# Asymptotics of $p$-torsion subgroup sizes in class groups of monogenized
cubic fields

Mikaeel Yunus

###### Abstract

Bhargava, Hanke, and Shankar have recently shown that the asymptotic average
$2$-torsion subgroup size of the family of class groups of monogenized cubic
fields with positive and negative discriminants is $3/2$ and $2$,
respectively. In this paper, we provide strong computational evidence for
these asymptotes. We then develop a pair of novel conjectures that predicts,
for $p$ prime, the asymptotic average $p$-torsion subgroup size in class
groups of monogenized cubic fields.

##  1 Introduction

Asymptotics of class groups of number fields over the rationals have been
studied for hundreds of years by a wide variety of mathematicians, including
Gauss [12] in the 18th century; Davenport, Heilbronn, Cohen, Lenstra, and
Martinet [8, 6, 7] in the late 20th century; and Bhargava, Fouvry, and Klüners
in the 21st century [2, 10]. In 2014, Bhargava and Varma [5] showed that the
asymptotic average $2$-torsion subgroup size of the family of class groups of
cubic fields ordered by discriminant remains the same regardless of any local
conditions imposed on these cubic fields. In 2018, Ho, Shankar, and Varma [14]
showed that the same average size remains even when ordering by height rather
than discriminant.

However, Bhargava, Hanke, and Shankar [3] have recently shown that mandating
these cubic fields to have monogenic rings of integers – a global condition –
suprisingly changes the asymptotic average in question when ordering by
height. This result is unexpected and interesting, and little is known about
it aside from the calculation of the average $2$-torsion subgroup size in [3]
(see also [20] and [21]).

In this paper, we provide computational evidence that imposing the global
monogenicity condition mentioned above does indeed increase the asymptotic
average in question. We then provide models that verify the results of [3] on
the average $2$-torsion subgroup size of the family of class groups of cubic
fields with monogenic rings of integers.

Our general approach is as follows. We first use SageMath [18] to calculate
the average $2$-torsion subgroup size of the family of class groups for
irreducible, monogenic, and maximal binary cubic forms with height less than
$Y$. Here, $Y=10^{11}$ for positive-discriminant binary cubic forms, and
$Y=10^{10}$ for negative-discriminant binary cubic forms. We then employ
genetic programming [15, 19] to predict the necessary asymptotes and provide
evidence for their agreement with the theoretical values [3].

We subsequently present our main result: a pair of new conjectures predicting,
for prime $p$, the asymptotic averages of $p$-torsion subgroup sizes of class
groups of monogenized cubic fields ordered by height.111Note that ordering
these monogenized cubic fields by height instead of discriminant should not
change the asymptotic averages in question (compare the results of [2] and
[14]). For positive discriminants, we predict the averages size to be:

$1+\dfrac{1}{p(p-1)}$ (1)

For negative discriminants:

$1+\dfrac{1}{p-1}$ (2)

We develop these conjectures using similar computational methods to the ones
we use to provide evidence for the results of [3].

##  2 Theoretical Background

Here, we present definitions that are necessary for the result of [3] that we
work with in this paper.

###  2.1 Preliminary Definitions

Our goal is to study class groups222Note that we will be using the same
definition of a class group (also known as an ideal class group) as the one
provided in [1]. of number fields. We begin by reviewing number fields.

######  Definition 2.1.1.

Let $K_{0}$ be a field. Then, the degree of the field extension $K/K_{0}$ is
the dimension of $K$ as a vector space over $K_{0}$.

######  Definition 2.1.2.

A number field $K$ is a field extension of $\mathbb{Q}$ that has a finite
degree.

Number fields are a generalization of the rational numbers, $\mathbb{Q}$. Just
as the rational numbers contain the ring $\mathbb{Z}$, there is an analogous
ring of integers of a number field.

######  Definition 2.1.3.

A ring of integers $\mathcal{O}_{K}$ of a number field $K$ is the ring of all
$\alpha$ in $K$ for which there exists a minimal polynomial $f$ such that
$f(\alpha)=0$ and $f(x)$ has coefficients in $\mathbb{Z}$.

The ring of integers of a number field is always a Dedekind domain. An
important difference between rings of integers and $\mathbb{Z}$ is that the
rational integers are a principal ideal domain (in fact, $\mathbb{Z}$ is a
Euclidean domain).

Additionally, the ring of integers of a number field is maximal in the sense
that it is the maximal order of a number field $K$.

######  Definition 2.1.4.

Let $K$ be a number field. Then, an order of $K$ is a subring of
$\mathcal{O}_{K}$ whose fraction field is equal to $K$.

While orders may not be Dedekind domains, they are always integral domains.

######  Definition 2.1.5.

An integral domain $R$ has rank $n$ if its fraction field is a number field of
degree $n$.

######  Definition 2.1.6.

We say an integral domain $R$ of rank $n$ is monogenic if and only if

$R=\left\\{\sum_{i=0}^{n-1}a_{i}\gamma^{i}\mid a_{i}\in\mathbbm{Z}\right\\}$

for some $\gamma\in R$. In other words, all elements of $R$ can be represented
as a sum of integer multiples of the powers of exactly one element.

We now recall two definitions from [3]:

######  Definition 2.1.7.

We define a monogenized cubic integral domain to be a pair $(R,\alpha)$, where
$R$ is a cubic integral domain, and $\alpha\in R$ such that
$R=\mathbb{Z}[\alpha]$.

######  Remark 2.1.

A monogenized cubic integral domain is monogenic by definition.

######  Definition 2.1.8.

We define a monogenized cubic field to be a pair $(K,\alpha)$, where $K$ is a
cubic field, and $\alpha\in\mathcal{O}_{K}$ such that
$\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

Finally, we recall a definition from [3] that is essential to Theorem 2.9
below.

######  Definition 2.1.9.

Let $(R,\alpha)$ and $(R^{\prime},\alpha^{\prime})$ be two monogenized cubic
integral domains. Then, $(R,\alpha)$ and $(R^{\prime},\alpha^{\prime})$ are
isomorphic if there exists an isomorphism $R\rightarrow R^{\prime}$ under
which $\alpha$ is mapped to $\alpha+m$ for some $m\in\mathbb{Z}$.

###  2.2 The Delone-Faddeev Correspondence

######  Definition 2.2.1.

A binary cubic form $f$ is a cubic homogeneous polynomial in two variables
with integer coefficients, i.e.
$f(x,y)=f_{0}x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ where
$f_{0},f_{1},f_{2},f_{3}\in\mathbb{Z}$.

We define the action of

$\mathrm{GL}_{2}(\mathbb{Z})=\left\\{\gamma=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\ \middle|\ a,b,c,d\in\mathbb{Z}\ \mbox{ and
}\det(\gamma)=ad-bc=\pm 1\right\\}$

on the space of all integral binary cubic forms by:

$\displaystyle\gamma\cdot f(x,y)$
$\displaystyle\coloneqq\frac{1}{\det(\gamma)}\
f\left(\begin{matrix}(x&y)\end{matrix}\cdot\gamma\right)$
$\displaystyle=\frac{1}{ad-bc}\
f\left(\begin{matrix}(x&y)\end{matrix}\cdot\left(\begin{matrix}a&b\\\ c&d\\\
\end{matrix}\right)\right)$

For the remainder of this paper, we refer to an integral domain of rank 3 as a
cubic integral domain.

######  Definition 2.2.2.

The discriminant of a binary cubic form is

$\emph{Disc}(f):=f_{1}^{2}f_{2}^{2}-4f_{0}f_{2}^{3}-4f_{3}f_{1}^{3}-27f_{0}^{2}f_{3}^{3}+18f_{0}f_{1}f_{2}f_{3}$

We now recall the Delone-Faddeev correspondence as given in [4].

######  Theorem 2.2 (Delone-Faddeev [9]).

There is a natural bijection between the set of
$\mathrm{GL}_{2}(\mathbb{Z})$-equivalence classes of irreducible integral
binary cubic forms and the set of isomorphism classes of cubic integral
domains. Furthermore, the discriminant of an integral binary cubic form $f$ is
equal to the discriminant of the corresponding cubic integral domain $R(f)$.

Although we do not define the discriminant of a cubic integral domain here, we
can utilize the above theorem to define

$\textrm{Disc}(R(f)):=\textrm{Disc}(f)$

The construction of a cubic integral domain from a binary cubic form can be
described quite explicitly in the form of a multiplication table. Given a
binary cubic form $f(x,y)=f_{0}x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$, the
cubic integral domain associated to $f(x,y)$ under Theorem 2.2 can be written
as

$R(f):=\\{a+b\cdot\omega+c\cdot\theta\mid a,b,c\in\mathbbm{Z}\\}$

where:

$\displaystyle\omega\cdot\theta$ $\displaystyle=$
$\displaystyle\theta\cdot\omega=-f_{0}\cdot f_{3}$ (3)
$\displaystyle\omega^{2}$ $\displaystyle=$ $\displaystyle-f_{0}\cdot
f_{2}-f_{1}\cdot\omega+f_{0}\cdot\theta$ (4) $\displaystyle\theta^{2}$
$\displaystyle=$ $\displaystyle-f_{1}\cdot
f_{3}-f_{3}\cdot\omega+f_{2}\cdot\theta$ (5)

Now, building on Theorem 2.2, we can establish a correspondence between binary
cubic forms and fraction fields.

######  Lemma 2.3.

Let $f(x,y)=f_{0}x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ be a binary cubic
form, and let $R(f)$ be its corresponding integral domain (cf. Theorem 2.2).
Then the fraction field $\textrm{Frac}(R(f))$ is precisely the fraction field
of the polynomial $g(x)=x^{3}+f_{1}x^{2}+f_{0}f_{2}x+f_{0}^{2}f_{3}$.

###### Proof.

If $f(x,y)=f_{0}x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$, then the basis
elements of $R(f)$ are $1,\omega,\theta$, where $\omega,\theta$ are determined
via Equations (1), (2), and (3) in the discussion following Theorem 2.2.

This means that

$\begin{cases}\theta=\frac{-f_{0}f_{3}}{\omega}\hskip 113.81102pt\\\
\omega^{2}=-f_{0}f_{2}-f_{1}\omega+f_{0}\theta\hskip 38.41121pt\\\
\end{cases}$

Thus, substituting the first equation above into the second equation,

$\omega^{2}=-f_{0}f_{2}-f_{1}\omega-\frac{f_{0}^{2}f_{3}}{\omega}$

Multiplying both sides by $\omega$ and rearranging, we have

$\omega^{3}+f_{1}\omega^{2}+f_{0}f_{2}\omega+f_{0}^{2}f_{3}=0$

Since the solution to the above equation is the second basis element of the
cubic integral domain corresponding to the given binary cubic form, the
fraction field of this binary cubic form is precisely the fraction field of
the following polynomial:

$g(x)=x^{3}+f_{1}x^{2}+f_{0}f_{2}x+f_{0}^{2}f_{3}$

∎

###  2.3 Special Properties of Binary Cubic Forms

We start with the following definition:

######  Definition 2.3.1.

A monogenic binary cubic form is a binary cubic form that corresponds to a
monogenic cubic integral domain through the Delone-Faddeev correspondence.

In this subsection, we present two constraints that we are allowed to make on
the coefficients of monogenic binary cubic forms. These constraints prove
useful in the computational methodology (Section 3).

We begin with a constraint on the $x^{3}$-coefficient $f_{0}$ of an arbitrary
monogenic binary cubic form.

######  Lemma 2.4.

Given $f_{0},f_{1},f_{2},f_{3}\in\mathbb{Z}$, the binary cubic form
$f_{0}x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ is monogenic if and only if we
can perform a $\mathrm{GL}_{2}(\mathbb{Z})$-action on it to achieve another
binary cubic form
$f_{0}^{\prime}x^{3}+f_{1}^{\prime}x^{2}y+f_{2}^{\prime}xy^{2}+f_{3}^{\prime}y^{3}$
with
$f_{0}^{\prime},f_{1}^{\prime},f_{2}^{\prime},f_{3}^{\prime}\in\mathbb{Z}$ and
$f_{0}^{\prime}=1$.

