defined over
$\mathbb{F}_{p}$ is smooth. Let $d(p)$ be the number of pairs
$\kappa=(a,b)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$ such that

  1. (1)

$\Delta(\kappa)\neq 0$.

  2. (2)

$E_{\kappa}\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}+ax+b$ has a point over
$\mathbb{F}_{p}$ of order $p$.

For the primes $p$ in the range $5\leq p<500$, computations on sage show that
$d(p)\leq 1$ and $d(p)=1$ for $p\in\\{5,7,61\\}$. We remark that $d(p)$ is
closely related to the Kronecker class number of $1-4p$ (see [29, p. 184]).
The estimate $\eqref{cremonasadek}$ follows from the method of M. Sadek [26],
or the results of J. Cremona and Sadek, see [7].

######  Theorem 4.2.

Let $p\geq 5$ be a fixed prime number. Then

(4.4) $\mathfrak{d}_{p}^{(2)}\leq\sum_{l\neq p}\frac{(l-1)^{2}}{l^{p+2}},$

where the sum is taken over prime numbers $l\neq p$, and

(4.5) $\mathfrak{d}_{p}^{(3)}\leq\zeta(10)\cdot\frac{d(p)}{p^{2}}.$

We provide a proof of (4.4) in Theorem 4.11 and of (4.5) in Theorem 4.14. Note
that $\zeta(10)=\frac{\pi^{10}}{93555}$ is approximately equal to $1.001$.
This quantity arises since the proportion of Weierstrass equations ordered by
height which are minimal is $\frac{1}{\zeta(10)}$ (see [7]). To avoid
confusion, we state the results for good ordinary and good supersingular
elliptic curves separately. First, we state the result for elliptic curves
with good ordinary reduction at $p$.

######  Theorem 4.3.

Let $p\geq 5$ be a fixed prime number. Let $\mathcal{Z}^{\operatorname{ord}}$
denote the set of rank 0 elliptic curves $E$ with good ordinary reduction at
$p$ for which the following equivalent conditions are satisfied

  1. (1)

$\chi(\Gamma,E[p^{\infty}])=1$,

  2. (2)

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

Then,

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{Z}^{\operatorname{ord}}(X)}{\\#\mathcal{E}(X)}\geq\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{J}^{\operatorname{ord}}(X)}{\\#\mathcal{E}(X)}-\mathfrak{d}_{p}^{(1)}-\sum_{l\neq
p}\frac{(l-1)^{2}}{l^{p+2}}-\zeta(10)\cdot\frac{d(p)}{p^{2}}.$

###### Proof.

It follows from Corollary 3.6 that $\chi(\Gamma,E[p^{\infty}])=1$ and
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ are
equivalent. By the Euler characteristic formula $\eqref{ecfrdinary}$,

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{Z}^{\operatorname{ord}}(X)}{\\#\mathcal{E}(X)}\geq\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{J}^{\operatorname{ord}}(X)}{\\#\mathcal{E}(X)}-\mathfrak{d}_{p}^{(1)}-\mathfrak{d}_{p}^{(2)}-\mathfrak{d}_{p}^{(3)}.$

The result follows from Theorem 4.2. ∎

Next, we prove an analogous result in the case when $E$ varies over elliptic
curves with good supersingular reduction at $p$.

######  Theorem 4.4.

Let $p\geq 5$ be a fixed prime number. Let $\mathcal{Z}^{\operatorname{ss}}$
be the rank 0 elliptic curves $E$ with good supersingular reduction at $p$,
for which the following equivalent conditions are satisfied

  1. (1)

$\chi^{\pm}(\Gamma,E[p^{\infty}])=1$

  2. (2)

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

Then,

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{Z}^{\operatorname{ss}}(X)}{\\#\mathcal{E}(X)}\geq\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{J}^{\operatorname{ss}}(X)}{\\#\mathcal{E}(X)}-\mathfrak{d}_{p}^{(1)}-\sum_{l\neq
p}\frac{(l-1)^{2}}{l^{p+2}}.$

###### Proof.

The proof is identical to that of Theorem 4.3. It is a direct consequence of
Corollary 3.6, Theorem 4.2, and the Euler characteristic formula
$\eqref{ecfss}$. ∎

We prove a result which applies for all elliptic curves with good reduction.

######  Theorem 4.5.

Let $p\geq 5$ be a fixed prime number. Let $\mathcal{Z}$ be the set of rank 0
elliptic curves $E$ with good reduction at $p$, for which
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})=0$.
Then,

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{Z}(X)}{\\#\mathcal{E}(X)}\geq\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{J}(X)}{\\#\mathcal{E}(X)}-\mathfrak{d}_{p}^{(1)}-\sum_{l\neq
p}\frac{(l-1)^{2}}{l^{p+2}}-\zeta(10)\cdot\frac{d(p)}{p^{2}}.$

###### Proof.

Let $\mathcal{Y}\subset\mathcal{E}$ consist of the elliptic curves $E$ for
which $s_{p}(E)\tau_{p}(E)\delta_{p}(E)\neq 1$. It follows from the Euler
characteristic formulas $\eqref{ecfrdinary}$ and $\eqref{ecfss}$ that
$\mathcal{Z}$ is contained in $\mathcal{E}\setminus\mathcal{Y}$. By Theorem
4.3 that

