# On the squarefree values of $a^{4}+b^{3}$

Gian Cordana Sanjaya and Xiaoheng Wang

###### Abstract

In this article, we prove that the density of integers $a,b$ such that
$a^{4}+b^{3}$ is squarefree, when ordered by ${\rm
max}\\{|a|^{1/3},|b|^{1/4}\\}$, equals the conjectured product of the local
densities.

We show that the same is true for polynomials of the form $\beta a^{4}+\alpha
b^{3}$ for any fixed integers $\alpha$ and $\beta$. We give an exact count for
the number of pairs $(a,b)$ of integers with ${\rm
max}\\{|a|^{1/3},|b|^{1/4}\\}<X$ such that $\beta a^{4}+\alpha b^{3}$ is
squarefree, with a power-saving error term.

##  1 Introduction

A classical question in analytic number theory is to determine the probability
that a given polynomial $F$ with integer coefficients takes squarefree values
when evaluated at random integers. The simplest case of one-variable and
degree-one asks for the probability that a random integer is squarefree, which
is well-known to be $6/\pi^{2}$. The one-variable degree-two case can also be
solved by elementary methods. The one-variable degree-three case was solved by
Hooley [14]. For homogeneous polynomials of two variables, the question is
known up to degree 6 due to Greaves [8]. For non-homogeneous polynomials of
two variables that factor completely into a product of linear factors over
some extension of ${\mathbb{Q}}$, the question is known also up to degree 6
due to Hooley [13]. Very recently, Kowalski [16] proved the case where $F$ is
a sum of at least $3$ cubic polynomials in different variables. The cases when
$F$ is the discriminant of monic polynomials or when $F$ is the discriminant
of general polynomials were solved by Bhargava-Shankar-Wang [4, 5].

Conditionally, Granville [7] showed that the one-variable case in general
follows from the $abc$-conjecture and a bound on the error term was later
obtained by Murty-Pasten [17]. Poonen [18] generalized it to the multi-
variable case where the variables are growing to infinity one by one.

Very little is known otherwise. In most cases, even the infinitude of
squarefree values is open—the most famous example being $a^{4}+2.$ There is a
handful of results on the number of prime values taken by polynomials:
Friedlander-Iwaniec [6] for $a^{4}+b^{2}$; Heath-Brown [10] for $a^{3}+2b^{3}$
and its generalization by Heath-Brown-Moroz [12] for irreducible binary cubic
forms; and Health-Brown-Li [11] for $a^{2}+p^{4}$ where $p$ is a prime.

In this paper, we consider for the first time the polynomial $a^{4}+b^{3}$.
Our method in fact allows us to consider all polynomials of the form $\beta
a^{4}+\alpha b^{3}$ for any fixed integers $\alpha$ and $\beta$. We prove:

###### Theorem 1.1

Let $\alpha$ and $\beta$ be fixed nonzero integers such that
$\gcd(\alpha,\beta)$ is squarefree. Let

$N(X;\alpha,\beta)=\\#\\{(a,b)\in{\mathbb{Z}}^{2}\colon{\rm
max}\\{|a|^{1/3},|b|^{1/4}\\}<X,\beta a^{4}+\alpha b^{3}\mbox{ is
squarefree}\\}.$

For any positive integer $m$, let
$\rho_{\alpha,\beta}(m)=\\#\\{(a,b)\bmod{m}\colon m\mid\beta a^{4}+\alpha
b^{3}\\}$ and let

$C(\alpha,\beta)=\prod_{p}(1-\rho_{\alpha,\beta}(p^{2})p^{-4}).$

Then

$N(X;\alpha,\beta)=C(\alpha,\beta)X^{7}+O_{\epsilon}(X^{6.992+\epsilon}).$

The implied constant depends on $\alpha$ and $\beta$.

The case $\alpha=256$ and $\beta=-27$ is of special importance since
$256b^{3}-27a^{4}$ is the discriminant of the quartic polynomial $x^{4}+ax+b.$
An elementary calculation shows that $\rho_{256,-27}(p^{2})$ equals $p^{3}$
for $p=2,3$; and equals $2p^{2}-p$ for $p\geq 5$. Therefore, we have:

###### Theorem 1.2

When pairs $(a,b)$ of integers are ordered by $H(a,b)={\rm
max}\\{|a|^{1/3},|b|^{1/4}\\}$, the density of quartic polynomials of the form
$x^{4}+ax+b$ having squarefree discriminant exists and is equal to

$\frac{1}{3}\prod_{p\geq 5}(1-\frac{2}{p^{2}}+\frac{1}{p^{3}})$

which is approximately $28.03\%$.

It is easy to see that the Euler product $C(\alpha,\beta)$ gives an upper
bound for the desired density, if it exists, by applying the Chinese Remainder
Theorem to more and more primes. As is standard in sieve theory, to
demonstrate the lower bound, a “tail estimate” is required to show that there
are not too many pairs $(a,b)$ of integers such that $\beta a^{4}+\alpha
b^{3}$ is divisible by $m^{2}$ for some squarefree integer $m$. More
precisely, we prove:

###### Theorem 1.3

Let $\alpha$ and $\beta$ be fixed nonzero integers such that
$\gcd(\alpha,\beta)$ is squarefree. For any squarefree integer $m$, let

$N_{m}(X;\alpha,\beta)=\\#\\{(a,b)\in{\mathbb{Z}}^{2}\colon|a|\leq
X^{3},|b|\leq X^{4},m^{2}\mid\beta a^{4}+\alpha b^{3}\\}.$

Then for any positive real number $M$ and $\epsilon>0$,

$\sum_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}N_{m}(X;\alpha,\beta)=O_{\epsilon}\left(\frac{X^{7+\epsilon}}{\sqrt{M}}\right)+O_{\epsilon}(X^{6.992+\epsilon})$
(1)

The implied constants depend on $\alpha$ and $\beta$.

We note that since the exponents $3$ and $4$ are coprime, it is enough to
prove Theorem 1.3 for one choice of $\alpha$, $\beta$. Indeed, we have

$-256\cdot 27\cdot\alpha^{8}\beta^{3}(\beta a^{4}+\alpha
b^{3})=256(-3\alpha^{3}\beta b)^{3}-27(4\alpha^{2}\beta a)^{4},$

which implies that,

$N_{m}(X;\alpha,\beta)\leq N_{m}(c_{\alpha,\beta}X;\,256,-27)$

for some constant $c_{\alpha,\beta}$ depending only on $\alpha,\beta.$ Hence
the power saving bound (1) for $\alpha=256$ and $\beta=-27$ implies it for all
other $\alpha$ and $\beta.$ We simplify notation by writing $\Delta(a,b)$ for
$256b^{3}-27a^{4}$.