###### Proof.

First, we write the proof of the reverse direction. Assume that
$f_{0}^{\prime}=1$. Then, by the Delone-Faddeev correspondence, the following
system of equations is satisfied:

$\displaystyle\omega\cdot\theta$ $\displaystyle=$
$\displaystyle\theta\cdot\omega=-f_{3}^{\prime}$ $\displaystyle\omega^{2}$
$\displaystyle=$ $\displaystyle-
f_{2}^{\prime}-f_{1}^{\prime}\cdot\omega+\theta$ $\displaystyle\theta^{2}$
$\displaystyle=$ $\displaystyle-f_{1}^{\prime}\cdot
f_{3}^{\prime}-f_{3}^{\prime}\cdot\omega+f_{2}^{\prime}\cdot\theta$

From the second equation in the system above, we have
$\theta=\omega^{2}+f_{2}^{\prime}+f_{1}^{\prime}\omega$. As a result, we have
that any element in $R(f)$ can be written as a linear combination of powers of
$\omega$:

$\displaystyle a+b\omega+c\theta$ $\displaystyle=$ $\displaystyle
a+b\omega+c(\omega^{2}+f_{2}^{\prime}+f_{1}^{\prime}\omega)$ $\displaystyle=$
$\displaystyle(a+cf_{2}^{\prime})+(b+cf_{1}^{\prime})\omega+c\omega^{2}$

Let $a^{\prime}=a+cf_{2}^{\prime}$, $b^{\prime}=b+cf_{1}^{\prime}$, and
$c^{\prime}=c$. Then, we have
$R(f)=\\{a^{\prime}+b^{\prime}\omega+c^{\prime}\omega^{2}\mid
a^{\prime},b^{\prime},c^{\prime}\in\mathbb{Z}\\}$. Thus, $R(f)$ is monogenic.

Now, we write the proof of the forward direction. Since the binary cubic form
$f(x,y)$ is monogenic, the corresponding integral domain $R(f)$ is monogenic.
For some $\alpha\in R(f)$, we can write $R(f)=\langle
1,\alpha,\alpha^{2}\rangle$.

Since $\alpha$ is a member of a cubic integral domain, there exists a monic
polynomial $g(x)=x^{3}+g_{1}x^{2}+g_{2}x+g_{3}$, where
$g_{1},\cdots,g_{3}\in\mathbb{Z}$, such that $g(\alpha)=0$. Keeping this in
mind, we can redefine $R(f)$ as follows:

$R(f)=\langle 1,\alpha+g_{1},\alpha^{2}+g_{2}\rangle$

We note that

$(\alpha+g_{1})(\alpha^{2}+g_{2})=\alpha^{3}+g_{1}\alpha^{2}+g_{2}\alpha+g_{1}g_{2}=-g_{3}+g_{1}g_{2}\in\mathbb{Z}$

Thus, the basis $\langle 1,\alpha,\alpha^{2}\rangle$ corresponds to a binary
cubic form through the Delone-Faddeev correspondence.

Let $\omega=\alpha+g_{1}$, and $\theta=\alpha^{2}+g_{2}$. We compute the
following:

$\omega^{2}=(\alpha+g_{1})^{2}=\alpha^{2}+2\alpha g_{1}+g_{1}^{2}$

Note that since $\omega=\alpha+g_{1}$ and $\theta=\alpha^{2}+g_{2}$, we have
$\alpha=\omega-g_{1}$ and $\alpha^{2}=\theta-g_{2}$. Thus,

$\displaystyle\omega^{2}$ $\displaystyle=$ $\displaystyle(\theta-
g_{2})+2(\omega-g_{1})g_{1}+g_{1}^{2}$ $\displaystyle=$ $\displaystyle\theta-
g_{2}+2\omega g_{1}-2g_{1}^{2}+g_{1}^{2}$ $\displaystyle=$
$\displaystyle-(g_{1}^{2}+g_{2})+2g_{1}\omega+\theta$

In conjunction with the above, (4) implies that $f_{0}=1$.

∎

For the remainder of this paper, we assume any monogenic binary cubic form has
$x^{3}$-coefficient $1$, because out interest lies in cubic domains, not their
bases, and we can choose whichever binary cubic form we want to represent its
own $\mathrm{GL}_{2}(\mathbb{Z})$-equivalence class.

Before we proceed, we define the following property of monogenic binary cubic
forms, which we call “height.” This property lets us not only assign a weak
ordering to binary cubic forms, but also a weak ordering to cubic fields.

######  Definition 2.3.2.

Let $f=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ be a monogenic binary cubic
form, and let $I(f)=f_{1}^{2}-3f_{2}$, $J(f)=-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}$.
Then the height $H$ of $f$ is as follows:

$H(f):=\max\left(|I|^{3},\frac{J^{2}}{4}\right)$

If $f$ is clear from context, we will simply let $I=I(f)$ and $J=J(f)$.

Now, having defined the height invariant, we set a constraint on the second
coefficient $f_{1}$ of an arbitrary binary cubic form. But first, we must
introduce a little more theory.

######  Definition 2.3.3.

We define the subgroup $F(\mathbb{Z})<\mathrm{GL}_{2}(\mathbb{Z})$ as follows:

$F(\mathbb{Z})=\left\\{\gamma_{a}=\left(\begin{matrix}1&0\\\
a&1\end{matrix}\right)\ \middle|\ a\in\mathbb{Z}\right\\}$

Note that every $\gamma_{a}\in F(\mathbb{Z})$ has determinant $1$.
Furthermore, note that for some monogenic binary cubic form
$f(x,y)=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ and $\gamma_{a}\in
F(\mathbb{Z})$,
$\begin{smallmatrix}(x&y)\end{smallmatrix}\cdot\gamma_{a}=\left(\begin{smallmatrix}x+ay\\\
y\end{smallmatrix}\right)$. Thus, $\gamma_{a}\cdot f(x,y)=f(x+ay,y)$.

######  Lemma 2.5.

Let $f(x,y)$ be a binary cubic form with $x^{3}$-coefficient $1$, and let
$\gamma_{a}\in F(\mathbb{Z})$. Then, the $x^{3}$-coefficient of
$\gamma_{a}\cdot f(x,y)$ is also $1$.

###### Proof.

Let $f(x,y)=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$. In the discussion
following Definition 2.3.3, we show that $\gamma_{a}\cdot f(x,y)=f(x+ay,y)$.
We now compute the following:

$\displaystyle f(x+ay,y)$ $\displaystyle=$
$\displaystyle(x+ay)^{3}+f_{1}(x+ay)^{2}y+f_{2}(x+ay)y^{2}+f_{3}y^{3}$
$\displaystyle=$ $\displaystyle
x^{3}+3x^{2}ay+3xa^{2}y^{2}+a^{3}y^{3}+f_{1}x^{2}y$
$\displaystyle+2f_{1}xay^{2}+f_{1}a^{2}y^{3}+f_{2}xy^{2}+f_{2}ay^{3}+f_{3}y^{3}$
$\displaystyle=$ $\displaystyle
x^{3}+(3a+f_{1})x^{2}y+(3a^{2}+2f_{1}a+f_{2})xy^{2}$
$\displaystyle+(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})y^{3}$

Upon observation, we can see that the $x^{3}$-coefficient of this polynomial
is still $1$. ∎

######  Lemma 2.6.

Let $f(x,y)$ be a binary cubic form with $x^{3}$-coefficient $1$, and let
$\gamma_{a}\in F(\mathbb{Z})$. Then $I(\gamma_{a}\cdot f)=I(f)$ and
$J(\gamma_{a}\cdot f)=J(f)$.

###### Proof.

Let $f(x,y)=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$. In the proof of Lemma
2.5, we show that

$\gamma_{a}\cdot
f=f(x+ay,y)=x^{3}+(3a+f_{1})x^{2}y+(3a^{2}+2f_{1}a+f_{2})xy^{2}+(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})y^{3}$

We can see that the action of $\gamma_{a}$ sends $f_{1}$ to $3a+f_{1}$,
$f_{2}$ to $3a^{2}+2f_{1}a+f_{2}$, and $f_{3}$ to
$a^{3}+f_{1}a^{2}+f_{2}a+f_{3}$. Thus, this action sends
$I(f)=f_{1}^{2}-3f_{2}$ to the following:

$I(\gamma_{a}\cdot
f)=(3a+f_{1})^{2}-3(3a^{2}+2f_{1}a+f_{2})=9a^{2}+6af_{1}+f_{1}^{2}-9a^{2}-6f_{1}a-3f_{2}$

Thus, by cancellation, $I(\gamma_{a}\cdot f)=f_{1}^{2}-3f_{2}$, so
$I(\gamma_{a}\cdot f)=I(f)$.

Now, we turn towards $J(f)=-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}$. We compute the
following:

$\displaystyle J(\gamma_{a}\cdot f)$ $\displaystyle=$
$\displaystyle-2(3a+f_{1})^{3}+9(3a+f_{1})(3a^{2}+2f_{1}a+f_{2})$
$\displaystyle-27(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})$ $\displaystyle=$
$\displaystyle-2(27a^{3}+27a^{2}f_{1}+9af_{1}^{2}+f_{1}^{3})$
$\displaystyle+9(9a^{3}+9a^{2}f_{1}+2af_{1}^{2}+3af_{2}+f_{1}f_{2})$
$\displaystyle-27(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})$ $\displaystyle=$
$\displaystyle-54a^{3}-54a^{2}f_{1}-18af_{1}^{2}-2f_{1}^{3}$
$\displaystyle+81a^{3}+81a^{2}f_{1}+18af_{1}^{2}+27af_{2}+9f_{1}f_{2}$
$\displaystyle+-27a^{3}-27a^{2}f_{1}-27af_{2}-27f_{3}$

Thus, by cancellation, $J(\gamma_{a}\cdot f)=-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}$,
so $J(\gamma_{a}\cdot f)=J(f)$. This concludes the proof. ∎

######  Definition 2.3.4.

We define the height of a monogenized cubic integral domain to be equivalent
to the height of its corresponding orbit of binary cubic forms.

######  Lemma 2.7.

In every $F(\mathbb{Z})$-equivalence class of monogenic binary cubic forms,
there exists exactly one binary cubic form
$f(x,y)=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$ with $f_{1}\in\\{-1,0,1\\}$.
Additionally, $f_{1}\equiv J(f)\ (\mathrm{mod}\ 3)$.

###### Proof.

To prove the first part of the lemma, we will start by demonstrating that the
binary cubic form $f(x,y)$, as described in the lemma, exists in any arbitrary
$F(\mathbb{Z})$-equivalence class. Then, we will show that only one such
binary cubic form exists in this equivalence class.

We have shown in the proof of Lemma 2.5 that under the action of an
$F(\mathbb{Z})$-matrix, the $x^{2}y$-coefficient $f_{1}$ of $f(x,y)$ becomes
$f_{1}+3a$. Now, let $f_{1}^{\prime}\in\\{-1,0,1\\}$ such that
$f_{1}^{\prime}\equiv f_{1}\ (\mathrm{mod}\ 3)$. We want to find an
$a\in\mathbb{Z}$ such that $f_{1}+3a=f_{1}^{\prime}$. We have
$3a=f_{1}^{\prime}-f_{1}$. Thus,

$a=\frac{f_{1}^{\prime}-f_{1}}{3}$

Since $f_{1}^{\prime}\equiv f_{1}\ (\mathrm{mod}\ 3)$ – in other words,
$f_{1}^{\prime}-f_{1}\equiv 0\ (\mathrm{mod}\ 3)$ – $a$ is always an integer.
Hence, we are able to reduce the original binary cubic form to one whose
$x^{2}y$-coefficient is $-1$, $0$, or $1$.

Now that we have shown that such a binary cubic form exists, we will
demonstrate that only one such binary cubic form exists. Let
$f_{2}^{\prime}=3a^{2}+2f_{1}a+f_{2}$ and
$f_{3}^{\prime}=a^{3}+f_{1}a^{2}+f_{2}a+f_{3}$, where
$a=\frac{f_{1}^{\prime}-f_{1}}{3}$ and $f_{1}^{\prime}$ is defined as above.
Let $g(x,y)$ be the “reduced” form of $f(x,y)$ – i.e.
$g(x,y)=x^{3}+f_{1}^{\prime}x^{2}y+f_{2}^{\prime}xy^{2}+f_{3}^{\prime}y^{3}$.

By Lemma 2.6, $I(f)$ and $J(f)$ are invariant in this
$F(\mathbb{Z})$-equivalence class. Thus, we can derive the following
equations:

$\begin{cases}I(f)=f_{1}^{\prime 2}-3f_{2}^{\prime}\implies
f_{2}^{\prime}=\frac{f_{1}^{\prime 2}-I(f)}{3}\\\ J(f)=-2f_{1}^{\prime
3}+9f_{1}^{\prime}f_{2}^{\prime}-27f_{3}^{\prime}\implies
f_{3}^{\prime}=-\frac{2f_{1}^{\prime
3}}{27}+\frac{f_{1}^{\prime}f_{2}^{\prime}}{9}-\frac{J(f)}{27}\end{cases}$

We can see from the above equations that there is only one possible value for
$f_{2}^{\prime}$ and $f_{3}^{\prime}$. Thus, there is exactly one binary cubic
form with $x^{2}y$-coefficient in the set $\\{-1,0,1\\}$ in every
$F(\mathbb{Z})$-equivalence class.

Finally, we show that $f_{1}\equiv J(f)\ (\mathrm{mod}\ 3)$.

$\displaystyle J(f)$ $\displaystyle\equiv$
$\displaystyle-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}$ $\displaystyle\equiv$
$\displaystyle-2f_{1}^{3}$ $\displaystyle\equiv$ $\displaystyle-2f_{1}$
$\displaystyle\equiv$ $\displaystyle f_{1}\ (\mathrm{mod}\ 3)$

∎

As before, we can treat any arbitrary monogenic binary cubic form as being in
an $F(\mathbb{Z})$-equivalence class with a binary cubic form with
$x^{2}y$-coefficient in the set $\\{-1,0,1\\}$. Thus, we can disregard all
binary cubic forms with other $x^{2}y$-coefficients.