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{Y}(X)}{\\#\mathcal{E}(X)}\leq\mathfrak{d}_{p}^{(1)}+\sum_{l\neq
p}\frac{(l-1)^{2}}{l^{p+2}}+\zeta(10)\cdot\frac{d(p)}{p^{2}}$

and the result follows. ∎

######  Remark 4.6.

On average, the proportion of elliptic curves over $\mathbb{Z}_{p}$ with good
reduction at $p$ (ordered by height) is $(1-\frac{1}{p})$, see [7]. Also, it
is expected that $1/2$ the elliptic curves have rank $1$ when ordered by
height. Therefore, it is reasonable to expect that

$\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{J}(X)}{\\#\mathcal{E}(X)}=\frac{1}{2}\left(1-\frac{1}{p}\right).$

Heuristics of Delauney suggest that $\mathfrak{d}_{p}^{(1)}$ should approach
zero quite rapidly as $p\rightarrow\infty$. The result indicates that the
proportion of elliptic curves for which the Selmer group is zero is $>0$ and
the proportion approaches $1/2$ as $p\rightarrow\infty$.

We are led to make the following conjecture.

######  Conjecture 4.7.

Let $p$ be a fixed prime. Denote by $\mathcal{J}_{p}$ the set of rank 0
elliptic curves with good reduction at $p$, and by $\mathcal{Z}_{p}$ the
subset of elliptic curves for which
$\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})=0$.
Then,

$\liminf_{p\rightarrow\infty}\left(\limsup_{X\rightarrow\infty}\frac{\mathcal{Z}_{p}(X)}{\mathcal{J}_{p}(X)}\right)=1.$

######  Remark 4.8.

In the rank one case, such an analysis is difficult. This is because of the
term arising from the $p$-adic regulator in the formula for the truncated
Euler characteristic. At the time of writing, the authors are not aware of any
results or heuristics for the average behaviour of the $p$-adic valuation of
$R_{p}(E/\mathbb{Q})$ as $E$ ranges over all elliptic curves of rank $1$ with
good ordinary reduction at $p$.

Theorem 4.2 is proved in the remainder of the section.

###  4.1. Average results on Tamagawa numbers

Let $p\geq 5$ be a fixed prime, and $l$ be a prime different from $p$. In this
section, we estimate the proportion of elliptic curves $E/\mathbb{Q}$ up to
height $X$ with Kodaira type $I_{p}$ at $l$. These estimates are well known,
but we include them for the sake of completeness, see [26, 7]. Recall that
when the Kodaira symbol at the prime $l$ is $I_{p}$, the Tamagawa number
$c_{l}$ is divisible by $p$ [31, p. 448]. Let $\mathcal{E}(X)$ be the set of
isomorphism classes of all elliptic curves over $\mathbb{Q}$ with height $\leq
X$. This is in one-to-one correspondence with the set

$\begin{Bmatrix}&\mathinner{\\!\left\lvert A\right\rvert}\leq\sqrt[3]{X},\
\mathinner{\\!\left\lvert B\right\rvert}\leq\sqrt{X}\\\
(A,B)\in\mathbb{Z}\times\mathbb{Z}\mathrel{\mathop{\ordinarycolon}}&\quad
4A^{3}+27B^{2}\neq 0\\\ &\textrm{for all primes }q\textrm{ if
}q^{4}|A,\textrm{then }q^{6}\nmid B\end{Bmatrix}.$

######  Lemma 4.9 (A. Brumer).

With notation as above,

$\\#\mathcal{E}(X)=\frac{4X^{5/6}}{\zeta(10)}+O\left(\sqrt{X}\right).$

###### Proof.

See [3, Lemma 4.3]. ∎

Consider the set $\mathcal{E}^{I_{p}}_{l}(X)$, i.e. the set of elliptic curves
over $\mathbb{Q}$ with bad reduction at $l$, height $\leq X$, and Kodaira type
$I_{p}$. The Kodaira symbol forces the bad reduction to be of multiplicative
type. It follows from Tate’s algorithm that this set is in one-to-one
correspondence with

(4.6) $\begin{Bmatrix}&\mathinner{\\!\left\lvert
A\right\rvert}\leq\sqrt[3]{X},\ \mathinner{\\!\left\lvert
B\right\rvert}\leq\sqrt{X}\\\
(A,B)\in\mathbb{Z}\times\mathbb{Z}\mathrel{\mathop{\ordinarycolon}}&l\nmid A,\
l\nmid B,\ l^{p}\mathrel{\|}4A^{3}+27B^{3}\\\
&(A,B)\neq(0,0)\in\mathbb{Z}/q^{4}\times\mathbb{Z}/q^{6}\textrm{ for any prime
}q\end{Bmatrix}.$

We include both upper and lower bounds, however, we only apply upper bounds in
our analysis. The following calculations have been done in the preprint [26,
Lemma 4.1]. We clarify the arguments and include them here for completeness.