For any prime $p$ and pair $(a,b)$ of integers such that
$p^{2}\mid\Delta(a,b)$, we say $p^{2}$ _strongly divides_ $\Delta(a,b)$ if
$p^{2}\mid\Delta(a^{\prime},b^{\prime})$ for any integers $a^{\prime}\equiv
a\pmod{p}$ and $b^{\prime}\equiv b\pmod{p}$; otherwise, we say $p^{2}$ _weakly
divides_ $\Delta(a,b)$. Note in this case, for $p\geq 5$, $p^{2}$ strongly
divides $\Delta(a,b)$ if and only if $p\mid a$ and $p\mid b.$ For any
squarefree integer $m$, let ${\mathcal{W}}_{m}^{(1)}$ (respectively
${\mathcal{W}}_{m}^{(2)}$) denote the set of pairs $(a,b)$ of integers such
that $p^{2}$ strongly divides (respectively weakly divides) $\Delta(a,b)$ for
every prime $p\mid m$. Then we prove:

###### Theorem 1.4

For any positive real number $M$ and $\epsilon>0$,

$\displaystyle{\rm(a)}\quad\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{(a,b)\in{\mathcal{W}}_{m}^{(1)}\colon
H(a,b)<X\\}$ $\displaystyle=$ $\displaystyle
O\Big{(}\frac{X^{7}}{M}\Big{)}+O\Big{(}X^{4}\log X\Big{)};$ (2)
$\displaystyle{\rm(b)}\quad\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{(a,b)\in{\mathcal{W}}_{m}^{(2)}\colon
H(a,b)<X\\}$ $\displaystyle=$ $\displaystyle
O_{\epsilon}\Big{(}X^{6.992+\epsilon}\Big{)}+O_{\epsilon}\left(\frac{X^{7+\epsilon}}{M}\right),$
(3)

where the implied constants are independent of $M$ and $X$.

We now briefly describe our methods. Theorem 1.4(a) is immediate with the
first term counting the contribution from $a\neq 0$ and the second term
counting the contributiom from $a=0$. We devote the rest of the paper to
proving Theorem 1.4(b). We follow the strategy of [4] to embed
${\mathcal{W}}_{m}^{(2)}$ into the space $W$ of $4\times 4$ symmetric
matrices. More precisely, let $A_{0}$ denote the $4\times 4$ matrix with $1$’s
on the anti-diagonal and $0$’s elsewhere. The group $G={\rm PSO}(A_{0})={\rm
SO}(A_{0})/\langle\pm I\rangle$ acts on $W$ via the action $g\cdot B=gBg^{t}$
for $g\in G$ and $B\in W$. Define the invariant polynomial of an element $B\in
W$ by

$f_{B}(x)=\det(A_{0}x-B).$

Then $f_{B}$ is a monic quartic polynomial. We extend the definition $H(a,b)$
to arbitrary monic quartic polynomials by

$H(x^{4}+c_{1}x^{3}+c_{2}x^{2}+c_{3}x+c_{4})={\rm
max}\\{|c_{1}|,|c_{2}|^{1/2},|c_{3}|^{1/3},|c_{4}|^{1/4}\\}.$

Define the discriminant and height of an element $B\in W$ by the discriminant
and height of $f_{B}$, respectively. We then construct a map

$\sigma_{m}:{\mathcal{W}}_{m}^{(2)}\rightarrow\frac{1}{4}W({\mathbb{Z}})$

with $f_{\sigma_{m}(a,b)}=x^{4}+ax+b$ as in [4], where
$\frac{1}{4}W({\mathbb{Z}})$ is the lattice of elements $B$ whose coefficients
have denominators dividing $4$. It thus remains to count
$G({\mathbb{Z}})$-orbits in $\frac{1}{4}W({\mathbb{Z}})$ that intersect the
image of $\sigma_{m}$ for some squarefree $m>M$, and have height bounded by
$X$.

The space $W$ has several subspaces: $W_{00}$ consisting of $B\in W$ whose
$(1,1)$\- and $(1,2)$-entries are $0$; $W_{01}$ consisting of $B\in W$ whose
$(1,1)$\- and $(1,3)$-entries are $0$; and $W_{0}$ consisting of $B\in W$
whose $(1,1)$-entry is 0. The map $\sigma_{p}$ in fact lands in $W_{00}$, and
so are guaranteed to be _distinguished_ in the sense of [20]. We obtain a
bound of $O_{\epsilon}(X^{7+\epsilon}/M)$ for the distinguished cusps $W_{00}$
and $W_{01}$ and a bound of $O_{\epsilon}(X^{6+\epsilon})$ for the “thick”
cusp $W_{0}$.

The main novelty of this paper is on counting the distinguished orbits in the
main body. We use the circle method to handle the condition that the invariant
polynomials have vanishing $x^{2}$-coefficients, combined with the Selberg
sieve to impose the distinguished condition to obtain the desired power
saving.

We remark that Heath-Brown’s result [9] on $k$-free values of the polynomial
$n^{d}+c$ specializes to squarefree values of the cubic polynomial $n^{3}+c$
where $c$ is a constant. A major observation of [9] is that counting triples
$(n,s,t)$ with $n^{3}+c=s^{2}t$ when $n$ and $s$ are large and $c$ is fixed,
is akin to counting points close to the projective curve $N^{3}=S^{2}T$. The
bigger $c$ is, which in our case can be as big as $n^{3}$, the worse the
estimate gets. As such, we cannot patch the results of [9] together to prove
Theorem 1.1. One does immediately obtain the infinitude of squarefree values.

This paper is organized as follows. In Section 2, we set up the embedding into
$W$ and collect some results on the invariant theory for the action of $G$ on
$W$, which allows us to reduce Theorem 1.4(b) to a result on counting
$G({\mathbb{Z}})$-orbits in $\frac{1}{4}W({\mathbb{Z}})$. In Section 3, we
apply Bhargava’s averaging trick and count in the cusp. In Section 4, we use
the circle method and the Selberg sieve to count in the main body. Finally, in
Section 5, we show how Theorem 1.1 and Theorem 1.3 follow from Theorem 1.4.

##  2 Embedding into the space of $4\times 4$ symmetric matrices

Let $A_{0}$ be the $4\times 4$ matrix with $1$’s on the anti-diagonal and
$0$’s elsewhere. The group $G={\rm PSO}(A_{0})={\rm SO}(A_{0})/\langle\pm
I\rangle$ acts on the space $W$ of symmetric $4\times 4$ matrices via the
action $g\cdot B=gBg^{t}$ for $g\in G$ and $B\in W$. The ring of polynomial
invariants over ${\mathbb{C}}$ is freely generated by the coefficients of the
invariant polynomial $f_{B}(x)=\det(A_{0}x-B)$, which is a monic quartic
polynomial. Define $G$-invariant discriminant $\Delta(B)$ and height $H(B)$ of
an element $B\in W$ by $\Delta(B)=\Delta(f_{B})$ and $H(B)=H(f_{B})$. We
recall some of the arithmetic invariant theory for this representation. See
[20, 4] for more detail.