######  Lemma 2.8.

Let $f(x,y)$ and $g(x,y)$ be binary cubic forms with $x^{3}$-coefficient $1$.
If $I(f)=I(g)$ and $J(f)=J(g)$, then $f$ and $g$ are
$F(\mathbb{Z})$-equivalent.

###### Proof.

Let $f(x,y)=x^{3}+f_{1}x^{2}y+f_{2}xy^{2}+f_{3}y^{3}$, and let
$g(x,y)=x^{3}+g_{1}x^{2}y+g_{2}xy^{2}+g_{3}y^{3}$. Moreover, suppose that
$I(f)=I(g)$ and $J(f)=J(g)$.

According to Lemma 2.7, $f_{1}\equiv J\ (\mathrm{mod}\ 3)$ and $g_{1}\equiv J\
(\mathrm{mod}\ 3)$ – hence, $f_{1}\equiv g_{1}\ (\mathrm{mod}\ 3)$. We can
pick an $a\in\mathbb{Z}$ such that $g_{1}=3a+f_{1}$.

Since $I(f)=I(g)$, we have

$\displaystyle f_{1}^{2}-3f_{2}$ $\displaystyle=$ $\displaystyle
g_{1}^{2}-3g_{2}$ $\displaystyle=$ $\displaystyle(3a+f_{1})^{2}-3g_{2}$
$\displaystyle=$ $\displaystyle 9a^{2}+6f_{1}a+f_{1}^{2}-3g_{2}$

After cancelling the $f_{1}^{2}$ terms and dividing both sides by $3$, we have

$-f_{2}=3a^{2}+2f_{1}a-g_{2}$

Rearranging, we get

$g_{2}=3a^{2}+2f_{1}a+f_{2}$

Since $J(f)=J(g)$, we have

$\displaystyle-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}$ $\displaystyle=$
$\displaystyle-2(3a+f_{1})^{3}+9(3a+f_{1})(3a^{2}+2f_{1}a+f_{2})-27g_{3}$
$\displaystyle=$ $\displaystyle-2(27a^{3}+27a^{2}f_{1}+9af_{1}^{2}+f_{1}^{3})$
$\displaystyle+9(9a^{3}+9a^{2}f_{1}+2af_{1}^{2}+3af_{2}+f_{1}f_{2})-27g_{3}$
$\displaystyle=$ $\displaystyle-54a^{3}-54a^{2}f_{1}-18af_{1}^{2}-2f_{1}^{3}$
$\displaystyle+81a^{3}+81a^{2}f_{1}+18af_{1}^{2}+27af_{2}+9f_{1}f_{2}-27g_{3}$
$\displaystyle=$ $\displaystyle 27a^{3}+27a^{2}f_{1}+27af_{2}-27g_{3}$

Dividing both sides by $27$ and rearranging, we get

$g_{3}=a^{3}+a^{2}f_{1}+af_{2}+f_{3}$

Combining the expressions that we have obtained for $g_{1}$, $g_{2}$, and
$g_{3}$, we have the following:

$g(x,y)=x^{3}+(3a+f_{1})x^{2}y+(3a^{2}+2f_{1}a+f_{2})xy^{2}+(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})y^{3}$

In the proof of Lemma 2.5, we show that if $\gamma_{a}\in F(\mathbb{Z})$ such
that $\gamma_{a}=\left(\begin{smallmatrix}1&0\\\ a&1\end{smallmatrix}\right)$,
then

$\gamma_{a}\cdot
f(x,y)=x^{3}+(3a+f_{1})x^{2}y+(3a^{2}+2f_{1}a+f_{2})xy^{2}+(a^{3}+f_{1}a^{2}+f_{2}a+f_{3})y^{3}$

Thus, $g(x,y)=\gamma_{a}\cdot f(x,y)$, so $f$ and $g$ are
$F(\mathbb{Z})$-equivalent.

∎

Finally, we state a property of monogenic binary cubic forms that proves to be
crucial in our computational methodology (see Section 3).

######  Theorem 2.9 (Theorem 2.2 of [3]).

There is a natural bijection between $F(\mathbb{Z})$-equivalence classes of
binary cubic forms with $x^{3}$-coefficient $1$ and isomorphism classes of
monogenized cubic integral domains.

###  2.4 Average $2$-Torsion Sizes of Class Groups of Cubic Fields

Before stating Theorem 1.2 of [3], we define the following:

######  Definition 2.4.1.

Given a finite abelian group $G$ and a prime number $p$, the $p$-torsion
subgroup of G, denoted as $G[p]$, is defined as follows:

$G[p]:=\\{g\in G\ |\ g^{p}=1\\}$

Moreover, the p-torsion size of $G$ is defined to be the cardinality of the
$p$-torsion subgroup of $G$.

A fact about $p$-torsion subgroups, which follows from Lagrange’s Theorem,
proves to be useful in our later discussion of the computational methodology:

######  Lemma 2.10.

Let $G$ be a finite abelian group. For a prime number $p$,

$G[p]=\\{g\in G\ |\ g^{p}=g_{0}^{|G|}\quad\forall g_{0}\in G\\}$

Now we have enough theoretical background to state the theorems in question
from [2] and [3]. First, the theorem from [2]:

######  Theorem 2.11 (Theorem 5 of [2]).

Let us denote the $2$-torsion subgroup of the class group of a cubic field $K$
as $Cl(K)[2]$. Then,

  * •

The average size, when ordering by discriminant, of $Cl(K)[2]$ for cubic
fields $K$ of positive discriminant is $5/4$.

  * •

The average size, when ordering by discriminant, of $Cl(K)[2]$ for cubic
fields $K$ of negative discriminant is $3/2$.

Note that in [14], the authors prove the same mean values, but when ordering
by height.

Now, the theorem from [3]:

######  Theorem 2.12 (Theorem 1.2 of [3]).

Again, let us denote the $2$-torsion subgroup of the class group of a cubic
field $K$ as $Cl(K)[2]$. Then,

  * •

The average size, when ordering by height, of $Cl(K)[2]$ for cubic fields $K$
having monogenized rings of integers with positive discriminant is 3/2.

  * •

The average size, when ordering by height, of $Cl(K)[2]$ for cubic fields $K$
having monogenized rings of integers of negative discriminant is 2.

This is the result that we provide computational evidence for in this paper.

It is surprising that the average $2$-torsion sizes from Theorem 2.11 change
when we mandate that the cubic fields in question have monogenized rings of
integers. Based on local behavior, we would expect these average $2$-torsion
sizes not to change under the added restriction.

Before we continue, we introduce some notation that can be used to express the
above theorem:

######  Definition 2.4.2.

For some $Y\in\mathbb{Z}$, $Y\geq 0$, let $\mathcal{F}^{+}_{Y}$ denote the set
of positive-discriminant, maximal, and monogenized cubic integral domains
$(R,\alpha)$ with height less than $Y$. Similarly, let $\mathcal{F}^{-}_{Y}$
denote the set of negative-discriminant, maximal, and monogenized cubic
integral domains $(R,\alpha)$ with height less than $Y$.

Let $K$ denote the fraction field $\textrm{Frac}(R)$. Then, for $p$ prime, we
define the following:

$\mu_{p}(\mathcal{F}^{+}(Y)):=\frac{\sum\limits_{(R,\alpha)\in\mathcal{F}^{+}_{Y}}|Cl(K)[p]|}{|\mathcal{F}^{+}_{Y}|}$
(6)

$\mu_{p}(\mathcal{F}^{-}(Y)):=\frac{\sum\limits_{(R,\alpha)\in\mathcal{F}^{-}_{Y}}|Cl(K)[p]|}{|\mathcal{F}^{-}_{Y}|}$
(7)

With this new notation, Theorem 2.12 can be rewritten as

$\lim_{Y\rightarrow\infty}\mu_{2}(\mathcal{F}^{+}(Y))=3/2$

$\lim_{Y\rightarrow\infty}\mu_{2}(\mathcal{F}^{-}(Y))=2$

##  3 Computational Methodology

Our computational methodology for verifying Theorem 2.12 is summarized below.
Our goal is to compute the averages $\mu_{2}(\mathcal{F}^{+}(Y))$ and
$\mu_{2}(\mathcal{F}^{-}(Y))$ for very large values of $Y$.

  * •

Step 1: Generate all monogenic binary cubic forms of bounded height and
positive/negative discriminant

We can assume that any monogenic binary cubic form $f(x,y)$ in our computation
has $f_{0}=1$ by Lemma 2.4. Moreover, by Lemma 2.7, we can assume that
$f_{1}\in\\{-1,0,1\\}$ and $f_{1}\equiv J(f)\ (\mathrm{mod}\ 3)$. By looping
through all of the possible values of $I$ and $J$, we can see that every
$(I,J)$ pair corresponds to a single $(f_{0},f_{1})$ pair (by Lemma 2.8). This
approach requires us to impose bounds on $I$ and $J$ so that the number of
binary cubic forms we generate remains finite. Upon calculation, we find that
given a positive integer $Y$, for all binary cubic forms $f(x,y)$ such that
$H(f)<Y$, $|I|<\sqrt[3]{Y}$ and $|J|<2\sqrt{Y}$.

We now construct the following nested loop: We loop from
$I=-\lfloor{\sqrt[3]{Y}\rfloor}$ to $I=\lfloor{\sqrt[3]{Y}\rfloor}$, and for
each individual value of $I$ in this loop, we loop from
$J=-2\lfloor{\sqrt{Y}\rfloor}$ to $J=2\lfloor{\sqrt{Y}\rfloor}$. For any
$(I,J)$ pair, we will always have $f_{0}=1$ and $f_{1}=-1$, $0$, or $1$
depending on the value of $J$. According to Definition 2.3.2, we also have

$\begin{cases}I=f_{1}^{2}-3f_{2}\implies f_{2}=\frac{f_{1}^{2}-I}{3}\\\
J=-2f_{1}^{3}+9f_{1}f_{2}-27f_{3}\implies
f_{3}=-\frac{2f_{1}^{3}}{27}+\frac{f_{1}f_{2}}{9}-\frac{J}{27}\end{cases}$

If $f_{2},f_{3}$ are integers, then we have generated a valid binary cubic
form; otherwise, the $(I,J)$ pair does not have a corresponding binary cubic
form.

  * •

Step 2: Calculate the fraction field $K$ of the monogenized cubic integral
domain corresponding to each monogenic binary cubic form: Before starting this
step, note that we can only construct a correspondence between monogenized
cubic integral domains and binary cubic forms with $x^{3}$-coefficient $1$ and
$x^{2}y$-coefficient in the set $\\{-1,0,1\\}$ because of Theorem 2.9,
together with Remark 2.1.

We now have a large list of binary cubic forms that we must loop through.
Given a specific binary cubic form within our loop, we can find the minimal
polynomial for the corresponding fraction field using Lemma 2.3. But since the
binary cubic forms we are concerned with are all monogenic, by Lemma 2.4, we
can simplify the minimal polynomial for the fraction field to the following:

$g(x)=x^{3}+f_{1}x^{2}+f_{2}x+f_{3}$

We can now compute the fraction field corresponding to the given binary cubic
form. (Note that if the above polynomial is reducible, the fraction field is
no longer a field – thus, the corresponding binary cubic form will be ejected
from the computation.)

  * •

Step 3: Ensure that each binary cubic form is maximal: Within our large binary
cubic form loop described in Step 2, we must ensure that every binary cubic
form’s corresponding cubic integral domain is maximal inside its fraction
field. In order to do so, we must compute the discriminant of each binary
cubic form and its corresponding fraction field, and then ensure that the two
discriminants are equal. This will imply that the cubic integral domain
corresponding to each binary cubic form is isomorphic to the ring of integers
inside the binary cubic form’s fraction field, implying that the cubic
integral domain is indeed maximal. Any binary cubic form whose cubic integral
domain is not maximal will be ejected from the computation – otherwise, it
will remain until the end of the computation.

  * •

Step 4: Find the class group of each fraction field $K$ resulting from Step 2:
We do this using SageMath [18].

  * •

Step 5: Calculate the number of 2-torsion elements in class group of each
fraction field $K$: By Lemma 2.10, we simply have to find all the elements of

$Cl(K)[2]=\\{g\in Cl(K)\ |\ g^{2}=e=g_{0}^{|Cl(K)|}\\}$

where $e$ is the multiplicative identity of $Cl(K)$ and $g_{0}$ is an
arbitrary element of $Cl(K)$. We perform this computation for every surviving
fraction field $K$ in the large binary cubic form loop described in Step 2.