######  Lemma 4.10 (Sadek).

Let $X>0$ and $l_{i}$ be the $i$-th prime. Let $k>0$ be the largest positive
integer such that $L_{k}=\prod_{i=1}^{k}l_{i}\leq\sqrt[12]{X}$. Then,

$\displaystyle
4l^{p}(l-1)^{2}\prod_{i=1}^{k}(l_{i}^{10}-1)\left(\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}L_{k}^{6}}\right\rfloor-\frac{X^{5/6}}{9l^{2p+2}L_{k}^{10}l_{k}^{9}}\right)\leq\\#\mathcal{E}_{l}^{I_{p}}(X)$
$\displaystyle\leq
4l^{p}(l-1)^{2}\prod_{i=1}^{k}(l_{i}^{10}-1)\left(\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}L_{k}^{6}}\right\rfloor+\frac{\sqrt[3]{X}}{3l^{p+1}L_{k}^{4}l_{k}^{3}}+\frac{\sqrt{X}}{5l^{p+1}L_{k}^{6}l_{k}^{5}}\right)$
.

###### Proof.

To obtain the estimate on the size of the set $\mathcal{E}_{l}^{I_{p}}(X)$, we
use the description of the set in (4.6). Observe that $l^{p+1}\nmid
4A^{3}+27B^{2}$, hence the condition $4A^{3}+27B^{2}\neq 0$ is inherent in the
definition of $\mathcal{E}_{l}^{I_{p}}(X)$.

Consider the congruence equation $4A^{3}+27B^{2}\equiv 0\pmod{l}$. It has
$l-1$ non-singular solutions, which lift to $l^{p-1}(l-1)$ solutions modulo
$l^{p}$ (see for example [34, §3.4.1]). Note that the description of the set
in (4.6) says that $l\nmid A$ and $l\nmid B$; thereby allowing us to ignore
the point $(0,0)$.

In view of (4.6), we are interested in solutions modulo $l^{p}$ that fail to
satisfy the congruence equation modulo $l^{p+1}$. Since the $l-1$ non-singular
solutions lift to $l^{p-1}(l-1)$ solutions modulo $l^{p}$ and $l^{p}(l-1)$
solutions modulo $l^{p+1}$, it follows that the number of pairs
$(A,B)\in\mathbb{Z}/l^{p+1}\times\mathbb{Z}/l^{p+1}$ such that

  1. (1)

$(A,B)\not\equiv(0,0)\mod{l}$,

  2. (2)

$4A^{3}+27B^{2}\equiv 0\mod{l^{p}}$,

  3. (3)

$4A^{3}+27B^{2}\not\equiv 0\mod{l^{p+1}}$,

is equal to

$l^{2}\cdot(l^{p-1}(l-1))-l^{p}(l-1)=l^{p}(l-1)^{2}.$

Therefore, $\mathcal{E}_{l}^{I_{p}}(X)$ has $l^{p}(l-1)^{2}$ pairs of residue
classes in $\mathbb{Z}/l^{p+1}\times\mathbb{Z}/l^{p+1}$. Next, we need to
count the number of lifts of each such pair under the additional condition
that $(A,B)\neq(0,0)\in\mathbb{Z}/l_{i}^{4}\times\mathbb{Z}/l_{i}^{6}$ for
each prime $l_{i}$. Note that the number of pairs $(A,B)$ satisfying this
additional condition is $(l_{i}^{10}-1)$.

It follows that the number of pairs $(A,B)$ in the box
$[-\sqrt[3]{X},\sqrt[3]{X}]\times[-\sqrt{X},\sqrt{X}]$ such that
$(A,B)\neq(0,0)\in\mathbb{Z}/l^{p+1}\times\mathbb{Z}/l^{p+1}$,
$(A,B)\neq(0,0)\in\mathbb{Z}/l_{i}^{4}\times\mathbb{Z}/l_{i}^{6}$ and
$4A^{3}+27B^{2}\equiv 0\mod l^{p}$ is

$4l^{p}(l-1)^{2}\prod_{i=1}^{k}\left(l_{i}^{10}-1\right)\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}L_{k}^{6}}\right\rfloor.$

Our estimate so far might include pairs $(A,B)$ such that it is
$(0,0)\in\mathbb{Z}/q^{4}\times\mathbb{Z}/q^{6}$ when
$l_{k}<q\leq\sqrt[12]{X}$. So, we must exclude the integral pairs which reduce
to $(0,0)\in\mathbb{Z}/q^{4}\times\mathbb{Z}/q^{6}$ for
$l_{k}<q\leq\sqrt[12]{X}$. Therefore, we need to remove

$4l^{p}(l-1)^{2}\prod_{i=1}^{k}\left(l_{i}^{10}-1\right)\sum_{l_{k}<q\leq\sqrt[12]{X}}\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}\right\rfloor$

many pairs from our count. Putting this together, we get

$\displaystyle\\#\mathcal{E}_{l}^{I_{p}}(X)$
$\displaystyle=4l^{p}(l-1)^{2}\prod_{i=1}^{k}\left(l_{i}^{10}-1\right)\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}L_{k}^{6}}\right\rfloor$
$\displaystyle-4l^{p}(l-1)^{2}\prod_{i=1}^{k}\left(l_{i}^{10}-1\right)\sum_{l_{k}<q\leq\sqrt[12]{X}}\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}\right\rfloor.$

To obtain the expression as in the statement of the lemma, we will need to
manipulate the following term

$\sum_{l_{k}<q\leq\sqrt[12]{X}}\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}\right\rfloor.$

First we consider its upper bound.