###  2.1 Invariant theory for the representation $W$ of $G$

Let $k$ be a field of characteristic not $2$. For any monic quartic polynomial
$f(x)\in k[x]$ such that $\Delta(f)\neq 0$, let $C_{f}$ denote the smooth
hyperelliptic curve $y^{2}=f(x)$ of genus $1$ and let $J_{f}$ denote its
Jacobian (which is an elliptic curve). The stabilizer in $G(k)$ of an element
$B\in W(k)$ with $f_{B}(x)=f(x)$ is naturally isomorphic to $J_{f}[2](k)$,
which in turn is in bijection with the set of _even factorization_ of $f(x)$
over $k$. An even factorization of $f(x)$ over $k$ is an unordered pair
$(g(x),h(x))$ of quadratic polynomials with $g(x)h(x)=f(x)$ such that either
$g$ and $h$ are both defined over $k$, or they are (defined and) conjugate
over a quadratic extension of $k$.

An element $B\in W(k)$ or its $G(k)$-orbit with $\Delta(B)\neq 0$ is said to
be: $k$-_soluble_ if there exists a nonzero vector $v\in k^{4}$ such that

$v^{t}A_{0}v=0=v^{t}Bv;$ (4)

$k$-_distinguished_ if there exist linearly independent vectors $v,w\in k^{4}$
such that

$v^{t}A_{0}v=v^{t}Bv=w^{t}A_{0}w=v^{t}A_{0}w=v^{t}Bw=0.$ (5)

Moreover, the set of $k$-lines ${\rm Span}(v)$ satisfying (4), if nonempty, is
in bijection with $J_{f_{B}}(k)$; and the set of $k$-flags ${\rm
Span}(v)\subset{\rm Span}(v,w)$ satisfying (5), if nonempty, is in bijection
with $J_{f_{B}}[2](k).$ The set of $k$-soluble orbits with $f_{B}(x)=f(x)$ is
in bijection with $J(k)/2J(k)$. The number of $k$-distinguished orbits with
$f_{B}(x)=f(x)$ is $1$ if $f(x)$ has a linear factor over $k$ or if $f(x)$
admits a factorization of the form $g(x)h(x)$ where $g$ and $h$ are not
rational over $k$ but are conjugate over a quadratic extension of $k$; and is
$2$ otherwise.

Let $W_{00}$ denote the subspace of $W$ consisting of matrices $B$ whose
$(1,1)$\- and $(1,2)$-entries are $0$. Let $W_{01}$ denote the subspace of $W$
consisting of matrices $B$ whose $(1,1)$\- and $(1,3)$-entries are $0$. Let
$W_{0}$ denote the subspace of $W$ consisting of matrices $B$ whose
$(1,1)$-entry is $0$. Then elements in $W_{00}(k)$ or $W_{01}(k)$ with nonzero
discriminants are $k$-distinguished and elements in $W_{0}(k)$ with nonzero
discriminants are $k$-soluble. A further polynomial invariant, called the
$Q$-invariant, is defined on $W_{00}$ in [4, §3.1]. For the case of $4\times
4$ matrices $B$, this is simply the $(1,3)$-entry $b_{13}$. The $Q$-invariant
has the following important property:

###### Proposition 2.1

Let $B\in W_{00}({\mathbb{Q}})$ be an element whose invariant polynomial
$f_{B}(x)$ is irreducible over ${\mathbb{Q}}$ and does not factor as
$g(x)h(x)$ where $g$ and $h$ are conjugate over some quadratic extension of
${\mathbb{Q}}$. If $B^{\prime}\in W_{00}({\mathbb{Q}})$ is any element that is
$G({\mathbb{Z}})$-equivalent to $B$, then the $(1,3)$-entries of $B^{\prime}$
and $B$ are equal up to sign. If $B^{\prime}\in W_{01}({\mathbb{Q}})$ is any
element that is $G({\mathbb{Z}})$-equivalent to $B$, then the $(1,2)$-entry of
$B^{\prime}$ equals the $(1,3)$-entry of $B$ up to sign.

Proof: We prove the statement for $B^{\prime}\in W_{01}({\mathbb{Q}})$. The
statement for $W_{00}({\mathbb{Q}})$ follows by a similar argument (see also
[4, Proposition 3.1]).

Let $\\{e_{1},e_{2},e_{3},e_{4}\\}$ denote the standard basis for
${\mathbb{Q}}^{4}.$ Let $\gamma_{0}$ be the element of ${\rm
SO}(A_{0})({\mathbb{Z}}[i])$ defined by

$\gamma_{0}(e_{1})=ie_{1},\quad\gamma_{0}(e_{2})=ie_{2},\quad\gamma_{0}(e_{3})=-ie_{3},\quad\gamma_{0}(e_{4})=-ie_{4},$

where $i=\sqrt{-1}$ is a root to $x^{2}+1=0$. Then any $\gamma\in{\rm
PSO}(A_{0})({\mathbb{Z}})$ can either be lifted to some
$\widetilde{\gamma}\in{\rm SO}(A_{0})({\mathbb{Z}})$ or to
$\gamma_{0}\widetilde{\gamma}\in{\rm SO}(A_{0})({\mathbb{Z}}[i])$ for some
$\widetilde{\gamma}\in{\rm SO}(A_{0})({\mathbb{Z}})$.

Suppose $B^{\prime}=\gamma B\gamma^{t}$ for some $\gamma\in{\rm
PSO}(A_{0})({\mathbb{Z}})$. Since $\gamma_{0}$ only scales $e_{1}$ and
$e_{3}$, we see that there is some $\widetilde{\gamma}\in{\rm
SO}(A_{0})({\mathbb{Z}})$ such that ${\rm
Span}(\widetilde{\gamma}^{t}e_{1})\subset{\rm
Span}(\widetilde{\gamma}^{t}e_{1},\widetilde{\gamma}^{t}e_{3})$ is a flag
satisfying (5) for $B$, and $\gamma$ lifts either to $\widetilde{\gamma}$ or
to $\gamma_{0}\widetilde{\gamma}$. The assumption on $f_{B}$ implies that the
flag ${\rm Span}(\widetilde{\gamma}^{t}e_{1})\subset{\rm
Span}(\widetilde{\gamma}^{t}e_{1},\widetilde{\gamma}^{t}e_{3})$ coincides with
the flag ${\rm Span}(e_{1})\subset{\rm Span}(e_{1},e_{2}).$ Hence, there are
integers $\alpha_{1},\alpha_{2},\alpha_{3}$ such that

$\displaystyle\widetilde{\gamma}^{t}e_{1}$ $\displaystyle=$
$\displaystyle\alpha_{1}e_{1},$ $\displaystyle\widetilde{\gamma}^{t}e_{3}$
$\displaystyle=$ $\displaystyle\alpha_{2}e_{2}+\alpha_{3}e_{1}.$