  * •

Step 6: Compute averages $\mu_{2}(\mathcal{F}^{+}(Y))$ and
$\mu_{2}(\mathcal{F}^{-}(Y))$

Our ultimate goal now is to increase the height restriction $Y$ from Step 1 to
a large enough number so that we can predict the values of
$\mu_{2}(\mathcal{F}^{+}(Y))$ and $\mu_{2}(\mathcal{F}^{-}(Y))$. We were able
to reach a $Y$-value of 100 billion for positive-discriminant monogenic binary
cubic forms, and 10 billion for negative-discriminant monogenic binary cubic
forms. (The limiting factor here was computation time – our longest run took
about a week to complete.)

##  4 Computational Data Towards Theorem 2.12

In this section, we present our computational data on the average $2$-torsion
size of class groups of monogenized cubic fields, ordered by height. First, we
will present two propositions from [3] and compare them with computational
data.

###  4.1 Asymptotics on Binary Cubic Form Counts

The following proposition is equivalent to Proposition 3.2 of [3].

######  Proposition 4.1.

Let the number of $F(\mathbb{Z})$-equivalence classes of positive-discriminant
monogenic binary cubic forms with height less than $Y$ be $N^{+}(Y)$, and let
the number of $F(\mathbb{Z})$-equivalence classes of negative-discriminant
monogenic binary cubic forms with height less than $Y$ be $N^{-}(Y)$. Then,

$N^{+}(Y)=\frac{8}{135}Y^{5/6}+O(Y^{1/2+\epsilon})$
$N^{-}(Y)=\frac{32}{135}Y^{5/6}+O(Y^{1/2+\epsilon})$

Figure 1 below shows $N^{+}(Y)$ in a dotted blue line (computed by our
program) and $\frac{8}{135}Y^{5/6}$ in a solid red line. Figure 2 shows
$N^{-}(Y)$ in a dotted blue line (also computed by our program) and
$\frac{32}{135}Y^{5/6}$ in a solid red line. In Figure 1, $Y$ goes up to
$10^{11}$; in Figure 2, $Y$ goes up to $10^{10}$.

$1$$2\cdot 10^{10}$$4\cdot 10^{10}$$6\cdot 10^{10}$$8\cdot 10^{10}$$1\cdot
10^{11}$$0$$2\cdot 10^{7}$$4\cdot 10^{7}$$6\cdot 10^{7}$$8\cdot
10^{7}$$Y$$N^{+}(Y)$$\frac{8}{135}Y^{5/6}$ Figure 1: The number of positive-
discriminant monogenic binary cubic forms bounded by height $Y$, compared to
the equation $\frac{8}{135}Y^{5/6}$ $1$$2\cdot 10^{9}$$4\cdot 10^{9}$$6\cdot
10^{9}$$8\cdot 10^{9}$$1\cdot 10^{10}$$0$$2\cdot 10^{7}$$4\cdot
10^{7}$$Y$$N^{-}(Y)$$\frac{32}{135}Y^{5/6}$ Figure 2: The number of negative-
discriminant monogenic binary cubic forms bounded by height $Y$, compared to
the equation $\frac{32}{135}Y^{5/6}$

Note that in both Figure 1 and Figure 2, from a visual standpoint, the
theoretical and computational results are in agreement. Below is a more
detailed error analysis.

For $Y=10^{11}$, our program calculated that $N^{+}(Y)=86,961,377$. The
expected theoretical value of $N^{+}(Y)$ is
$\frac{8}{135}(10^{11})^{5/6}=86,980,697.341$, implying our calculated value
has an error of $0.022\%$.

For $Y=10^{10}$, our program calcualted that $N^{-}(Y)=51,074,450$. The
expected theoretical value of $N^{-}(Y)$ is
$\frac{32}{135}(10^{10})^{5/6}=51,068,081.542$, implying our calculated value
has an error of $0.012\%$.

From these low error percentages, it is clear that our computational results
quickly agree with the corresponding theoretical results.

###  4.2 Asymptotics on Binary Cubic Form Maximality Ratios

The following proposition can be deduced from Proposition 4.1 above, in
conjunction with Theorem 3.6 of [3].

######  Proposition 4.2.

Let $N_{max}^{+}(Y)$ be the number of $F(\mathbb{Z})$-equivalence classes of
maximal positive-discriminant binary cubic forms with height less than $Y$,
and let $N_{max}^{-}(Y)$ be the number of $F(\mathbb{Z})$-equivalence classes
of maximal negative-discriminant binary cubic forms with height less than $Y$.
In addition, let $N^{+}(Y)$, $N^{-}(Y)$ be as defined in Proposition 4.1.
Then,

$\frac{N_{max}^{+}(Y)}{N^{+}(Y)}=\frac{N_{max}^{-}(Y)}{N^{-}(Y)}=\frac{1}{\zeta(2)}$

Figure 3 below demonstrates the convergence of both
$\frac{N_{max}^{+}(Y)}{N^{+}(Y)}$ and $\frac{N_{max}^{-}(Y)}{N^{-}(Y)}$ to
$\frac{1}{\zeta(2)}$.

$1$$2\cdot 10^{6}$$4\cdot 10^{6}$$6\cdot 10^{6}$$8\cdot 10^{6}$$1\cdot
10^{7}$$0.5$$0.55$$0.6$$0.65$$Y$$N_{max}^{+}(Y)/N^{+}(Y)$$N_{max}^{-}(Y)/N^{-}(Y)$$1/\zeta(2)$
Figure 3: The fractions of positive-discriminant and negative-discriminant
monogenic binary cubic forms that are maximal, compared with
$\frac{1}{\zeta(2)}$

In Figure 3, as early as $Y=2\cdot 10^{6}$ , we have

$\dfrac{N_{max}^{+}(Y)}{N^{+}(Y)}=\dfrac{6318}{10405}=0.607$

$\dfrac{N_{max}^{-}(Y)}{N^{-}(Y)}=\dfrac{25662}{42143}=0.609$

Both of these values are within $0.2$ percent of $\frac{1}{\zeta(2)}$.

In order to ensure that a maximality constraint on our binary cubic forms
would not introduce too much additional error to our computation, let us
perform an error analysis similar to the one we conducted at the end of
Section 4.1, with the requirement that our binary cubic forms be maximal.

For $Y=2\cdot 10^{6}$, our program calculated that $N_{max}^{+}(Y)=6,318$. The
expected theoretical value of $N_{max}^{+}(Y)$ is
$\frac{8}{135\zeta(2)}(2\cdot 10^{6})^{5/6}=6,418.980$, implying our
calculated value has an error of $1.573\%$.

For the same value of $Y$, our program calculated that
$N_{max}^{-}(Y)=25,662$. The expected theoretical value of $N_{max}^{-}(Y)$ is
$\frac{32}{135\zeta(2)}(2\cdot 10^{6})^{5/6}=25,675.922$, implying our
calculated value has an error of $0.043\%$.

Since these error percentages are once again low, we are assured that our
computational results agree with the corresponding theoretical predictions.

###  4.3 Asymptotic Averages of Binary Cubic Forms

We are now ready to give evidence towards Theorem 2.12. The computational
verification process relied heavily on Eureqa, a computer application that
employs genetic programming [15] to find regression curves that optimally fit
the data given. See section 5.1 for more details on how Eureqa works.

Figures 4 and 5 below display $\mu_{2}(\mathcal{F}^{+}(Y))$ and
$\mu_{2}(\mathcal{F}^{-}(Y))$ with respect to $Y$:

$1$$2\cdot 10^{10}$$4\cdot 10^{10}$$6\cdot 10^{10}$$8\cdot 10^{10}$$1\cdot
10^{11}$$1$$1.2$$1.4$$Y=787,702,088$$Y$$\mu_{2}(\mathcal{F}^{+}(Y))$ Figure 4:
$\mu_{2}(\mathcal{F}^{+}(Y))$, compared to the general binary cubic form
asymptote in [2] (in black), as well as the monogenic binary cubic form
asymptote predicted in [3] (in red) $1$$2\cdot 10^{9}$$4\cdot 10^{9}$$6\cdot
10^{9}$$8\cdot 10^{9}$$1\cdot
10^{10}$$1$$1.2$$1.4$$1.6$$1.8$$2$$Y=17,382,351$$Y$$\mu_{2}(\mathcal{F}^{-}(Y))$
Figure 5: $\mu_{2}(\mathcal{F}^{-}(Y))$, compared to the general binary cubic
form asymptote in [2] (in black), as well as the monogenic binary cubic form
asymptote predicted in [3] (in red)

Note that upon examination of the above two graphs, we can instantly determine
that $\lim_{Y\rightarrow\infty}\mu_{2}(\mathcal{F}^{+}(Y))$ and
$\lim_{Y\rightarrow\infty}\mu_{2}(\mathcal{F}^{-}(Y))$ are different from the
asymptotic averages in Theorem 2.11 (where no monogenicity condition is
imposed). This is undoubtedly a surprising result.

After we inserted all of the data comprising the above two graphs into a
spreadsheet, Eureqa formulated the following pair of equations nearly
instantaneously:

$\begin{cases}\mu_{2}(\mathcal{F}^{+}(Y))\approx 1.51-1.34Y^{-0.0810}\\\
\mu_{2}(\mathcal{F}^{-}(Y))\approx 1.99-2.05Y^{-0.0878}\end{cases}$

These equations have the exact form we are looking for: asymptotes of
approximately $3/2$ and $2$, respectively, followed by second-order terms that
vanish to zero as $Y$ approaches infinity. It is also worth noting that, when
compared against the data, both of the above equations have mean-squared error
(MSE) values below $10^{-7}$.

It is evident now that this model independently relates the computational data
to the asymptotics given in Theorem 2.12. The next step, as mentioned
previously, is to extend our approach for computing class group $2$-torsion
sizes to computing class group $p$-torsion sizes, where $p$ is a prime
integer. This extension is detailed in the next section, along with our main
results.

##  5 Main Results

###  5.1 Introduction to Genetic Programming

As mentioned in Section 4.3, Eureqa [19] is a computer application that uses
genetic programming [15] to find regression curves that optimally fit the data
given. Here we will discuss what genetic programming is, and why it is useful
for solving regression problems like the ones we are working with.

Genetic programming (GP) is an artificial intelligence-based problem-solving
technique. Broadly speaking, if a GP-based algorithm is given a problem to
solve, it will start by developing randomly generated solutions that often do
not solve the problem well. It will then conduct two types of operations,
namely “reproduction” operations and “mutation” operations, to try to modify
the original solutions and develop new ones. A “reproduction” operation will
take two proposed solutions and combine them together somehow to form a new
solution. A “mutation” operation will take a proposed solution and change it
in some way.333It may be obvious by now that the name “genetic programming,”
as well as the names of the two types of operations involved in GP, are
derived from Darwin’s theory of evolution.

Through combinations of these two types of operations, the algorithm will
improve the original randomly-generated solutions and develop new solutions
that solve the problem better. We are able to discern whether or not a
proposed solution solves the given problem well based on some evaluation
metric (e.g. mean-squared error) – this metric is usually referred too as a
loss function. Generally, a solution with a lower loss (i.e. a lower loss-
function value) is a better solution.

In our case, we are using a GP-based algorithm – i.e. Eureqa – to solve a
regression problem. We start with a set of random solutions, and develop
better solutions by conducting “reproduction” and “mutation” operations on the
original solutions. We determine which solutions are better by looking at
their mean-squared errors with respect to the original data (i.e. the
$\mu_{p}(\mathcal{F}^{\pm}(Y))$ values that we provide Eureqa, for fixed $p$
and increasing $Y$). It is worth noting, however, that we often do not choose
the solutions with the best mean-squared errors, because these solutions are
often so complex that they over-fit the given data. In other words, they have
poor predictive power for high $Y$-values that are not supplied to Eureqa due
to computational limitations. Thus, we tend to look for solutions with
sufficiently low mean-squared errors whose forms are as simple as possible.
These solutions often take the same form as the
$\mu_{2}(\mathcal{F}^{\pm}(Y))$ equations stated at the end of Section 4.3:

$\mu_{p}(\mathcal{F}^{\pm}(Y))\approx\alpha^{\pm}-\beta^{\pm}Y^{\gamma^{\pm}}$

where $\alpha^{\pm},\beta^{\pm},\gamma^{\pm}\in\mathbb{R}$.

Note that GP-based algorithms are often highly probabilistic, so running the
same program several times in a row may yield multiple different answers. When
comparing the results presented at the end of Section 5.3 with the conjectures
stated in Section 6, bear in mind that any differences between computed and
conjectured values of $\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{\pm}(Y))$
can be at least partially attributed to the noise inherent to Eureqa’s
training algorithm.

###  5.2 Eureqa Methodology

In principle, the approach for calculating class group $p$-torsion sizes is
highly similar to the approach for calculating class group $2$-torsion sizes.
The only theoretical difference lies in Step 5 of the methodology detailed in
Section 3. By Lemma 2.10, all we have to do is find all the elements of
$Cl(K)[p]$.s

However, due to Proposition 4.1, when counting monogenic binary cubic forms
bounded by height $Y$, there are far more negative-discriminant binary cubic
forms than positive-discriminant binary cubic forms. Thus, it is easier to
push the positive-discriminant height bound to a high value than the negative-
discriminant height bound. In our computation, we pushed the positive-
discriminant height bound to $10^{11}$, and we pushed the negative-
discriminant height bound to $10^{10}$.