$\displaystyle\sum_{l_{k}<q\leq\sqrt[12]{X}}\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}\right\rfloor$
$\displaystyle\leq\frac{X^{5/6}}{l^{2(p+1)}L_{k}^{10}}\sum_{l_{k}<q\leq\sqrt[12]{X}}1/q^{10}$
$\displaystyle\leq\frac{X^{5/6}}{l^{2(p+1)}L_{k}^{10}}\int_{l_{k}}^{\infty}1/x^{10}dx$
$\displaystyle\leq\frac{X^{5/6}}{9l^{2(p+1)}l_{k}^{9}L_{k}^{10}}.$

Now for the lower bound, observe that

$\displaystyle\sum_{l_{k}<q\leq\sqrt[12]{X}}\left\lfloor\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}\right\rfloor\left\lfloor\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}\right\rfloor$
$\displaystyle\geq\sum_{l_{k}<q\leq\sqrt[12]{X}}\left(\frac{\sqrt[3]{X}}{l^{p+1}q^{4}L_{k}^{4}}-1\right)\left(\frac{\sqrt{X}}{l^{p+1}q^{6}L_{k}^{6}}-1\right)$
$\displaystyle\geq-\frac{\sqrt[3]{X}}{l^{p+1}L_{k}^{4}}\sum_{l_{k}<q\leq\sqrt[12]{X}}1/q^{4}-\frac{\sqrt{X}}{l^{p+1}L_{k}^{6}}\sum_{l_{k}<q\leq\sqrt[12]{X}}1/q^{6}$
$\displaystyle\geq-\frac{\sqrt[3]{X}}{3l^{p+1}L_{k}^{4}l_{k}^{3}}-\frac{\sqrt{X}}{5l^{p+1}L_{k}^{6}l_{k}^{5}}.$

The result is now immediate. ∎

The following result follows from the previous lemmas in this section.

######  Theorem 4.11.

With notation as above,

$\displaystyle\limsup_{X\rightarrow\infty}\frac{\\#\mathcal{E}^{I_{p}}_{l}(X)}{\\#\mathcal{E}(X)}$
$\displaystyle\leq\frac{(l-1)^{2}}{l^{p+2}}.$

###### Proof.

The result follows from using the upper bound of $\mathcal{E}_{l}^{I_{p}}(X)$
and Lemma 4.9. ∎

######  Remark 4.12.

When the elliptic curves are ordered by conductor (rather than height), the
same bounds have been obtained in [30, Theorem 1.6].

###  4.2. Average results on anomalous primes

We fix a prime $p\geq 5$. Let $\mathcal{W}$ consist of tuples
$(A,B)\in\mathbb{Z}\times\mathbb{Z}$, where $(A,B)$ is identified with the
(minimal) Weierstrass equation

$y^{2}=x^{3}+Ax+B.$

Denote by $\mathcal{W}(X)$ the set of Weierstrass equations for which the
height is $\leq X$. Note that $\mathcal{E}(X)$ is a subset of $\mathcal{W}(X)$
such that

(4.7)
$\lim_{X\rightarrow\infty}\frac{\mathcal{E}(X)}{\mathcal{W}(X)}=\frac{1}{\zeta(10)}$

(see [7]). Thus $99.9\%$ of Weierstrass equations are globally minimal.

Let $\kappa=(a,b)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$ with
$\Delta(\kappa)\neq 0$. Let $E_{\kappa}$ be the elliptic curve defined by the
Weierstrass equation

$E_{\kappa}\mathrel{\mathop{\ordinarycolon}}y^{2}=x^{3}+ax+b.$

Note that $\kappa$ is not uniquely determined by $E_{\kappa}$.

######  Lemma 4.13.

Let $\kappa$ be a pair and $\mathcal{W}_{\kappa}(X)\subset\mathcal{W}(X)$ be
the subset of Weierstrass equations $y^{2}=x^{3}+Ax+B$ such that the pair
$(A,B)$ reduces to $\kappa$. Then,

$\limsup_{X\rightarrow\infty}\frac{\mathcal{W}_{\kappa}(X)}{\mathcal{E}(X)}\leq\frac{\zeta(10)}{p^{2}}.$

###### Proof.

Observe that $\mathcal{W}_{(0,0)}$ is the lattice in
$\mathbb{Z}\times\mathbb{Z}$ with lattice basis $(p,0)$ and $(0,p)$. Since
$\mathcal{W}_{\kappa}$ is simply a translation of $\mathcal{W}_{(0,0)}$, it
follows that

$\lim_{X\rightarrow\infty}\frac{\mathcal{W}_{\kappa}(X)}{\mathcal{W}(X)}=\frac{1}{p^{2}}.$

The result follows from $\eqref{99.9 are glob min}$. ∎

Denote by $\mathfrak{S}$ the set of pairs
$\kappa=(a,b)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$ such that $E_{\kappa}$
contains a point of order $p$ over $\mathbb{F}_{p}$. Recall that
$d(p)\mathrel{\mathop{\ordinarycolon}}=\\#\mathfrak{S}$. Let
$\mathcal{W}^{\prime}(X)\subset\mathcal{W}(X)$ be the set of Weierstrass
equations $y^{2}=x^{3}+Ax+B$ which reduce to $E_{\kappa}$ for some
$\kappa\in\mathfrak{S}$.