Since $\widetilde{\gamma}\in{\rm SO}(A_{0})({\mathbb{Z}})$, we must then have

$\displaystyle\widetilde{\gamma}^{t}e_{4}$ $\displaystyle=$
$\displaystyle\alpha_{1}^{-1}e_{4}-\alpha_{3}\alpha_{1}^{-1}\alpha_{2}^{-1}e_{3},$
$\displaystyle\widetilde{\gamma}^{t}e_{2}$ $\displaystyle=$
$\displaystyle\alpha_{2}^{-1}e_{3},$

with $\alpha_{1}=\pm 1$ and $\alpha_{2}=\pm 1.$ The $(1,2)$-entry
$b^{\prime}_{12}$ of $B^{\prime}$ is then either
$(\widetilde{\gamma}^{t}e_{1})^{t}B(\widetilde{\gamma}^{t}e_{2})$ or
$(i\widetilde{\gamma}^{t}e_{1})^{t}B(i\widetilde{\gamma}^{t}e_{2})$. In both
cases, we have
$b^{\prime}_{12}=\pm\alpha_{1}\alpha_{2}^{-1}e_{1}^{t}Be_{3}=\pm b_{13}$.
$\Box$

Let $U\simeq{\mathbb{A}}^{2}\backslash\\{\Delta=0\\}$ be the space of monic
quartic polynomials of the form $x^{4}+ax+b$ with nonzero discriminant. The
next result shows that the number of elements of $U({\mathbb{Z}})$ failing the
condition of Proposition 2.1 is negligible.

###### Proposition 2.2

The number of elements $f\in U({\mathbb{Z}})$ with $H(f)<X$ such that $f(x)$
is either reducible over ${\mathbb{Q}}$ or factors as $g(x)h(x)$ where $g$ and
$h$ are conjugate over some quadratic extension of ${\mathbb{Q}}$ is
$O(X^{4}\log X)$.

Proof: Throughout this proof, we use repeatedly the classical result that the
sum $\sum_{|n|<X}d(n)$ of the divisor function is $O(X\log X)$ and that the
sum $\sum_{|n|<X}\tau_{3}(n)$ of the triple-divisor function is
$O(X\log^{2}X).$ See for example [1, §3.5].

Suppose first $f(x)=x^{4}+ax+b$ has a linear factor $x-r$ over ${\mathbb{Q}}$.
Then there are $O(X^{3})$ of them with $b=0$. Suppose now $b\neq 0$. Then
since $r\mid b$, we get $O(X^{4}\log X)$ choices for the pair $(r,b)$, which
uniquely determines $a$. Hence, there are $O(X^{4}\log X)$ such $f(x)$ with a
linear factor.

Next we consider the case where $f(x)=x^{4}+ax+b$ does not have a linear
factor but factors as $(x^{2}+cx+d)(x^{2}-cx+e)$ over ${\mathbb{Q}}$. Then
$de=b\neq 0$, which gives us $O(X^{4}\log X)$ choices for the triple
$(d,e,b)$. Comparing the $x^{2}$-coefficients gives $c^{2}=d+e$, and so $c$ is
determined given $d$ and $e$. Comparing the $x$-coefficients then uniquely
determines $a$. Hence, there are $O(X^{4}\log X)$ such $f(x)$ that factors as
a product of two irreducible quadratic polynomials.

Finally, we consider the case where $f(x)=x^{4}+ax+b$ is irreducible over
${\mathbb{Q}}$ but factors as

$(x^{2}+e_{1}\sqrt{d}x+\frac{c_{2}+e_{2}\sqrt{d}}{2})(x^{2}-e_{1}\sqrt{d}x+\frac{c_{2}-e_{2}\sqrt{d}}{2})$

over the ring of integers in ${\mathbb{Q}}(\sqrt{d})$ for some $d$. If $a=0$,
then we have $O(X^{4})$ choices for $b$. Suppose now $a\neq 0$. Comparing the
$x$-coefficients gives $e_{1}e_{2}d=a$. Hence there are $O(X^{3}\log^{2}X)$
choices for the tuple $(e_{1},e_{2},d,a)$. Comparing the $x^{2}$-coefficients
gives $c_{2}-e_{1}^{2}d=0$, and so $c_{2}$ is determined given $e_{1}$ and
$d$. Comparing the constant terms then uniquely determines $b$. Hence, there
are $O(X^{3}\log^{2}X)$ such $f(x)$ that factors into conjugate quadratic
polynomials over some quadratic extension of ${\mathbb{Q}}$. $\Box$

We end this section with a bound on distinguished elements over finite fields,
which will be used in the Selberg sieve in Section 4.

###### Proposition 2.3

Let $p\geq 7$ be a prime. Then the number $d_{p}$ of elements $B\in
W({\mathbb{F}}_{p})$ with $f_{B}\in U({\mathbb{F}}_{p})$ and are not
${\mathbb{F}}_{p}$-distinguished satisfies

$\frac{1}{16}p^{8}+O(p^{7})\leq d_{p}\leq\frac{3}{4}p^{8}+O(p^{7}).$

Proof: Over the finite field ${\mathbb{F}}_{p}$, every orbit is
${\mathbb{F}}_{p}$-soluble. Moreover, for any monic quartic polynomial
$f(x)\in{\mathbb{F}}_{p}[x]$ with nonzero discriminant, the number
$\\#J_{f}({\mathbb{F}}_{p})/2J_{f}({\mathbb{F}}_{p})$ of
${\mathbb{F}}_{p}$-orbits with invariant polynomial $f$ equals the size
$\\#J_{f}[2]({\mathbb{F}}_{p})$ of any stabilizer with invariant polynomial
$f$. Hence, the number of $B\in W({\mathbb{F}}_{p})$ with $f_{B}=f$ equals
$\\#G({\mathbb{F}}_{p})=p^{2}(p^{2}-1)^{2}.$ There are $p^{2}+O(p)$
polynomials $f\in U({\mathbb{F}}_{p})$ and so a total of $p^{8}+O(p^{7})$
elements $B\in W({\mathbb{F}}_{p})$ with $f_{B}\in U({\mathbb{F}}_{p})$.
Moreover, for any $f\in U({\mathbb{F}}_{p})$, there is at least one
${\mathbb{F}}_{p}$-distinguished orbit with stabilizer having size at most
$4$. Hence, we have the upper bound $d_{p}\leq\frac{3}{4}p^{8}+O(p^{7})$.