As we saw in Section 4.3, the formulas Eureqa discovered were of the form

$\mu_{p}(\mathcal{F}^{\pm}(Y))\approx\alpha_{i}^{\pm}-\beta_{i}^{\pm}Y^{\gamma_{i}^{\pm}}$

where $\alpha_{i}^{\pm},\beta_{i}^{\pm},\gamma_{i}^{\pm}\in\mathbb{R}$. It is
safe to assume that $\gamma_{i}^{+}=\gamma_{i}^{-}$ [17]. Thus, since we have
a higher height bound for positive-discriminant binary cubic forms than for
negative-discriminant binary cubic forms, we can give $\gamma_{i}^{+}$ to the
negative-discriminant Eureqa model as a fixed constant. In other words, we can
set $\gamma_{i}^{-}$ to always be equal to $\gamma_{i}^{+}$.

With this in mind, we can develop a new Eureqa strategy. Let $p$ be a prime
number. First, we train the positive-discriminant model as usual. This yields
our formula for $\mu_{p}(\mathcal{F}^{+}(Y))$, which has the form
$\alpha_{i}^{+}-\beta_{i}^{+}Y^{\gamma_{i}^{+}}$. We then train a second
positive discriminant model where Eureqa is told to use the form
$\alpha_{f}^{+}-\beta_{f}^{+}Y^{\gamma_{f}^{+}}$. In addition, the
$\alpha_{f}^{+}$ value – i.e. the asymptote – is fixed. This way, we can
eliminate some of the unwanted noise inherent to Eureqa’s training algorithm
and focus on finding a more accurate value of the exponent $\gamma_{f}^{+}$.
(This is discussed in Section 5.1 above.) We refer to this process as “fine-
tuning” the exponent.

After all of this is finished, we begin training the negative model. We
mandate that Eureqa uses the form
$\alpha_{f}^{-}-\beta_{f}^{-}Y^{\gamma_{f}^{-}}$, and we set $\gamma_{f}^{-}$
to be equal to $\gamma_{f}$, i.e. the fine-tuned version of $\gamma_{i}^{+}$.
After Eureqa discovers $\alpha_{f}^{-}$ and $\beta_{f}^{-}$ for us, we have
our formula for $\mu_{p}(\mathcal{F}^{-}(Y))$.444Note that we use the
$\gamma_{i}^{+}$ exponents to state our $\mu_{p}(\mathcal{F}^{+}(Y))$
formulas, and we use the $\gamma_{f}^{-}$ exponents to state our
$\mu_{p}(\mathcal{F}^{-}(Y))$ formulas. Thus, for a given $p$, the exponent in
the $\mu_{p}(\mathcal{F}^{+}(Y))$ expression will not necessarily be the same
as the exponent in the $\mu_{p}(\mathcal{F}^{-}(Y))$ expression.

###  5.3 Results

Below are the graphs that we obtained for the average class group $p$-torsion
sizes – the first graph is for positive-discriminant binary cubic forms, and
the second is for negative-discriminant binary cubic forms.

$1$$2\cdot 10^{10}$$4\cdot 10^{10}$$6\cdot 10^{10}$$8\cdot 10^{10}$$1\cdot
10^{11}$$1$$1.1$$1.2$$1.3$$Y$$p$-Torsion Size Figure 6:
$\mu_{p}(\mathcal{F}^{+}(Y))$ for
$p={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3},5,{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}7},$
and ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}11}$
$1$$2\cdot 10^{9}$$4\cdot 10^{9}$$6\cdot 10^{9}$$8\cdot 10^{9}$$1\cdot
10^{10}$$1$$1.2$$1.4$$1.6$$Y$$p$-Torsion Size Figure 7:
$\mu_{p}(\mathcal{F}^{-}(Y))$ for
$p={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3},5,{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}7},$
and ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}11}$.

Note that at $Y=10^{11}$,

$\begin{cases}\mu_{2}(\mathcal{F}^{+}(10^{11}))=1.333\\\
\mu_{3}(\mathcal{F}^{+}(10^{11}))=1.259\\\
\mu_{5}(\mathcal{F}^{+}(10^{11}))=1.039\\\
\mu_{7}(\mathcal{F}^{+}(10^{11}))=1.020\\\
\mu_{11}(\mathcal{F}^{+}(10^{11}))=1.008\end{cases}$

Also note that at $Y=10^{10}$,

$\begin{cases}\mu_{2}(\mathcal{F}^{-}(10^{10}))=1.714\\\
\mu_{3}(\mathcal{F}^{-}(10^{10}))=1.645\\\
\mu_{5}(\mathcal{F}^{-}(10^{10}))=1.203\\\
\mu_{7}(\mathcal{F}^{-}(10^{10}))=1.142\\\
\mu_{11}(\mathcal{F}^{-}(10^{10}))=1.090\end{cases}$

Using the new Eureqa strategy described in Section 5.2, we obtained the
following generalized asymptotic average predictions. Note that we used mean-
squared error (MSE) as our error metric, since it is commonly used in the
machine learning community.

Positive Discriminants:

$\begin{cases}\mu_{2}(\mathcal{F}^{+}(Y))\approx
1.51-1.34Y^{-0.0810}\longrightarrow\text{MSE}=5.620\times 10^{-9}\\\
\mu_{3}(\mathcal{F}^{+}(Y))\approx 1.30-1.72Y^{-0.143}\
\longrightarrow\text{MSE}=1.106\times 10^{-7}\\\
\mu_{5}(\mathcal{F}^{+}(Y))\approx 1.04-0.595Y^{-0.196}\
\longrightarrow\text{MSE}=8.359\times 10^{-9}\\\
\mu_{7}(\mathcal{F}^{+}(Y))\approx 1.02-0.248Y^{-0.171}\
\longrightarrow\text{MSE}=7.595\times 10^{-9}\\\
\mu_{11}(\mathcal{F}^{+}(Y))\approx 1.01-0.548Y^{-0.261}\
\longrightarrow\text{MSE}=3.108\times 10^{-9}\end{cases}$

Negative Discriminants555Note that the formula for
$\mu_{2}(\mathcal{F}^{-}(Y))$ is different from the one in Section 4.3 – this
is because we recalculated it using the new Eureqa strategy from Section 5.2.
:

$\begin{cases}\mu_{2}(\mathcal{F}^{-}(Y))\approx 2.00-1.95Y^{-0.0832}\
\longrightarrow\text{MSE}=1.746\times 10^{-7}\\\
\mu_{3}(\mathcal{F}^{-}(Y))\approx 1.45+0.02Y^{0.0939}\
\longrightarrow\text{MSE}=5.936\times 10^{-5}\\\
\mu_{5}(\mathcal{F}^{-}(Y))\approx 1.24-0.254Y^{-0.0872}\
\longrightarrow\text{MSE}=2.349\times 10^{-6}\\\
\mu_{7}(\mathcal{F}^{-}(Y))\approx 1.16-0.485Y^{-0.145}\
\longrightarrow\text{MSE}=2.995\times 10^{-6}\\\
\mu_{11}(\mathcal{F}^{-}(Y))\approx 1.10-0.643Y^{-0.178}\
\longrightarrow\text{MSE}=1.138\times 10^{-6}\end{cases}$

##  6 Conjectures on Class Group $p$-Torsion Sizes

Based on the set of Eureqa-predicted asymptotes from Section 5, we can now
formulate conjectures on the general form of
$\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{+}(Y))$ and
$\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{-}(Y))$ for $p$ prime, $p\neq
3$.666Theoretical results indicate that we should not include $p=3$ in the
statement of our conjecture [6]. The following conjectures were also predicted
by Eureqa.

######  Conjecture 6.1.

Let $p$ be a prime number such that $p\neq 3$. Then,

$\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{+}(Y))=1+\dfrac{1}{p(p-1)}$

######  Conjecture 6.2.

Let $p$ be a prime number such that $p\neq 3$. Then,

$\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{-}(Y))=1+\dfrac{1}{p-1}$

These conjectures predict most of the asymptotes in Section 5 with minimal
error. The one pair of exceptions is
$\lim_{Y\rightarrow\infty}\mu_{3}(\mathcal{F}^{\pm}(Y))$ – however, note that
these conjectures are not expected to hold for $p=3$ [6].

######  Remark 6.3.

It is worth noting that according to conjectures 6.1 and 6.2,

$\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{-}(Y))-\lim_{Y\rightarrow\infty}\mu_{p}(\mathcal{F}^{+}(Y))=\left(1+\dfrac{1}{p-1}\right)-\left(1+\dfrac{1}{p(p-1)}\right)=\dfrac{1}{p}$

##  7 Conclusions

Conjectures 6.1 and 6.2 are the main contributions of this paper. They are the
culminating result of our overall methodology – computationally rediscovering
the theoretical result in [3] on the average $2$-torsion size of class groups
of cubic fields, and then extending this result from average $2$-torsion sizes
to average $p$-torsion sizes, for all prime $p$. It is worth noting, however,
that these conjectures were made solely based on computational evidence. We
hope that future work in this field presents theoretical evidence for these
conjectures and provides deeper insight into what these conjectures truly
mean.

## Acknowledgements

Many thanks to Prof. Ila Varma (University of Toronto), whose guidance
throughout the course of this project was invaluable. Also, thanks to Prof.
Jon Hanke (Princeton University) and Dr. Dylan Yott (UC Berkeley) for their
mentorship during the early stages of this project, and to Prof. Arul Shankar
(University of Toronto) for his advice later in the project. Finally, thanks
to Hari Pingali and Stephen New, whose contributions during the early stages
of the project helped shape its future.

## References

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# Splitting Receiver with Joint Envelope and Coherent Detection

Yanyan Wang, _Member, IEEE_ , Wanchun Liu, _Member, IEEE_ , Xiangyun Zhou,
_Senior Member, IEEE_

###### Abstract

†† Y. Wang is with School of Information Science and Technology, Southwest
Jiaotong University, Chengdu, China (email: yanyanwang@swjtu.edu.cn). W. Liu
is with School of Electrical and Information Engineering, The University of
Sydney, Australia (email: wanchun.liu@sydney.edu.au). X. Zhou is with School
of Engineering, The Australian National University, Australia (email:
xiangyun.zhou@anu.edu.au).

This letter proposes a new splitting receiver design with joint envelope
detection (ED) and coherent detection (CD). To characterize its fundamental
performance limit, we conduct high signal-to-noise ratio (SNR) analysis on the
proposed ED-CD splitting receiver and obtain closed-form approximations of
both the achievable mutual information and the optimal splitting ratio (i.e.,
a key design parameter of the receiver). Our numerical results show that these
high SNR approximations are accurate over a wide range of moderate SNR values,
signifying the usefulness of the obtained analytical results. We also provide
insights on the conditions at which the proposed splitting receiver has
significant performance advantages over the traditional receivers.

###### Index Terms:

Splitting receiver, wireless receiver, coherent detection, envelope detection.

##  I Introduction

Receiver design is an essential element in the evolution of wireless
communication systems. Coherent receivers are the most commonly adopted design
in cellular communications and local area networks. Such receivers perform
coherent detection (CD) by extracting both the in-phase and quadrature
components of the input signal. Recently, non-coherent receivers have been
extensively utilized in the internet of things (IoT) and future communication
systems due to its potential for enabling low-cost, low-power and
straightforward implementation [1, 2].

Roughly speaking, there are two types of non-coherent receivers used in
communication systems: envelope detection (ED) receiver and power detection
(PD) receiver. ED is the most commonly used non-coherent detector in radio-
frequency (RF) communications [3, 4]. The widely seen ED circuits in practice
use either a half-wave or full-wave rectifier followed by a low-pass filter,
the output signal of which captures the envelope of the input RF signal [5].
The PD receiver is more commonly adopted in optical communications [6, 7, 8].
Note that while PD can still be used in RF communications, it commonly
requires active components. In contrast, ED circuits are usually passive and
hence have much lower cost and energy consumption.

Different from the traditional coherent and non-coherent receiver designs
mentioned above, a fundamentally new receiver architecture was recently
proposed in [9], [10], named as splitting receiver. This receiver architecture
jointly uses both coherent and non-coherent processing for signal detection.
In the original design of [10], PD is adopted for non-coherent processing,
resulting in the PD-CD splitting receiver. The work in [10] showed that the
PD-CD splitting receiver can achieve substantial performance improvement over
the traditional receivers. This was an exciting result demonstrating the huge
potential of the splitting receiver architecture. In the context of RF
wireless communication systems, ED is a much more suitable detection method
and has a much lower complexity than PD. This motivates us to propose the ED-
CD splitting receiver, which gives a more practical implementation option of
splitting receiver for RF communications, and investigate if the new splitting
receiver can still provide performance gain compared to conventional non-
splitting receivers.