######  Theorem 4.14.

We have that $\mathfrak{d}_{p}^{(3)}\leq\zeta(10)\cdot\frac{d(p)}{p^{2}}$.

###### Proof.

It follows from Lemma 4.13 that

$\limsup_{X\rightarrow\infty}\frac{\mathcal{W}^{\prime}(X)}{\mathcal{E}(X)}\leq\zeta(10)\cdot\frac{d(p)}{p^{2}}.$

Recall that $\mathcal{E}_{3}(X)\subseteq\mathcal{W}^{\prime}(X)$. The result
follows. ∎

## Acknowledgments

DK thanks J. Balakrishnan, I. Varma, and N. Kaplan for helpful discussions.
She acknowledges the support of the PIMS Postdoctoral Fellowship. AR is
grateful to R. Sujatha for informing him of the compelling role of the
generalized Euler characteristic in Iwasawa theory. We thank the referee for
timely reading of the paper and expert suggestions.

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# StyleGAN-induced data-driven regularization for inverse problems

###### Abstract

Recent advances in generative adversarial networks (GANs) have opened up the
possibility of generating high-resolution photo-realistic images that were
impossible to produce previously. The ability of GANs to sample from high-
dimensional distributions has naturally motivated researchers to leverage
their power for modeling the image prior in inverse problems. We extend this
line of research by developing a Bayesian image reconstruction framework that
utilizes the full potential of a pre-trained StyleGAN2 generator, which is the
currently dominant GAN architecture, for constructing the prior distribution
on the underlying image. Our proposed approach, which we refer to as learned
Bayesian reconstruction with generative models (L-BRGM), entails joint
optimization over the style-code and the input latent code, and enhances the
expressive power of a pre-trained StyleGAN2 generator by allowing the style-
codes to be different for different generator layers. Considering the inverse
problems of image inpainting and super-resolution, we demonstrate that the
proposed approach is competitive with, and sometimes superior to, state-of-
the-art GAN-based image reconstruction methods.

Index Terms—  Inverse problems, generative prior.

##  1 Introduction

Ill-posed inverse problems are encountered routinely in various imaging
applications, wherein one seeks to estimate an unknown image
$x\in\mathbb{R}^{n}$ from its degraded and noisy measurement
$y\in\mathbb{R}^{m}$. Variational reconstruction circumvents ill-posedness by
incorporating prior knowledge about $x$ via a regularizer
$\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}$, and computes an estimate $\hat{x}$
as

$\hat{x}\in\underset{x}{\operatorname*{arg\,min}}\,L\left(y,Ax\right)+\lambda\,\psi(x),$
(1)

where $A$ denotes the (typically ill-conditioned) measurement operator and $L$
measures data-fidelity. The role of $\psi$ is to penalize undesirable
solutions. Some popular choices for $\psi$ include the Tikhonov
regularization, total-variation (TV), and more recently, sparsity-promoting
regularizers that encourage the image to be sparse in a suitable (analytical
or learned) basis [1]. Notably, for a Gibbs-type prior
$p(x)\propto\exp\left(-\lambda\,\psi(x)\right)$, (1) can be interpreted as the
Bayesian maximum a-posteriori probability (MAP) estimate of the unknown image
$x$.

The success of deep generative modeling in recent years serves as a major
inspiration for learning the regularizer based on a dataset of training images
[2, 3, 4, 5, 6]. Such data-driven regularizers have been shown to
significantly outperform their hand-crafted variants on a wide array of
imaging inverse problems. Within the realm of data-driven regularizers, a
particularly promising approach has been to seek a reconstruction $\hat{x}$
that lies in the range of a pre-trained generator
$G:\mathcal{Z}\rightarrow\mathbb{R}^{n}$, where $\mathcal{Z}$ is a latent
space whose dimension is significantly smaller than $n$. This amounts to
reformulating (1) as

$\hat{x}=G(\hat{z}),\text{\,where\,}\hat{z}\in\underset{z}{\operatorname*{arg\,min}}\,L\left(y,AG(z)\right).$
(2)

Training such a generator $G$ generally tends to suffer from mode collapse, in
which case $\text{range}(G)$ might fail to represent the image manifold in its
entirety and there could be target images lying outside $\text{range}(G)$. In
such scenarios, solving (2) fails to recover the image of interest, and one
needs to allow for some flexibility for the target image to lie close to
$\text{range}(G)$ through soft constraints. Such methods typically require
computing a latent code for a given image, which calls for inverting $G$, and
are therefore referred to as GAN-inversion [7]. Methods for inverting a GAN
can be learning- or optimization-based, and we adopt the latter in our work.
In particular, we extend the recently proposed Bayesian reconstruction through
generative models (BRGM) approach [8] by introducing a data-driven modeling of
the regularizer on the latent space. Since our approach employs learned
networks for prior modeling, we refer to it as learned-BRGM (L-BRGM).