Consider next quartic polynomials of the form

$g_{a,b}(x):=(x-a)(x-b)(x^{2}+(a+b)x+(a^{2}+ab+b^{2}))\in
U({\mathbb{F}}_{p}).$

Since $g_{a,b}(x)$ has a linear factor, there is only one distinguished orbit
with invariant $g_{a,b}$. Moreover, we have
$2\leq\\#J_{g_{a,b}}[2]({\mathbb{F}}_{p})\leq 4$. Hence, there is at least one
non-distinguished orbit of size at least $|G({\mathbb{F}}_{p})|/4.$ It remains
to count the number of such $g_{a,b}(x)$ with nonzero discriminant, which is
equivalent to requiring that $a\neq b$, that $a$ is not a root of the
quadratic factor, and that the quadratic factor has nonzero discriminant. In
other words, we have $a\neq b$, $3(a+b/3)^{2}+(2/3)b^{2}\neq 0$ and
$3(a+b/3)^{2}+(8/3)b^{2}\neq 0$. Given any $b$, there are at least $p-5$
choices for $a$. Finally, given any $g_{a,b}$ with nonzero discriminant, we
see that $g_{a,b}=g_{a^{\prime},b^{\prime}}$ if and only if
$x^{2}+(a+b)x+(a^{2}+ab+b^{2})=(x-a^{\prime})(x-b^{\prime})$ or
$(x-a)(x-b)=(x-a^{\prime})(x-b^{\prime})$, as any other possibility
contradicts $\Delta(g_{a,b})\neq 0$. Hence, there are at least $p(p-5)/4$
quartic polynomials with nonzero discriminant of the form $g_{a,b}$ for some
$a,b\in{\mathbb{F}}_{p}.$ Therefore, we have at least
$\frac{1}{16}p^{8}+O(p^{7})$ non-distinguished elements $B$ in
$W({\mathbb{F}}_{p})$ with $f_{B}\in U({\mathbb{F}}_{p}).$ $\Box$

###  2.2 Embedding ${\mathcal{W}}_{m}^{(2)}$ into $\frac{1}{4}W({\mathbb{Z}})$

In light of Proposition 2.2, it is sufficient to prove Theorem 1.4 with
${\mathcal{W}}_{m}^{(2)}$ replaced by the set ${\mathcal{W}}_{m}^{(2),{\rm
irr}}$ of pairs $(a,b)\in{\mathcal{W}}_{m}^{(2)}$ such that
$f_{a,b}(x):=x^{4}+ax+b$ is irreducible and does not factor into a product of
quadratic polynomials conjugate over some quadratic extension of
${\mathbb{Q}}$. We prove some preliminary results in order to use the map
$\sigma_{m}$ defined in [4, §3.2].

Fix $(a,b)\in{\mathcal{W}}_{m}^{(2)}$ and fix any prime $p\mid m$. For any
$(a^{\prime},b^{\prime})\in{\mathbb{Z}}^{2}$ with $a^{\prime}\equiv a\pmod{p}$
and $b^{\prime}\equiv b\pmod{p}$, we have $f_{a^{\prime},b^{\prime}}(x)\equiv
f_{a,b}(x)\pmod{p}$. Since $p^{2}$ weakly divides $\Delta(a,b)$, we see that
$f_{a,b}(x)$ has a unique double root mod $p$. Let $r\in{\mathbb{Z}}$ be an
integer such that $f_{a,b}(x+r)=x^{4}+b_{1}x^{3}+b_{2}x^{2}+b_{3}x+b_{4}$ with
$p\mid b_{3}$ and $p\mid b_{4}.$ We claim that $p^{2}\mid b_{4}$. Note the
discriminant of a quartic polynomial is of the form

$\Delta(x^{4}+b_{1}x^{3}+b_{2}x^{2}+b_{3}x+b_{4})=b_{4}\Delta^{\prime}(b_{1},b_{2},b_{3},b_{4})+b_{3}^{2}\Delta(x^{3}+b_{1}x^{2}+b_{2}x+b_{3})$

where $\Delta^{\prime}$ is some polynomial with integer coefficients. Suppose
for a contradiction that $p^{2}\nmid b_{4}$. Then, since
$p^{2}\mid\Delta(f_{a,b}(x+r))$, we have
$p\mid\Delta^{\prime}(b_{1},b_{2},b_{3},b_{4}).$ Hence,
$p^{2}\mid\Delta(g(x))$ for any monic quartic polynomial $g(x)$ congruent to
$f_{a,b}(x+r)$. Now for any $(a^{\prime},b^{\prime})\in{\mathbb{Z}}^{2}$ with
$a^{\prime}\equiv a\pmod{p}$ and $b^{\prime}\equiv b\pmod{p}$, we have
$f_{a^{\prime},b^{\prime}}(x+r)\equiv f_{a,b}(x+r)\pmod{p}$ and so
$p^{2}\mid\Delta(f_{a^{\prime},b^{\prime}}(x+r))$. Since
$\Delta(f_{a^{\prime},b^{\prime}}(x+r))=\Delta(a^{\prime},b^{\prime})$, this
contradicts the assumption that $p^{2}$ weakly divides $\Delta(a,b).$

By the Chinese Remainder Theorem, there exists an integer $r$ such that
$f_{a,b}(x+r)=x^{4}+c_{1}x^{3}+c_{2}x^{2}+mc_{3}x+m^{2}c_{4}$ for some
integers $c_{1},c_{2},c_{3},c_{4}$. Consider the following matrix:

$B(c_{1},c_{2},c_{3},c_{4})=\begin{pmatrix}0&0&m&0\\\ 0&1&-c_{1}/2&0\\\
m&-c_{1}/2&c_{1}^{2}/4-c_{2}&-c_{3}/2\\\ 0&0&-c_{3}/2&-c_{4}\end{pmatrix}.$

A direct computation shows that
$f_{B(c_{1},c_{2},c_{3},c_{4})}=x^{4}+c_{1}x^{3}+c_{2}x^{2}+mc_{3}x+m^{2}c_{4}.$
We now set
$\sigma_{m}(a,b)=B(c_{1},c_{2},c_{3},c_{4})+rA_{0}\in\frac{1}{4}W({\mathbb{Z}}).$
Then $f_{\sigma_{m}(a,b)}=f_{a,b}(x)$. Note in fact that the image of
$\sigma_{m}$ lies inside $W_{00}({\mathbb{Q}})$ and the $(1,3)$-entry of any
element in the image of $\sigma_{m}$ is $m$. We summarize the above in the
following theorem.

###### Theorem 2.4

Let $m$ be any squarefree integer. There is a map
$\sigma_{m}:{\mathcal{W}}_{m}^{(2)}\rightarrow\frac{1}{4}W({\mathbb{Z}})$ such
that $f_{\sigma_{m}(a,b)}=f_{a,b}(x)$ for any
$(a,b)\in{\mathcal{W}}_{m}^{(2)}$. Moreover, the $(1,3)$-entry (respectively
the $(1,2)$-entry) of any element in $W_{00}({\mathbb{Q}})$ (respectively
$W_{01}({\mathbb{Q}})$) that is $G({\mathbb{Z}})$-equivalent to some element
in $\sigma_{m}({\mathcal{W}}_{m}^{(2),{\rm irr}})$ equals $p$ in absolute
value.

##  3 Averaging and counting in the cusp

Let $\mathcal{L}_{M}$ denote the set of elements in
$\frac{1}{4}W({\mathbb{Z}})$ that are $G({\mathbb{Z}})$-equivalent to some
elements in $\sigma_{m}({\mathcal{W}}_{m,M}^{(2),{\rm irr}})$ for some prime
$p>M$. Write $N(\mathcal{L}_{M},X)$ for the number of $G({\mathbb{Z}})$-orbits
in $\mathcal{L}_{M}$ having height at most $X$. Theorem 2.4 implies that
Theorem 1.4 follows from the following bound on $N(\mathcal{L}_{M},X)$.