In this letter, we propose the new ED-CD splitting receiver and establish its
performance limit through analyzing the achievable mutual information.
Specifically, we conduct a high signal-to-noise ratio (SNR) analysis and
derive an approximation of the achievable mutual information. This
approximation also enables us to analytically obtain the optimal design of the
splitting ratio which is the key design parameter of the splitting receiver.
Our numerical results verify the accuracy of the analytical approximation even
at moderately low SNR. This also implies that the optimal splitting ratio
analytically obtained at high SNR can directly be used at moderate SNR for
achieving best data rate performance. The numerical results also identify the
conditions at which the ED-CD splitting receiver gives significant performance
advantages over the traditional receivers that only use either ED or CD for
signal detection.

##  II The Proposed ED-CD Splitting receiver

The proposed ED-CD splitting receiver design is presented in Fig. 1. The RF
transmitted signal propagates through a channel with coefficient $\tilde{h}$.
In this work, we assume that $\tilde{h}$ is perfectly known by the receiver.
The received RF signal is divided into two streams with a power splitting
ratio $\rho$, where $\rho\in[0,1]$. One stream goes through the CD processing,
while the other stream goes through the ED processing.

The baseband digitized signals after the CD and ED processing are respectively
given by

${{{\tilde{Y}}}^{{}^{\prime}}_{1}}=\sqrt{\rho}\,(\sqrt{P}\tilde{h}{{\tilde{X}}}+{{\tilde{W}^{{}^{\prime}}}})+{{\tilde{Z}^{{}^{\prime}}}},$
(1)
${{{Y}}_{2}}=\sqrt{1-\rho}\,\big{|}\sqrt{P}\tilde{h}{{\tilde{X}}}+{{\tilde{W}^{{}^{\prime}}}}\big{|}+{{N}},$
(2)

where ${{\tilde{X}}}$ is the transmitted signal with normalized variance, $P$
is the average transmit power of the signal, ${\tilde{W}^{{}^{\prime}}}$
denotes the antenna noise, and the processing noises introduced in the CD and
ED streams are denoted by $\tilde{Z}^{{}^{\prime}}$ and ${N}$ respectively.
Note that ${{\tilde{Y}}}^{{}^{\prime}}_{1}$, $\tilde{h}$, $\tilde{X}$,
${\tilde{W}^{{}^{\prime}}}$ and $\tilde{Z}^{{}^{\prime}}$ are complex-valued,
while $Y_{2}$ and $N$ are real-valued.

For the convenience of the later analysis, we rewrite the channel coefficient
as $\tilde{h}\triangleq|\tilde{h}|e^{j\phi}$ and
${{{Y}}_{2}}\triangleq\sqrt{1-\rho}\big{|}\sqrt{P}{|\tilde{h}|e^{j\phi}}{{\tilde{X}}}+e^{j\phi}{{\tilde{W}}}\big{|}+{{N}}$,
and define ${{{\tilde{Y}}}_{1}}\triangleq
e^{-j\phi}{{{\tilde{Y}}}^{{}^{\prime}}_{1}}$, ${\tilde{W}}\triangleq
e^{-j\phi}{{\tilde{W}}^{{}^{\prime}}}$ and $\tilde{Z}\triangleq
e^{-j\phi}{{\tilde{Z}^{{}^{\prime}}}}$. The equivalent baseband signals are
given by

${{{\tilde{Y}}}_{1}}=\sqrt{\rho}\,(\sqrt{P}|\tilde{h}|{{\tilde{X}}}+{{\tilde{W}}})+{{\tilde{Z}}},$
(3)
${{{Y}}_{2}}=\sqrt{1-\rho}\,\big{|}\sqrt{P}{|\tilde{h}|}{{\tilde{X}}}+{{\tilde{W}}}\big{|}+{{N}}.$
(4)

For ease of reference, we call the variables ${\tilde{W}}$, $\tilde{Z}$ and
${{N}}$ as the equivalent antenna noise, (CD) conversion noise, and (ED)
rectifier noise, respectively, and they follow zero-mean Gaussian distribution
with variances $\sigma_{\textrm{A}}^{2}$, $\sigma_{\textrm{cov}}^{2}$ and
$\sigma_{\textrm{rec}}^{2}$, respectively. Again, note that
${{{\tilde{Y}}}_{1}}$, ${\tilde{W}}$ and $\tilde{Z}$ are complex-valued.

In the practical communication systems, the conversion noise power
$\sigma_{\textrm{cov}}^{2}$ is commonly much greater than the rectifier noise
power $\sigma_{\textrm{rec}}^{2}$ [11, 12]. For example, the value of
$\sigma_{\textrm{rec}}^{2}$ is -85 dBm is -85 dBm [12], while
$\sigma_{\textrm{cov}}^{2}$ is -70 dBm [13, 14]. Apart from the relatively
strength of the ED and CD processing noises, we should also put them in
comparison with the antenna noise. From [13], a typical antenna noise power is
approximately $-104$ dBm. Of course this antenna noise power accounts for only
the thermal noise, while the antenna noise in our system model should also
include any potential interference signals received which can be much stronger
than thermal noise.

Compared with the previous PD-CD splitting receiver proposed in [10], the
difference in the ED-CD splitting receiver lies in (4) due to the ED
processing. Although this may seem to be a small difference in terms of the
mathematical model, it does represent a different receiver design in practice.
Also, as we will discuss at the end of the next section, there are some
fundamental differences in the parameter design and performance between the
previous PD-CD splitting receiver and the new ED-CD splitting receiver.

Figure 1: The proposed ED-CD splitting receiver.

##  III Mutual Information Performance Analysis

To characterize the fundamental performance limit of the proposed receiver, we
aim to derive the mutual information achievable based on the signal model in
(3) and (4). Note that ${{{\tilde{Y}}}_{1}}$ and ${{{Y}}_{2}}$ are jointed
used for making the detection. Therefore, the mutual information between the
input ${{\tilde{X}}}$ and output $({{{\tilde{Y}}}_{1}},{{{Y}}_{2}})$ is
expressed as

$\displaystyle\mathcal{I}\left({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}\right)=\mathcal{H}\left({{{\tilde{Y}}}_{1}},{{{Y}}_{2}}\right)-\mathcal{H}\left({{{\tilde{Y}}}_{1}},{{{Y}}_{2}}\big{|}{{\tilde{X}}}\right)$
$\displaystyle=-\int_{{{{Y}}_{2}}}\int_{{{{\tilde{Y}}}_{1}}}f_{{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}}({{{\tilde{y}}}_{1}},{{{y}}_{2}})\log_{2}\left(f_{{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}}({{{\tilde{y}}}_{1}},{{{y}}_{2}})\right)\,\mathrm{d}{{{\tilde{y}}}_{1}}\mathrm{d}{{{y}}_{2}}$
$\displaystyle\,\,\,\,\,\,\,\,+\int_{{{\tilde{X}}}}\int_{{{\tilde{Y}_{1}}}}\int_{{{{Y_{2}}}}}f_{{{\tilde{X}}}}(\tilde{x})f_{{{\tilde{Y}}_{1}},{{{Y_{2}}}}}(\tilde{y}_{1},{y_{2}}|\tilde{x})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\log_{2}\left(f_{{{\tilde{Y}_{1}}},{{{Y_{2}}}}}(\tilde{y}_{1},{y_{2}}|\tilde{x})\right)\,\mathrm{d}{y_{2}}\mathrm{d}\tilde{y}_{1}\mathrm{d}\tilde{x},$
(5)

where the joint probability density function (PDF) of
$({{{\tilde{Y}}}_{1}},{{{Y}}_{2}})$ is given by

$\displaystyle
f_{{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}}({{{\tilde{y}}}_{1}},{{{y}}_{2}})$
$\displaystyle=\int_{\tilde{X}}\int_{\tilde{W}}f_{{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}}({{{\tilde{y}}}_{1}},{{{y}}_{2}}|\tilde{x},\tilde{w})f_{{{\tilde{X}}}}(\tilde{x})f_{{{\tilde{W}}}}(\tilde{w})\,\mathrm{d}{\tilde{w}}\mathrm{d}{\tilde{x}}$
$\displaystyle=\\!\\!\int_{\tilde{X}}\int_{\tilde{W}}\\!\\!f_{{{{\tilde{Y}}}_{1}}}({{{\tilde{y}}}_{1}}|\tilde{x},\tilde{w})f_{{{{Y}}_{2}}}({{{y}}_{2}}|\tilde{x},\tilde{w})f_{{{\tilde{X}}}}(\tilde{x})f_{{{\tilde{W}}}}(\tilde{w})\,\mathrm{d}{\tilde{w}}\mathrm{d}{\tilde{x}},$
(6)

and the conditional joint PDF
$f_{{{\tilde{Y}_{1}}},{{{Y_{2}}}}}(\tilde{y}_{1},{y_{2}}|\tilde{x})$ is
$\int_{\tilde{W}}\\!\\!f_{{{{\tilde{Y}}}_{1}}}({{{\tilde{y}}}_{1}}|\tilde{x},\tilde{w})f_{{{{Y}}_{2}}}({{{y}}_{2}}|\tilde{x},\tilde{w})f_{{{\tilde{W}}}}(\tilde{w})\,\mathrm{d}{\tilde{w}}$,
where $f_{{{\tilde{X}}}}(\tilde{x})$ is the PDF of the normalized input
signal, $f_{{{\tilde{W}}}}(\tilde{w})$ follows the complex Gaussian
distribution, which is expressed as
$\mathcal{CN}\big{(}0,\sigma_{\textrm{A}}^{2}\big{)}$. The conditional PDFs
$f_{{{{\tilde{Y}}}_{1}}}({{{\tilde{y}}}_{1}}|\tilde{x},\tilde{w})$ and
$f_{{{{Y}}_{2}}}({{{y}}_{2}}|\tilde{x},\tilde{w})$ are
$\mathcal{CN}\big{(}\sqrt{\rho}(\sqrt{P}|\tilde{h}|\tilde{x}+\tilde{w}),\sigma_{\textrm{cov}}^{2}\big{)}$
and
$\mathcal{N}\,\big{(}\sqrt{1-\rho}\big{|}\sqrt{P}{|\tilde{h}|}{{\tilde{x}}}+{{\tilde{w}}}\big{|},\sigma_{\textrm{rec}}^{2}\big{)}$,
respectively.

We observe that the mutual information expression in (III) needs evaluation of
a large number of integrals and the complexity is extremely high. In addition,
the mutual information depends on the distribution of $\tilde{X}$. As finding
the optimal input distribution is an extremely challenging task, in this work,
we only consider a widely adopted approach of assuming Gaussian input signal,
i.e., $\tilde{X}\sim\mathcal{CN}(0,1)$. Note that Gaussian input is optimal
for the traditional CD receivers. In what follows, we aim to give an accurate
approximation of the mutual information in the high SNR regime.

(a)
$\sigma_{\textrm{{A}}}^{2}=\sigma_{\textrm{{cov}}}^{2}=1,\sigma_{\textrm{{rec}}}^{2}=0.01$.
(b)
$\sigma_{\textrm{{A}}}^{2}=\sigma_{\textrm{{rec}}}^{2}=0.01,\sigma_{\textrm{{cov}}}^{2}=1$.

Figure 2: Mutual information versus the power splitting ratio $\rho$.

######  Proposition 1.

In high SNR, the achievable mutual information of the splitting receiver with
$\rho\in(0,1]$ can be approximated as

$\displaystyle\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}})\approx\frac{1}{2}\log_{2}\left(\frac{\rho(P|\tilde{h}|^{2}+\sigma_{\textrm{{A}}}^{2})^{2}}{P|\tilde{h}|^{2}(\rho\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2})}\right)+$
$\displaystyle\frac{1}{2}\log_{2}\biggl{(}\frac{\sigma_{\textrm{{cov}}}^{2}(1-\rho)(P|\tilde{h}|^{2}+\sigma_{\textrm{{A}}}^{2})+2\rho\sigma_{\textrm{{rec}}}^{2}P|\tilde{h}|^{2}}{2\sigma_{\textrm{{rec}}}^{2}\sigma_{\textrm{A}}^{2}\rho+\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{A}}^{2}(1-\rho)+2\sigma_{\textrm{{rec}}}^{2}\sigma_{\textrm{cov}}^{2}}\biggr{)}.$
(7)

Importantly, this approximation is asymptotically tight as $P$ approaches
infinity. The optimal splitting ratio $\rho^{*}$ in the high SNR regime is

$\displaystyle\rho^{*}=\begin{cases}\Upsilon,&\sigma_{\textrm{{cov}}}^{2}>4\sigma_{\textrm{{rec}}}^{2},\\\
1,&\textrm{else},\end{cases}$ (8)

where

$\displaystyle\Upsilon=\frac{\sigma_{\textrm{{cov}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})-\sqrt{2\Psi}}{\sigma_{\textrm{{A}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{cov}}}^{2}-4\sigma_{\textrm{{rec}}}^{2})},$
(9)

and
$\Psi\\!=\\!\sigma_{\textrm{{cov}}}^{4}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})\sigma_{\textrm{{rec}}}^{2}(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})$.
_Proof_ : See Appendix A.