##  2 The proposed L-BRGM method

To construct the latent prior, L-BRGM relies on a pre-trained StyleGAN2, which
is a recent state-of-the-art GAN architecture developed by Karras et al. [9].
StyleGAN2 is an improved variant of its predecessor StyleGAN [10], named after
its architecture’s inspiration from the style transfer literature. In
StyleGAN, the input latent space $\mathcal{Z}$ is warped into an intermediate
disentangled feature space $\mathcal{W}$ (also called the style space) via a
eight-layer fully connected mapping network $M$. The StyleGAN generator $G$,
consisting of adaptive instance normalization (AdaIN) and convolution blocks,
subsequently produces photo-realistic images from these disentangled features.
Randomness is introduced in the generated samples by feeding noise into the
generator. StyleGAN2 enhances the sample quality by redesigning the generator
architecture and by introducing a novel path-length regularizer (PLR). The PLR
seeks to ensure that the generator approximately preserves length, i.e., a
fixed change in $w\in\mathcal{W}$ causes a fixed magnitude change in the
sampled image. The effect of PLR is to promote orthogonality in the Jacobian
matrix $J_{G}(w)=\frac{\partial G(w)}{\partial w}$ at any $w$. As remarked in
[9], the PLR makes it easier to invert $G$, motivating the usage of StyleGAN2
in our approach.

###  2.1 A Bayesian formulation of L-BRGM

The proposed L-BRGM approach has two key features: (i) formulating the
reconstruction problem as a joint optimization over the style-code $w$ and the
input latent code $z$, and (ii) using the pre-trained mapping network
$M:\mathcal{Z}\rightarrow\mathcal{W}$ to model the distribution of $w$. Both
of them emerge naturally from the Bayesian framework explained in the
following.

Let $p(y|w,z)$ be the probability density function (p.d.f.) of the measurement
$y$ conditioned on the style-code $w$ and the input noise $z$. Given $w$, the
distribution of $y$ is independent of $z$, and is determined by the generator
$G$ and the measurement operator $A$. Consequently, we have that
$p(y|w,z)=p(y|w)$. The posterior distribution of the joint latent code $(w,z)$
conditioned on $y$ can be expressed using Bayes’ rule:

$p(z,w|y)=\frac{p(y|w,z)p(w,z)}{p(y)}=\frac{p(y|w)p(w,z)}{p(y)}.$ (3)

Further, $p(w,z)$ factors as $p(w,z)=p(w|z)p(z)$, leading to

$\log p(z,w|y)=\log p(y|w)+\log p(w|z)+\log p(z)-\log p(y).$

Since $w$ is obtained by applying the mapping network $M$ on $z$, it is
natural to model $p(w|z)$ as a Gaussian perturbation around $M(z)$, i.e.,
$p(w|z)=\mathcal{N}\left(w;M(z),\lambda_{\text{map}}^{-1}\,I\right)$.
Subsequently, noting that
$p(z)=\mathcal{N}\left(z;0,\lambda_{\text{lat}}^{-1}I\right)$, and assuming
the same measurement noise model considered in [8, eq(4)] (i.e., Gaussian
noise in the pixel space as well as in the VGG-16 embedding space), the MAP
estimate of $(w,z)$ reduces to

$\displaystyle\underset{(z,w)}{\min}\lambda_{\text{pix}}\|y-A\,G(w)\|_{2}^{2}+\lambda_{\text{vgg}}\|\varphi(y)-\varphi(A\,G(w))\|_{2}^{2}$
$\displaystyle+\lambda_{\text{map}}\|w-M(z)\|_{2}^{2}+\lambda_{\text{lat}}\|z\|_{2}^{2},$
(4)

where $\varphi$ computes the VGG-16 features. Finally, we extend (4) by
optimizing over $w^{+}$, where $w^{+}=\left\\{w_{i}\right\\}_{i=1}^{L}$,
allowing for different style-codes $w_{i}$ at different layers $i$ of the
generator. The resulting optimization reads
$\underset{(z,w^{+})}{\min}J(w^{+},z)$, where

$\displaystyle J(w^{+},z)$
$\displaystyle:=\lambda_{\text{pix}}\|y-A\,G(w^{+})\|_{2}^{2}+\lambda_{\text{map}}\sum_{i=1}^{L}\|w_{i}-M(z)\|_{2}^{2}$
$\displaystyle+\lambda_{\text{vgg}}\|\varphi(y)-\varphi(A\,G(w^{+}))\|_{2}^{2}+\lambda_{\text{lat}}\|z\|_{2}^{2}.$
(5)

$J$ is minimized via an iterative optimization algorithm (namely, Adam [11])
starting from an appropriate initialization (c.f. Sec. 2.3); and the final
reconstructed image is computed as $\hat{x}=G(\hat{w}^{+})$, where
$\hat{w}^{+}$ is the optimal style-code.

Fig. 1: Effect of regularization. Without any regularization, the
reconstructed image tends to look unnatural. Fig. 2: Illustration of the
effect of early stopping for L-BRGM and BRGM. Further, despite strong
quantitative performance, we noted that GFP-GAN struggled to generate a
natural-looking image.