###### Theorem 3.1

For any positive real number $M$ and any $\epsilon>0$, we have

$N(\mathcal{L}_{M},X)=O_{\epsilon}\Big{(}X^{6.992+\epsilon}\Big{)}+O_{\epsilon}\left(\frac{X^{7+\epsilon}}{M}\right).$
(6)

In Section 3.1, we recall the set up in [20] for counting
$G({\mathbb{Z}})$-orbits in $\frac{1}{4}W({\mathbb{Z}})$ and divide up a
fundamental domain ${\mathcal{F}}$ for the left-multiplication action of
$G({\mathbb{Z}})$ on $G({\mathbb{R}})$ into the main body, the thick cusp, and
the distinguished cusps. In Section 3.2, we obtain bounds for the contribution
from the thick cusp and the distinguished cusps. Finally in Section 4, we
obtain bounds for the contribution from the main body and complete the proof
of Theorem 3.1.

###  3.1 Counting $G({\mathbb{Z}})$-orbits in $\frac{1}{4}W({\mathbb{Z}})$

The counting problem for the representation $W$ of $G$ is studied in [20]. In
this section, we recall some of the set up and results of [20].

Let $R$ be a fundamental domain for the action of $G({\mathbb{R}})$ on the
elements of $W({\mathbb{R}})$ having nonzero discriminant and height bounded
by $1$ as constructed in [20, §4.1]. Let ${\mathcal{F}}$ be a fundamental set
for the left-multiplication action of $G({\mathbb{Z}})$ on $G({\mathbb{R}})$
obtained using the Iwasawa decomposition of $G({\mathbb{R}})$. More
explicitly, we have

$G({\mathbb{R}})=N({\mathbb{R}})TK,$

where $N$ is a unipotent group consisting of lower triangular matrices, $K$ is
compact, and $T$ is the split torus of $G$ given by

$T=\left\\{\left(\begin{array}[]{cccc}t_{1}^{-1}&&&\\\ &t_{2}^{-1}&&\\\
&&t_{2}&\\\ &&&t_{1}\end{array}\right)\right\\}.$

We also make the following change of variables: set

$s_{1}=t_{1}/t_{2},\quad s_{2}=t_{1}t_{2}.$

We denote an element of $T$ with coordinates $t_{i}$ (resp. $s_{i}$) by $(t)$
(resp. $(s)$). We may take ${\mathcal{F}}$ to be contained in a Siegel set,
i.e., contained in $N^{\prime}T^{\prime}K$, where $N^{\prime}$ consists of
elements in $N({\mathbb{R}})$ whose entries are absolutely bounded and
$T^{\prime}\subset T$ consists of elements in $(s)\in T$ with $s_{1}\geq c$
and $s_{2}\geq c$ for some positive constant $c$.

For any $h\in G({\mathbb{R}})$, since ${\mathcal{F}}h$ remains a fundamental
domain for the action of $G({\mathbb{Z}})$ on $G({\mathbb{R}})$, the set
$({\mathcal{F}}h)\cdot(XR)$ (when viewed as a multiset) is a finite cover of a
fundamental domain for the action of $G({\mathbb{Z}})$ on the elements in
$W({\mathbb{R}})$ with nonzero discriminant and height bounded by $X$. The
degree of the cover depends only on the size of stabilizer in
$G({\mathbb{R}})$ and is thus absolutely bounded by $4$. The presence of these
stabilizers is in fact the reason we consider $({\mathcal{F}}h)\cdot(XR)$ as a
multiset. Hence, we have

$N(\mathcal{L}_{M},X)\ll\\#\big{\\{}\big{(}({\mathcal{F}}h)\cdot(XR)\big{)}\cap\mathcal{L}_{M}\big{\\}}.$
(7)

Let ${\mathcal{G}}_{1}$ be a compact left $K$-invariant set in
$G({\mathbb{R}})$ which is the closure of a nonempty open set. Averaging (7)
over $h\in{\mathcal{G}}_{1}$ and exchanging the order of integration as in [3,
Theorem 2.5], we obtain

$N(\mathcal{L}_{M},X)\ll\int_{\gamma\in{\mathcal{F}}}\\#\big{\\{}\big{(}(\gamma{\mathcal{G}}_{1})\cdot(XR)\big{)}\cap\mathcal{L}_{M}\big{\\}}d\gamma,$
(8)

where the implied constant depends only on ${\mathcal{G}}_{1}$ and $R$, and
where $d\gamma$ is a Haar measure on $G({\mathbb{R}})$ given by

$d\gamma=dn\,s_{1}^{-1}s_{2}^{-1}d^{\times}s\,dk,$

where $dn$ is a Haar measure on the unipotent group $N({\mathbb{R}})$, $dk$ is
a Haar measure on the compact group $K$, and
$d^{\times}s=s_{1}^{-1}ds_{1}\,s_{2}^{-1}ds_{2}$ is the standard Haar measure
on $\mathbb{G}_{m}^{2}$ (see [20, (20)]).

Since $s_{i}\geq c$ for every $i$, there exists a compact subset
$N^{\prime\prime}$ of $N({\mathbb{R}})$ containing $(t)^{-1}N^{\prime}\,(t)$
for all $t\in T^{\prime}$. Since $N^{\prime\prime}$, $K$, ${\mathcal{G}}_{1}$
are compact and $R$ is bounded, the set
$E=N^{\prime\prime}K{\mathcal{G}}_{1}R$ is bounded. Then we have

$N(\mathcal{L}_{M},X)\ll\int_{s_{i}\gg 1}\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}s_{1}^{-1}s_{2}^{-1}d^{\times}s.$ (9)

We denote the coordinates of $W$ by $b_{ij}$, for $1\leq i\leq j\leq 4$, and
we denote the $T$-weight of a coordinate $\alpha$ on $W$ by $w(\alpha)$. We
compute the weights of the coordinates $b_{ij}$ to be

$\begin{array}[]{rclrclrclrcl}w(b_{11})&=&s_{1}^{-1}s_{2}^{-1},&w(b_{12})&=&s_{2}^{-1},&w(b_{13})&=&s_{1}^{-1},&w(b_{14})&=&1,\\\
&&&w(b_{22})&=&s_{1}s_{2}^{-1},&w(b_{23})&=&1,&w(b_{24})&=&s_{1},\\\
&&&&&&w(b_{33})&=&s_{1}^{-1}s_{2},&w(b_{34})&=&s_{2},\\\
&&&&&&&&&w(b_{44})&=&s_{1}s_{2}.\\\ \end{array}$

Then the $(i,j)$-entry of any $B\in(s)\cdot XE$ is bounded by
$c_{0}Xw(b_{ij})$, where $c_{0}>0$ is a constant depending only on
${\mathcal{G}}_{1}$ and $R$.