Note that the result in Proposition 1 is derived under the assumption of
Gaussian input distribution, while it is not clear what is the optimal input
for the ED-CD splitting receiver. Therefore, finding the optimal input
distribution for the ED-CD splitting receiver is a very challenging open
problem.

TABLE I: Optimal Power Splitting Ratio and Mutual Information Performance Gain Noise Conditions | $P=10$ | $P=10^{2}$ | $P=10^{3}$ | $P=10^{4}$ |  $P\rightarrow\infty$ (analytical)
---|---|---|---|---|---
| $\rho^{*}$ | $G_{\textrm{MI}}$ | $G_{\textrm{MI}}\%$ | $\rho^{*}$ | $G_{\textrm{MI}}$ | $G_{\textrm{MI}}\%$ | $\rho^{*}$ | $G_{\textrm{MI}}$ | $G_{\textrm{MI}}\%$ | $\rho^{*}$ | $G_{\textrm{MI}}$ | $G_{\textrm{MI}}\%$ | $\rho^{*}$ | $G_{\textrm{MI}}$ | $G_{\textrm{MI}}\%$
| $\sigma_{\textrm{A}}^{2}=0.01$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=1$
1 |  0 |  $0$ |  1 |  0 |  $0$ |  1 |  0 |  $0$ |  1 |  0 |  $0$ |  $1$ |  0 | 0
| $\sigma_{\textrm{A}}^{2}=0.01$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.1$
0.63 |  0.18 |  $5.1\%$ |  0.63 |  0.30 |  $4.5\%$ |  0.63 |  0.31 |  $3.1\%$ |  0.63 |  0.31 |  $2.4\%$ |  0.63 |  0.31 | 0
| $\sigma_{\textrm{A}}^{2}=0.01$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.01$
0.56 |  1.56 |  $45.2\%$ |  0.56 |  1.68 |  $25.3\%$ |  0.56 |  1.68 |  $17.0\%$ |  0.56 |  1.68 |  $12.8\%$ |  0.56 |  1.69 | 0
| $\sigma_{\textrm{A}}^{2}=0.01$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.001$
0.71 |  2.57 |  $74.6\%$ |  0.71 |  2.69 |  $40.5\%$ |  0.71 |  2.71 |  $27.2\%$ |  0.71 |  2.71 |  $20.4\%$ |  0.71 |  2.71 | 0
| $\sigma_{\textrm{A}}^{2}=1$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=1$
1 |  0 |  $0$ |  1 |  0 |  $0$ |  1 |  0 |  $0$ |  1 |  0 |  $0$ |  1 |  0 | 0
| $\sigma_{\textrm{A}}^{2}=1$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.1$
0.74 |  0.01 |  $0.6\%$ |  0.77 |  0.09 |  $1.5\%$ |  0.77 |  0.10 |  $1.1\%$ |  0.77 |  0.10 |  $0.8\%$ |  0.77 |  0.10 | 0
| $\sigma_{\textrm{A}}^{2}=1$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.01$
0.84 |  0.29 |  $11.4\%$ |  0.85 |  0.35 |  $6.2\%$ |  0.85 |  0.36 |  $4.0\%$ |  0.85 |  0.36 |  $2.9\%$ |  0.85 |  0.36 | 0
| $\sigma_{\textrm{A}}^{2}=1$
---
$\sigma_{\textrm{cov}}^{2}=1$
$\sigma_{\textrm{rec}}^{2}=0.001$
0.94 |  0.40 |  $15.3\%$ |  0.94 |  0.45 |  $7.9\%$ |  0.94 |  0.45 |  $5.1\%$ |  0.94 |  0.45 |  $3.7\%$ |  0.94 |  0.45 | 0

To verify the accuracy of the approximation in (1), Fig. 2 (a) and Fig. 2 (b)
depict the mutual information approximation and the simulated mutual
information against the power splitting ratio with different signal power. We
observe that the analytical approximation in (1) matches with the simulation
for all different system parameter settings, with the only exception when the
splitting ratio is very small. In addition, with an appropriate choice of the
splitting ratio, the ED-CD splitting receiver achieves a higher mutual
information as compared to the traditional ED receiver ($\rho=0$) and CD
receiver ($\rho=1$). The optimal splitting ratio found in (8) is accurate in
all observed curves in Fig. 2. Therefore, the numerical results in Fig. 2 have
validated the accuracy of the analytical results in Proposition 1 for
different system parameter settings including moderate SNR values.

In order to further investigate how much performance gain the ED-CD splitting
receiver brings in as compared to the traditional ED and CD receivers, we
adopt the metrics defined in [10], namely the mutual information performance
gain $G_{\textrm{MI}}$ and its relative performance gain $G_{\textrm{MI}}\%$.
Specifically, $G_{\textrm{MI}}$ is the difference between the mutual
information achieved by the splitting receiver and that achieved by the best
of the traditional receivers, while $G_{\textrm{MI}}\%$ is the percentage
difference. If $G_{\textrm{MI}}>0$, the splitting receiver achieves higher
mutual information compared to the CD and ED receivers.

######  Proposition 2.

In high SNR and with the optimal splitting ratio, the mutual information
performance gain of the ED-CD splitting receiver is given by

$\displaystyle\mathop{\lim}\limits_{P\rightarrow\infty}G_{\textrm{MI}}=\begin{cases}\beta,&\sigma_{\textrm{{cov}}}^{2}>4\sigma_{\textrm{{rec}}}^{2},\\\
0,&\textrm{else},\end{cases}$ (10)
$\displaystyle\mathop{\lim}\limits_{P\rightarrow\infty}G_{\textrm{MI}}\%=0\%,$
(11)

where
$\beta=\log_{2}\left(\Upsilon\right)+\frac{1}{2}\log_{2}\left(\frac{K}{M}\right)>0$.
The variables $M$ and $K$ are determined by the noise variances and are
defined in Appendix B.

_Proof_ : See Appendix B. Note that between the traditional ED receiver and CD
receiver, it is intuitive to understand (and proven in the Appendix for high
SNR) that the CD receiver achieves a higher mutual information because it
makes use of both magnitude and phase information of the received signal.
Hence, the mutual information performance gain is the difference between the
mutual information achieved by the splitting receiver and that achieved by the
CD receiver.

In order to draw insights on the performance improvement under different SNRs
and different relative noise strengths, we provide numerical results in Table
I, which shows the results of the ED-CD splitting receivers in terms of the
optimal power splitting ratio $\rho^{*}$, mutual information gain
$G_{\textrm{MI}}$ and its percentage gain $G_{\textrm{MI}}\%$ for different
values of the noise variances and transmit power $P$.

We first discuss the trend in the optimal splitting ratio $\rho^{*}$ as the
transmit power $P$ increases. We observe that the value of $\rho^{*}$ in each
row almost does not change as the transmit power increases. That is, the
optimal splitting ratio $\rho^{*}$ of the ED-CD splitting receiver is
insensitive to the signal power $P$ and is only related to the noise
variances, which is consistent with the conclusions obtained from Proposition
1. It implies that the design of optimal splitting ratio obtained analytically
at high SNR also works very well and does not need to be changed for a wide
range of moderate SNRs.

Next we discuss the mutual information gain. We find that there is no
performance gain for the ED-CD splitting receiver when the conversion noise
variance $\sigma_{\textrm{{cov}}}^{2}$ is equal to the rectifier noise
variance $\sigma_{\textrm{{rec}}}^{2}$ as shown in the first and fifth rows of
Table I. When the rectifier noise variance $\sigma_{\textrm{{rec}}}^{2}$ is
much smaller than the conversion noise variance $\sigma_{\textrm{{cov}}}^{2}$,
the ED-CD splitting receiver can provide significant performance gain. These
observations are in agreement with Proposition 1, which says that the optimal
splitting ratio is less than 1 when the rectifier noise is much smaller than
the conversion noise (more specifically 4 times smaller). Clearly when the
optimal splitting ratio is 1, the splitting receiver becomes the same as the
CD receiver, hence, there is no performance gain. In addition, we observe that
as the antenna noise variance $\sigma_{\textrm{{A}}}^{2}$ decreases, the
performance gain increases. Therefore, one can conclude that the performance
gain of the ED-CD splitting receiver is most substantial when the rectifier
noise is much smaller than the conversion noise and the antenna noise is
relatively small as compared to these processing noises. For example, in the
third row when $\sigma_{\textrm{{A}}}^{2}=0.01$,
$\sigma_{\textrm{{cov}}}^{2}=1$, and $\sigma_{\textrm{{rec}}}^{2}=0.01$, the
ED-CD splitting receiver provides $45.2\%$ higher mutual information at $P=10$
and $25.3\%$ higher mutual information at $P=100$, compared to the traditional
CD receiver. These favorable noise conditions can be found in practical
scenarios: In an interference-free scenario, the antenna noise is almost at
the thermal noise level, which is much smaller than the processing noises.
Moreover, the rectifier noise variance $\sigma_{\textrm{{rec}}}^{2}$ is
commonly much lower than the conversion noise variance
$\sigma_{\textrm{{cov}}}^{2}$ in the practical circuits [11, 12]. Hence, the
ED-CD splitting receiver can provide significant performance improvement in
practical wireless communication systems.

######  Remark 1.

(_Comparison between the proposed ED-CD splitting receiver and the original
PD-CD splitting receiver in[10]_): In terms of similarity, both splitting
receiver designs can achieve higher mutual information than using either the
traditional coherent receiver or traditional non-coherent receiver. And this
performance gain is most substantial at low to moderate operating SNRs and
when the receiver noise is dominated by the processing noises rather than the
antenna noise. On the other hand, there are some major differences: (i) The
optimal splitting ratio of the proposed ED-CD receiver stays at a constant
value for a wide range of the operating SNRs and can be much smaller than 1
even at high SNRs, while the optimal splitting ratio of the PD-CD receiver
always increases towards 1 as the operating SNR increases. (ii) The
performance gain of the ED-CD receiver depends heavily on the relative
strength of the rectifier noise in the ED circuit and the conversion noise in
the CD circuit, while the performance gain of the PD-CD receiver has much less
dependence on the strength of the active rectifier noise in the PD circuit.
For these two aspects, the high SNR asymptotic behaviours of the two different
splitting receivers are fundamentally different. Therefore, many of the design
and performance insights previously obtained on the PD-CD splitting receiver
are not applicable to the newly proposed ED-CD splitting receiver in this
work.

In addition, the practical advantage of the ED-CD splitting receiver over the
PD-CD splitting receiver is that the passive ED circuit has a much lower
complexity than the active PD circuit and thus has a much lower cost and
energy consumption [15]. This advantage of the ED-based solution is more
significant when the receiver antennas scale up leading to the increasing of
detection circuits. The low complexity ED circuit introduces smaller
processing noise than the active PD. A detailed quantitative performance
comparison between the two different splitting receivers requires detailed
circuit-level modelling and noise analysis and is an interesting problem for
future work.

##  IV Conclusions

In this letter, we have presented a new ED-CD splitting receiver, which
jointly uses the envelope detection and coherent detection. Compared to the
original PD-CD splitting receiver, the newly proposed ED-CD splitting receiver
is more practically applicable in RF communications. Our analysis has provided
accurate closed-form approximation to understand the mutual information
performance of the proposed receiver. The optimal splitting ratio has also
been derived at high SNR and shown to be accurate for a wide range of
operating conditions. Numerical results have revealed that the proposed
splitting receiver can achieve significant performance gain over the
traditional receivers when the processing noise in the ED circuit is much
smaller than the processing noise in the CD circuit and the antenna noise is
relatively small as compared to these processing noises.

##  Appendix A  Proof of Proposition 1

To analyze the mutual information of the ED-CD splitting receiver, we utilize
the property of mutual information invariance under a change of coordinates.
Following the similar steps of the analysis of the PD-CD receiver in [10,
Appendix A], we firstly represent the Cartesian coordinate based random
variables as the cone-normal coordinate based random variables, and then
analyze the approximated mutual information in high SNR.

Different from the paraboloid-normal coordinate system adopted in [10], the
cone of the cone-normal coordination system of this letter is given below due
to the received signal expressions (3) and (4)

$c_{M}=k\sqrt{\frac{1-\rho}{\rho}}\sqrt{c_{I}^{2}+c_{Q}^{2}},$ (A.1)

where
$k\triangleq\frac{\sigma_{\textrm{cov}}}{\sqrt{2}\sigma_{\textrm{rec}}}$;
$c_{I}$, $c_{Q}$, and $c_{M}$ are the three axes of Cartesian coordinate
system of the in-phase-quadrature-magnitude (I-Q-M) space.