Since L-BRGM extends the BRGM approach by exploiting the StyleGAN2 mapping
network for modeling $p(w)$, it is imperative to compare and contrast the
Bayesian probability models for the two methods. As the ground-truth image $x$
can be modeled as $x=G(w)$ for an appropriate $w$, the BRGM approach seeks to
maximize $p(x|y)\propto p(y|G(w))p(G(w))$. Subsequently, the image prior in
BRGM is rewritten via the change of variable formula as
$p(G(w))=p(w)\left(\text{det}\left(J_{G}(w)\right)\right)^{-1}$. Finally,
making the simplifying assumption that the Jacobian $J_{G}(w)$ is not
dependent on $w$ (i.e., $G(w)$ varies approximately linearly in $w$, which can
be justified by the PLR during StyleGAN2 training), one ends up with the task
of solving eq(6) posed in [8]. To model $p(w)$ in BRGM, one allows for the
flexibility of having different $w_{i}$’s at different layers of $G$, while
encouraging the $w_{i}$s to be similar via a cosine similarity prior.
Additionally, each $w_{i}$ is modeled as a Gaussian random vector, with the
parameters empirically estimated from a large number of samples fed through
$M$.

L-BRGM extends this Bayesian formalism by formulating a Bayesian MAP
estimation problem over the joint code $(w,z)$ and alleviates the need to
hand-craft a suitable prior on $w$ by exploiting the mapping network $M$.
L-BRGM also allows for different style-codes $w_{i}$ for $L=18$ different
layers of $G$. Further, all $w_{i}$s are encouraged to be close to $M(z)$,
thus ensuring mutual similarity among each pair of $w_{i}$s. L-BRGM does not
require any Gaussian assumption on the style-codes $w_{i}$, but instead makes
this assumption on the latent $z$, thereby making the framework more realistic
(since $z$ is indeed sampled from a Gaussian distribution during training).

###  2.2 The effect of regularization

It is worth addressing the importance of modeling the distribution $p(w)$ of
the style-code, which acts as a regularizer in the reconstruction process. If
one optimizes the latent vector $w$ by solely minimizing the data-likelihood
term (i.e., a combination of the pixel-wise error and VGG perceptual loss),
the reconstructed images tend to have unnatural artifacts. This phenomenon is
demonstrated in Fig. 1 for inpainting on the FFHQ dataset consisting of human
face images of resolution $1024^{2}$. The reconstructed image with no
regularization on $w$ appears to be smooth around the inpainting boundary, but
fails to recover important facial features (such as hair color and texture)
far from this boundary. Notably, the L-BRGM reconstruction in Fig. 1 matches
the ground-truth more closely than BRGM. This highlights the fact that the
regularization term on the style-code can lead to realizing the full capacity
of the underlying generative model. While no regularization leads to worse
reconstructions, hand-crafting the regularizer also limits the potential of
the Bayesian framework considerably.

###  2.3 Key implementation details

The reconstruction loss $J(w^{+},z)$ arising from the Bayesian MAP formalism
is a non-convex objective of the joint latent code $(w^{+},z)$. Therefore, it
is generally challenging to devise an iterative algorithm that succeeds in
recovering the global minimum regardless of the initialization. We employ the
Adam algorithm developed by Kingma and Ba [11] in view of its wide
applicability for solving high-dimensional non-convex problems while requiring
minimal hyper-parameter tuning. Nevertheless, the success of Adam in solving
(5) depends to a great extent on careful initialization and the stopping rule,
both of which are elucidated in the following.

Multiple initialization: To initialize the latent code $z$ for (5), we draw
$N_{1}$ i.i.d. samples $\left\\{z_{(i)}\right\\}_{i=1}^{N_{1}}$ from the
standard normal distribution, and select the best $z_{(i)}$ that minimizes the
perceptual loss: $z_{\text{init}}=\underset{1\leq i\leq
N_{1}}{\operatorname*{arg\,min}}\,\|\varphi(y)-\varphi(A\,G(M(z_{(i)})))\|_{2}^{2}$.
Computing $z_{\text{init}}$ does not require any expensive gradient
computation and only entails forward passes through the generator and the
mapping network. Such an initialization approach prevents any bias that might
potentially arise from just one initial point and helps the reconstruction
algorithm avoid local minima in the loss landscape. We found that $N_{1}=100$
sufficed for our experiments.

Early stopping: We found that for both BRGM and L-BRGM, the iterative scheme
for reconstruction can potentially diverge if the iterations are not
terminated appropriately. This phenomenon is illustrated in Fig. 2 for the
task of image super-resolution. The example in the second row of Fig. 2
illustrates how early stopping might be necessary to prevent divergence of
BRGM and L-BRGM from the image manifold, whereas the first row shows a
representative example where such an early stopping may not be required.
Therefore, we track the learned perceptual image patch similarity (LPIPS)
metric [12] between the ground-truth and the reconstructed image over the
iterations and report the best LPIPS (i.e., the smallest) distance achieved
during the iterative optimization process. Empirically, we found that BRGM
also attained the optimal LPIPS distance around the same iteration as L-BRGM.
Such early stopping is not applicable to the GFP-GAN approach [13] considered
for numerical comparison in Section 3, since GFP-GAN is a learning-based
inversion method, as opposed to BRGM and L-BRGM that employ optimization for
inversion.

Fig. 3: Representative inpainting examples for the half mask (first three
rows) and the eye-patch mask (last row). L-BRGM visibly outperforms BRGM and
is competitive with DeepGIN.