We define two distinguished cusps: $T_{00}\subset T^{\prime}$ consisting of
elements $(s)$ such that $c_{0}Xw(b_{11})<1/4$ and $c_{0}Xw(b_{12})<1/4$; and
$T_{01}\subset T^{\prime}$ consisting of elements $(s)$ such that
$c_{0}Xw(b_{11})<1/4$ and $c_{0}Xw(b_{13})<1/4$. We define the thick cusp
$T_{0}$ to be the subset of $T^{\prime}$ consisting of elements $(s)$ such
that $c_{0}Xw(b_{11})<1/4$, $c_{0}Xw(b_{12})\geq 1/4$, and
$c_{0}Xw(b_{13})\geq 1/4$. We define the main body $T^{\prime\prime}$ to be
the complement $T^{\prime}\backslash(T_{00}\cup T_{01}\cup T_{0}).$ Then for
any $(s)\in T_{00}$, we have $\big{(}(s)\cdot
XE\big{)}\cap\frac{1}{4}W({\mathbb{Z}})\subset W_{00}({\mathbb{Q}})$; for any
$(s)\in T_{01}$, we have $\big{(}(s)\cdot
XE\big{)}\cap\frac{1}{4}W({\mathbb{Z}})\subset W_{01}({\mathbb{Q}})$; and for
any $(s)\in T_{0}$. we have $\big{(}(s)\cdot
XE\big{)}\cap\frac{1}{4}W({\mathbb{Z}})\subset W_{0}({\mathbb{Q}})$.

Since elements in $\mathcal{L}_{M}$ have invariant polynomials in $U$, we
express the conditions of the invariant polynomial having vanishing $x^{3}$\-
and $x^{2}$-coefficients in terms of the coordinates $b_{ij}$. The
$x^{3}$-coefficient is the anti-trace, and so we have

$b_{23}=-b_{14}.$

After replacing $b_{23}$ by $-b_{14}$, we see that the $x^{2}$-coefficient is
the following quadratic form:

$q(b_{ij}):=-b_{11}b_{44}-b_{22}b_{33}-2b_{12}b_{34}-2b_{13}b_{24}-2b_{14}^{2}.$

###  3.2 Counting in the cusps

In this section, we compute the contribution to (9) for $(s)\in T_{00}$,
$(s)\in T_{01}$ and for $(s)\in T_{0}$.

###### Proposition 3.2

For any positive real number $M$ and $\epsilon>0$, we have

$\displaystyle\int_{(s)\in T_{00}}\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}\,s_{1}^{-1}s_{2}^{-1}d^{\times}s$
$\displaystyle=$ $\displaystyle
O_{\epsilon}\Big{(}\frac{X^{7+\epsilon}}{M}\Big{)},$ (10)
$\displaystyle\int_{(s)\in T_{01}}\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}\,s_{1}^{-1}s_{2}^{-1}d^{\times}s$
$\displaystyle=$ $\displaystyle
O_{\epsilon}\Big{(}\frac{X^{7+\epsilon}}{M}\Big{)},$ (11)
$\displaystyle\int_{(s)\in T_{0}}\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}\,s_{1}^{-1}s_{2}^{-1}d^{\times}s$
$\displaystyle=$ $\displaystyle O_{\epsilon}\Big{(}X^{6+\epsilon}\Big{)}.$
(12)

Proof: Consider first the distinguished cusp $T_{00}$. In this case, any
element in $\big{(}(s)\cdot XE\big{)}\cap\mathcal{L}_{M}$ is an element in
$W_{00}({\mathbb{Q}})$ that is $G({\mathbb{Z}})$-equivalent to some element in
$\sigma_{p}({\mathcal{W}}_{p,M}^{(2),{\rm irr}})$ for some $p>M$. Hence, by
Theorem 2.4, we have $|b_{13}|>M$ for any element $B\in\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}$. In other words, we have $Xs_{1}^{-1}\gg M.$
Moreover, note that if $B\in W$ with $b_{11}=b_{12}=b_{13}=0$ or
$b_{11}=b_{12}=b_{22}=0$, then $\Delta(B)=0$. Hence we may assume that
$Xs_{1}s_{2}^{-1}\gg 1$. Let $T^{\prime}_{00}$ denote the subset of $T_{00}$
consisting of elements $(s)$ with $Xs_{1}^{-1}\gg M$ and $Xs_{1}s_{2}^{-1}\gg
1$. Note we also have $s_{1}\ll X$ and $s_{2}\ll X^{2}$ for $(s)\in
T^{\prime}_{00}$.

The quadratic form $q(b_{ij})$ when restricted to $W_{00}$ simplifies to
$q_{1}(b_{ij})=-b_{22}b_{33}-2b_{13}b_{24}-2b_{14}^{2}.$ Hence, we have

$\displaystyle\\#\big{(}(s)\cdot XE\cap\mathcal{L}_{M}\big{)}$
$\displaystyle\ll_{\epsilon}$
$\displaystyle\big{(}(Xw(b_{14}))^{1+\epsilon}(Xw(b_{22})+Xw(b_{33}))+(Xw(b_{14})Xw(b_{13})Xw(b_{24}))^{1+\epsilon}\big{)}Xw(b_{34})Xw(b_{44})$
$\displaystyle\ll$ $\displaystyle
X^{4+\epsilon}s_{1}^{2}s_{2}+X^{4+\epsilon}s_{2}^{3}+X^{5+\epsilon}s_{1}s_{2}^{2}$
$\displaystyle\ll$ $\displaystyle
X^{4+\epsilon}s_{1}^{2}s_{2}+X^{5+\epsilon}s_{1}s_{2}^{2}.$

Integrating these two terms separately gives

$\displaystyle\int_{(s)\in
T^{\prime}_{00}}X^{4+\epsilon}s_{1}^{2}s_{2}s_{1}^{-1}s_{2}^{-1}d^{\times}s$
$\displaystyle=$ $\displaystyle\int_{(s)\in
T^{\prime}_{00}}X^{4+\epsilon}s_{1}d^{\times}s\quad\\!\ll\quad\\!\frac{X^{5+\epsilon}\log
X}{M},$ $\displaystyle\int_{(s)\in
T^{\prime}_{00}}X^{5+\epsilon}s_{1}s_{2}^{2}s_{1}^{-1}s_{2}^{-1}d^{\times}s$
$\displaystyle=$ $\displaystyle\int_{(s)\in
T^{\prime}_{00}}X^{5+\epsilon}s_{2}d^{\times}s\quad\\!\ll\quad\\!\int_{(s)\in
T^{\prime}_{00}}X^{6+\epsilon}s_{1}d^{\times}s\quad\\!\ll\quad\\!\frac{X^{7+\epsilon}\log
X}{M}.$

The integral over the other distinguished cusp $T_{01}$ has the same bound via
the same analysis with $s_{1}$ and $s_{2}$ switched.