Based on the cone-normal coordinate system in (A.1) and following the same
steps in [10, Appendix A], the asymptotic mutual information can be obtained
as

$\displaystyle\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}})$
$\displaystyle=\log_{2}\left(\Theta_{1}\zeta^{2}\sqrt{\frac{k^{2}\Theta_{2}\zeta^{2}}{\Theta_{1}}+1}\right)$
$\displaystyle-\frac{1}{2}\log_{2}\left(\frac{\Theta_{1}\sigma_{\textrm{A}}^{2}+\sigma_{\textrm{cov}}^{2}}{(\Theta_{1}+\Theta_{2}k^{2})\sigma_{\textrm{A}}^{2}+\sigma_{\textrm{cov}}^{2}}\right),$
(A.2)

where $\zeta^{2}=1+\frac{\sigma_{\textrm{A}}^{2}}{P|\tilde{h}|^{2}}$,
$k^{2}=\frac{\sigma_{\textrm{cov}}^{2}}{2\sigma_{\textrm{rec}}^{2}}$,
$\Theta_{1}\triangleq\rho$ and $\Theta_{2}\triangleq 1-\rho$. (1) can be
obtained after simplifications.

The mutual information approximation in (1) can be rewritten as
$\frac{1}{2}\log_{2}(f(\rho))$. Knowing that the function $\log(\cdot)$ is
monotonic, we only need to find the optimal splitting ratio $\rho^{*}$ that
maximizes $f(\rho)$. In order to compute $\rho^{*}$, we first calculate
derivative function $f(\rho)$ as

$\displaystyle\\!\\!\\!\\!f^{\prime}\\!(\rho)\\!\\!\triangleq\\!\\!\frac{A\rho^{2}+B\rho+C}{\\!(\\!\rho\sigma_{\textrm{{A}}}^{2}\\!\\!+\\!\sigma_{\textrm{{cov}}}^{2}\\!)^{2}\\!\big{(}\\!(\rho\\!-\\!\\!1\\!)\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{2}\\!\\!-\\!\\!2\rho\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{rec}}}^{2}\\!\\!-\\!\\!2\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}\\!\big{)}^{2}},$
(A.3)

where

$\displaystyle
A=P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}-6P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}+8P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{4}$
$\displaystyle\,\,\,\,\,\,\,\,\,-2\sigma_{\textrm{{A}}}^{6}\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}+\sigma_{\textrm{{A}}}^{4}\sigma_{\textrm{{cov}}}^{6}-4\sigma_{\textrm{{A}}}^{4}\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2},$
$\displaystyle
B=-2P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}+4P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}-4P\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2}$
$\displaystyle\,\,\,\,\,\,\,\,\,+8P\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{4}-2\sigma_{\textrm{{A}}}^{4}\sigma_{\textrm{{cov}}}^{6}-4\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2},$
$\displaystyle
C=P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}+2P\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2}+\sigma_{\textrm{{A}}}^{4}\sigma_{\textrm{{cov}}}^{6}+2\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2}.$
(A.4)

Then, making the numerator of $f^{\prime}(\rho)=0$, the root that maximizes
$f(\rho)$ is $\rho^{*}$. Letting $P\rightarrow\infty$ and keeping the dominate
terms, the numerator of (A.3) is approximated as

$\displaystyle
f^{\prime}_{\textrm{num}}(\rho)\triangleq(P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}-6P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}+8P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{4})\rho^{2}+$
$\displaystyle(4P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}-2P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}-4P\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2}+8P\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{4})\rho$
$\displaystyle+P\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{6}+2P\sigma_{\textrm{{cov}}}^{6}\sigma_{\textrm{{rec}}}^{2}.$
(A.5)

When $\sigma_{\textrm{{A}}}^{2}\neq 0$ and
$\sigma_{\textrm{{cov}}}^{4}-6\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}+8\sigma_{\textrm{{rec}}}^{4}=(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{cov}}}^{2}-4\sigma_{\textrm{{rec}}}^{2})=0$,
we have the equality $f^{\prime}_{\textrm{num}}(\rho)\triangleq
2(2\sigma_{\textrm{{rec}}}^{2}-\sigma_{\textrm{{cov}}}^{2})(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})\rho+\sigma_{\textrm{{cov}}}^{2}(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})=0$,
which has the single root as $\rho^{*}=1$. When $\sigma_{\textrm{{A}}}^{2}\neq
0$ and
$(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{cov}}}^{2}-4\sigma_{\textrm{{rec}}}^{2})\neq
0$, the function $f^{\prime}_{\textrm{num}}(\rho)=0$ has two roots:

$\displaystyle\Upsilon=\frac{\sigma_{\textrm{{cov}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})-\sqrt{2\Psi}}{\sigma_{\textrm{{A}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{cov}}}^{2}-4\sigma_{\textrm{{rec}}}^{2})},$
(A.6)

and

$\displaystyle\Phi=\frac{\sigma_{\textrm{{cov}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2})+\sqrt{2\Psi}}{\sigma_{\textrm{{A}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{cov}}}^{2}-4\sigma_{\textrm{{rec}}}^{2})},$
(A.7)

where
$\Psi\\!=\\!\sigma_{\textrm{{cov}}}^{4}(\sigma_{\textrm{{cov}}}^{2}\\!\\!-\\!2\sigma_{\textrm{{rec}}}^{2})(\sigma_{\textrm{{A}}}^{2}\\!\\!+\\!\sigma_{\textrm{{cov}}}^{2}\\!\\!-\\!2\sigma_{\textrm{{rec}}}^{2})\sigma_{\textrm{{rec}}}^{2}(\sigma_{\textrm{{A}}}^{2}\\!\\!+\\!2\sigma_{\textrm{{rec}}}^{2})$.

From (A.6) and (A.7), if
$\sigma_{\textrm{{cov}}}^{2}<2\sigma_{\textrm{{rec}}}^{2}$, the numerators and
denominators of the $\Upsilon$ and $\Phi$ are smaller and larger than zero,
respectively. Then we can obtain that $\Upsilon<\Phi<0$, and easy to prove
that $f(\rho)$ is an increasing function in $0<\rho<1$, so $\rho^{*}=1$. When
$2\sigma_{\textrm{{rec}}}^{2}<\sigma_{\textrm{{cov}}}^{2}<4\sigma_{\textrm{{rec}}}^{2}$,
the roots $\Phi<0$, $\Upsilon>0$ and the value of the root $\Phi$ is smaller
than $\Upsilon$, and it can be verified that $f(\rho)$ is an increasing
function in $0<\rho<1$ and $\rho^{*}=1$. When
$\sigma_{\textrm{{cov}}}^{2}>4\sigma_{\textrm{{rec}}}^{2}$, we have
$\Phi>\Upsilon>0$. It can be proved that $\Upsilon$ maximizes $f(\rho)$ and
$0<\Upsilon<1$, Then the optimal splitting ratio $\rho^{*}=\Upsilon$.

##  Appendix B Proof of Proposition 2

According to the definition of the $G_{\textrm{MI}}$ shown in [10], for
analyzing the mutual information performance gain, we need to obtain the
asymptotic mutual information of the ED receiver ($\rho=0$), CD receiver
($\rho=1$) and the ED-CD splitting receiver. The mutual information of the ED-
CD splitting receiver has been achieved as shown in Proposition 1. The mutual
information of the CD receiver is [10]

$\displaystyle\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}\\!)|_{\rho=1}=\log_{2}\left(1+\frac{P|\tilde{h}|^{2}}{\sigma_{\textrm{A}}^{2}+\sigma_{\textrm{cov}}^{2}}\right).$
(A.8)

An upper bound on the mutual information of the ED receiver is given by [13,
16]

$\displaystyle\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}\\!)|_{\rho=0}$
$\displaystyle=\mathcal{I}({{\tilde{X}}};{{{Y}}_{2}}\\!)$
$\displaystyle\leq\frac{1}{2}\\!\log_{2}\\!\\!\left(\frac{P|\tilde{h}|^{2}}{\sigma_{\textrm{A}}^{2}}\right)\\!\\!+\\!\\!\frac{1}{2}\\!\\!\left(\\!\log_{2}\frac{e}{2\pi}\\!\right).$
(A.9)

Based on (B), the upper bound on mutual information of the ED receiver scales
with $P$ as $\frac{1}{2}\log_{2}P$. While the achievable mutual information of
the CD receiver scales with $P$ as $\log_{2}P$. Therefore, the CD receiver
achieves much higher mutual information than the ED receiver as
$P\rightarrow\infty$, i.e.,
$\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}})|_{\rho=0}\ll\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}})|_{\rho=1}$.
Thus, we have [10]

$\displaystyle G_{\textrm{MI}}\\!=\\!\sup\\{\mathcal{I}({{\tilde{X}}};$
$\displaystyle{{{\tilde{Y}}}_{1}},{{{Y}}_{2}}):\rho\in(0,1]\\}\\!-\\!\mathcal{I}({{\tilde{X}}};{{{\tilde{Y}}}_{1}},{{{Y}}_{2}})|_{\rho=1}.$
(A.10)

Then, the performance improvement of the splitting receiver is expressed as

$\displaystyle\mathop{\lim}\limits_{P\rightarrow\infty}G_{\textrm{MI}}\\!=\\!\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho=\rho^{*}}\\!-\\!\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho=1}$
$\displaystyle=\frac{1}{2}\log_{2}\\!\\!\left(\frac{2P|\tilde{h}|^{2}\rho^{*}\sigma_{\textrm{{rec}}}^{2}-(\rho^{*}-1)\sigma_{\textrm{{cov}}}^{2}(P|\tilde{h}|^{2}+\sigma_{\textrm{{A}}}^{2})}{\sigma_{\textrm{{A}}}^{2}(-\rho^{*}\sigma_{\textrm{{cov}}}^{2}+2\rho^{*}\sigma_{\textrm{{rec}}}^{2}+\sigma_{\textrm{{cov}}}^{2})+2\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}}\right)+$
$\displaystyle\frac{1}{2}\\!\log_{2}\\!\\!\left(\frac{(P|\tilde{h}|^{2}+\sigma_{\textrm{{A}}}^{2})^{2}\rho^{*}(\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2})^{2}}{P|\tilde{h}|^{2}(\rho^{*}\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2})(P|\tilde{h}|^{2}+\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2})^{2}}\\!\\!\right).$
(A.11)

Based on (B), when $\rho^{*}=1$, $G_{\textrm{MI}}=0$. When
$\rho^{*}=\Upsilon$, $G_{\textrm{MI}}$ is simplified as

$\displaystyle\mathop{\lim}\limits_{P\rightarrow\infty}G_{\textrm{MI}}\\!$
$\displaystyle=\\!\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho=\Upsilon}\\!-\\!\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho=1}$
$\displaystyle=\log_{2}\left(\Upsilon\right)+\frac{1}{2}\log_{2}(K/M),$ (A.12)

where

$\displaystyle
K=(\sigma_{\textrm{{A}}}^{2}+\sigma_{\textrm{{cov}}}^{2})^{2}(\sigma_{\textrm{{cov}}}^{4}-6\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}+8\sigma_{\textrm{{rec}}}^{4})^{2}(2\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{2}+$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,2\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}-4\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{4}-\sqrt{2\Psi}),$
$\displaystyle
M=\\!\\!(\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{cov}}}^{2}(\sigma_{\textrm{{cov}}}^{2}\\!\\!-\\!\\!2\sigma_{\textrm{{rec}}}^{2})\\!\\!+\\!\\!\sigma_{\textrm{{cov}}}^{6}\\!\\!-\\!4\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}\\!\\!+\\!4\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{4}\\!\\!-\\!\sqrt{2\Psi})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,(4\sigma_{\textrm{{A}}}^{2}\sigma_{\textrm{{rec}}}^{2}(\sigma_{\textrm{{cov}}}^{2}-2\sigma_{\textrm{{rec}}}^{2})+2\sigma_{\textrm{{cov}}}^{4}\sigma_{\textrm{{rec}}}^{2}-4\sigma_{\textrm{{cov}}}^{2}\sigma_{\textrm{{rec}}}^{4}-\sqrt{2\Psi})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\sigma_{\textrm{{cov}}}^{4}(\sigma_{\textrm{{A}}}^{2}+2\sigma_{\textrm{{rec}}}^{2}).$
(A.13)

Note that when $\rho^{*}=\Upsilon$, the splitting receiver can strictly
achieve higher mutual information than the CD receiver ($\rho=1$), as proved
in Appendix A, i.e.,
$\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho^{*}=\Upsilon}\\!>\\!\mathcal{I}(\\!{{\tilde{X}}};{{{\tilde{Y}}}_{1}},\\!{{{Y}}_{2}}\\!)|_{\rho=1}$
as $P\rightarrow\infty$. In addition, when $\rho=1$, the performance
improvement gain $G_{\textrm{MI}}=0$. Therefore, when $\rho^{*}=\Upsilon$, the
performance improvement $G_{\textrm{MI}}>0$, i.e., $\beta>0$. Then, we can
easily obtain the result related to $G_{\textrm{MI}}\%$.

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