Inpainting | Input | Mask | Method | LPIPS | SSIM
---|---|---|---|---|---
$256^{2}$ | Half | DeepGIN | 1 | 0.303 | 59 | 0.625
BRGM | 47 | 0.226 | 21 | 0.606
L-BRGM | 51 | 0.224 | 19 | 0.594
Super-res. | Input | Output | Method | LPIPS | SSIM
$64^{2}$ | $256^{2}$ | GFP-GAN | 72 | 0.231 | 89 | 0.534
BRGM | 6 | 0.294 | 10 | 0.480
L-BRGM | 21 | 0.277 | 0 | 0.404

Table 1: Comparison of L-BRGM with competing GAN-based approaches for inpainting and super-resolution on target images of size $256^{2}$. For both the LPIPS and SSIM, the first column indicates the number of test images (out of a total of 99) the corresponding method performed the best in terms of the respective metric, whereas the second column contains the average value over all the test images. The best performances are highlighted in boldface. | Inpainting | Super-resolution
---|---|---
Method | Mask | LPIPS | Input | LPIPS
BRGM | Half | 17 | 0.495 | $64^{2}$ | 32 | 0.426
L-BRGM | 82 | 0.460 | 67 | 0.414
BRGM | Eye-patch | 10 | 0.428 | $32^{2}$ | 33 | 0.470
L-BRGM | 89 | 0.393 | 66 | 0.445
Table 2: L-BRGM vs. BRGM on images of resolution $1024^{2}$. L-BRGM
outperforms BRGM on significantly more test images and results in superior
LPIPS distance on average.

##  3 Numerical results

We consider two prototypical inverse problems, namely (i) image inpainting and
(ii) super-resolution to validate L-BRGM quantitatively and compare it with
state-of-the-art GAN-based image reconstruction approaches. For inpainting, we
choose BRGM [8] and deep generative inpainting network (DeepGIN) [14] as the
competing methods. BRGM is a natural candidate for comparison since L-BRGM is
a direct extension of it, while the choice of DeepGIN, which was an entry in
the Aim-2020 inpainting challenge [15], was motivated by its applicability to
inpainting with extreme masking.

For super-resolution, we choose BRGM and the generative facial prior (GFP)
approach [13] that seeks to restore facial images by exploiting the prior
encapsulated by a GAN trained on face images (referred to as GFP-GAN) for
comparing with L-BRGM. GFP-GAN uses the existing architecture from Real-ESRGAN
[16] and is reported to yield state-of-the-art performance for facial image
restoration. Contrary to the generators in BRGM and L-BRGM, both GFP-GAN and
DeepGIN were trained on images of resolutions lower than $1024^{2}$.
Consequently, we include comparisons with GFP-GAN and DeepGIN for target
ground-truth images of resolution $256^{2}$, whereas for the images of
resolution $1024^{2}$, the comparison is restricted to only BRGM. Both L-BRGM
and BRGM reconstructions are computed with 2000 iterations. The Adam
parameters are chosen as $\eta,\beta_{,}\beta_{2}=0.1,0.96,0.9999$; whereas
the regularization penalties are:
$\lambda_{\text{pix}},\lambda_{\text{vgg}},\lambda_{\text{map}},\lambda_{\text{lat}}=2\times
10^{-5},2\times 10^{7},30,0.4$. A set of 99 images (not used during training)
are utilized for performance validation. The LPIPS scores are calculated using
AlexNet features [17] for a fair evaluation, since the perceptual loss based
on VGG-16 embedding is already a part of the reconstruction objective $J$.

For inpainting on images of size $256^{2}$, L-BRGM outperforms both BRGM and
DeepGIN in terms of LPIPS (c.f. Table 1). In case of super-resolution, L-BRGM
defeats BRGM, but performs slightly worse as compared with GFP-GAN. The
superiority of GFP-GAN can presumably be attributed to the fact it is trained
with bespoke loss terms that are particularly designed for face images (such
as the facial component and identity preserving losses [13]). For inpainting,
L-BRGM also emerges as the best performing method in terms of LPIPS on a
majority of the test images. The inpainting examples in Fig. 3 demonstrate
that L-BRGM is conspicuously superior to its competitors. However, L-BRGM
turns out to be slightly inferior in terms of SSIM, but we found that, unlike
LPIPS, SSIM does not correlate well with the visual image quality and we
therefore report only the LPIPS distances for inpainting and super-resolution
on $1024^{2}$ images (see Table 2). In this case, L-BRGM was found to
significantly surpass BRGM for both inpainting masks and for the task of
super-resolution. The superiority of L-BRGM over BRGM underscores the need for
a better regularizer on the latent space, and highlights the advantages of
data-driven modeling of such a regularizer.

##  4 Conclusions and outlook

We proposed a novel optimization-based approach for StyleGAN2 inversion, with
potential applications to imaging inverse problems. The proposed L-BRGM
algorithm leverages the full expressive power of the StyleGAN2 generator and
mapping network for modeling the joint latent- and style-code prior in a
Bayesian estimation framework. The proposed approach yields competitive
performance with state-of-the-art GAN-based approaches for prototypical
imaging inverse problems, such as inpainting and super-resolution.
Nevertheless, developing a generic method to outperform the state-of-the-art
on all possible target images is an ambitious and rather challenging
objective. Therefore, further research is needed to not only produce novel
regularization schemes that effectively make use of a pre-trained GAN, but
also to learn generative models with more interpretable (and hence
regularizable) latent spaces for image reconstruction problems.

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