Finally, we consider the thick cusp $T_{0}$. In this case, we have
$Xs_{1}^{-1}\gg 1$ and $Xs_{2}^{-1}\gg 1$. The quadratic form $q(b_{ij})$ when
restricted to $W_{0}$ simplifies to
$q_{2}(b_{ij})=-b_{22}b_{33}-2b_{12}b_{34}-2b_{13}b_{24}-2b_{14}^{2}.$ The
above analysis shows that the number of choices for
$(b_{22},b_{33},b_{13},b_{24},b_{14})$ such that $q_{1}(b_{ij})=0$ is
$O_{\epsilon}(X^{2+\epsilon}s_{1}s_{2}^{-1}+X^{2+\epsilon}s_{1}^{-1}s_{2}+X^{3+\epsilon})$.
Multiplying it by $(Xw(b_{12})+Xw(b_{34}))Xw(b_{44})$ gives a bound of
$O_{\epsilon}(X^{4+\epsilon}s_{1}^{2}s_{2}+X^{4+\epsilon}s_{2}^{3}+X^{5+\epsilon}s_{1}s_{2}^{2})$
for the number of $B\in\big{(}(s)\cdot XE\big{)}\cap\mathcal{L}_{M}$ with
$q_{1}(b_{ij})=0$. The contribution from $q_{1}(b_{ij})\neq 0$ is

$O_{\epsilon}\big{(}(Xw(b_{22})Xw(b_{33})Xw(b_{13})Xw(b_{24})Xw(b_{14}))^{1+\epsilon}Xw(b_{44})\big{)}=O_{\epsilon}(X^{6+\epsilon}s_{1}s_{2}).$

Using the bound $s_{1}\ll X$ and $s_{2}\ll X$, we have

$\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}\ll_{\epsilon}X^{6+\epsilon}s_{1}s_{2}.$

Multiplying by $s_{1}^{-1}s_{2}^{-1}$ and integrating then give the desired
bound (12). $\Box$

For the main body $T^{\prime\prime}$, we have $Xs_{1}^{-1}s_{2}^{-1}\gg 1.$
Since both $s_{1}$ and $s_{2}$ are bounded below by some absolute constant, we
still have the bound $s_{1}\ll X$ and $s_{2}\ll X$. The above analysis gives a
bound of

$\displaystyle
O_{\epsilon}\Big{(}\big{(}(X^{2+\epsilon}s_{1}s_{2}^{-1}+X^{2+\epsilon}s_{1}^{-1}s_{2}+X^{3+\epsilon})(Xw(b_{12})+Xw(b_{34}))+X^{5+\epsilon}\big{)}(Xw(b_{11})+Xw(b_{44}))\Big{)}$
$\displaystyle=$ $\displaystyle
O_{\epsilon}\Big{(}(X^{3+\epsilon}s_{1}^{-1}s_{2}^{2}+X^{4+\epsilon}s_{2}+X^{5+\epsilon})Xs_{1}s_{2}\Big{)}$
$\displaystyle=$ $\displaystyle
O_{\epsilon}\Big{(}X^{6+\epsilon}s_{1}s_{2}\Big{)}$

for the number of $B\in\big{(}(s)\cdot XE\big{)}\cap\mathcal{L}_{M}$ with
$q_{2}(b_{ij})=0$. Multiplying by $s_{1}^{-1}s_{2}^{-1}$ and integrating give
a bound of $O_{\epsilon}(X^{6+\epsilon})$. It remains to consider the
contribution to the main body integral from the number of $B\in\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}$ with $q_{2}(b_{ij})\neq 0$. We have a trivial
bound of $O_{\epsilon}(X^{7+\epsilon})$ for the number of such $B$.

For any positive real number $\delta$, let $T^{\prime\prime}_{\delta}$ denote
the subset of $T^{\prime\prime}$ where $s_{1}\gg X^{\delta}$ or $s_{2}\gg
X^{\delta}$. Then we have

$\int_{(s)\in T^{\prime\prime}_{\delta}}\\#\big{\\{}\big{(}(s)\cdot
XE\big{)}\cap\mathcal{L}_{M}\big{\\}}\,s_{1}^{-1}s_{2}^{-1}d^{\times}s=O_{\epsilon}(X^{7-\delta+\epsilon}).$
(13)

Therefore, it remains to consider the main body integral under the additional
assumption that $s_{1}\ll X^{\delta}$ and $s_{2}\ll X^{\delta}$ where $\delta$
is some small enough positive real number.

##  4 Counting in the main body using the circle method

In this section, we consider the contribution to (9) from the main body under
the additional assumption that $s_{1}\ll X^{\delta}$ and $s_{2}\ll
X^{\delta}$. Let $V\simeq{\mathbb{A}}^{9}$ denote the subspace of $W$ cut out
by $b_{14}=-b_{23}$. Since scaling an element $B\in V({\mathbb{Q}})$ by $4$
does not affect the vanishing of $q(B)$ or whether it is
${\mathbb{Q}}$-distinguished, it is enough to count points in a box in
$V({\mathbb{Z}})$ defined by $|b_{ij}|\leq 4c_{0}Xw(b_{ij})$. The assumption
on $s_{1}$ and $s_{2}$ implies that $4c_{0}Xw(b_{ij})=O(X^{1+2\delta})$ for
all $i,j$. The goal of this section is to prove the following theorem:

###### Theorem 4.1

Let $\delta<0.01$ be a positive real number. Let ${\mathcal{B}}$ be a box in
$V({\mathbb{R}})$ defined by $|b_{ij}|\leq X_{ij}$ for $(i,j)=(1,1),$ $(1,2),$
$(1,3),$ $(1,4),$ $(2,2),$ $(2,4),$ $(3,3),$ $(3,4),$ $(4,4)$ where $X_{ij}$
are real numbers satisfying $c_{1}^{-1}X^{1-2\delta}\leq X_{ij}\leq
c_{1}X^{1+2\delta}$, $X_{14}=c_{2}X$ and

$X_{11}X_{44}=X_{22}X_{33}=X_{12}X_{34}=X_{13}X_{24}=c_{2}^{2}X^{2},$

for some positive constants $c_{1},c_{2}.$ Let $N^{\rm
dist}_{q}({\mathcal{B}})$ denote the number of ${\mathbb{Q}}$-distinguished
elements $B\in{\mathcal{B}}\cap V({\mathbb{Z}})$ with $q(B)=0$. Then

$N_{q}^{\rm
dist}({\mathcal{B}})=O_{\epsilon}\left(X^{\frac{209}{30}+\frac{137}{45}\delta+\epsilon}\right).$
(14)

Multiplying the bound (14) by $s_{1}^{-1}s_{2}^{-1}$ and integrating over
$1\ll s_{1},s_{2}\ll X^{\delta}$, combining with (13), (9) and Proposition
3.2, and setting $\delta=3/364$ then complete the proof of Theorem 3.1.

We will prove Theorem 4.1 from Theorem 4.2 below by applying a Selberg sieve.

###### Theorem 4.2

Let $m$ be an odd squarefree positive integer with $m\ll X^{1/3}$. Let
$B_{0}\in V({\mathbb{Z}})$ be an element such that $m\mid q(B_{0})$ and
$B_{0}$ is nonzero modulo $p$ for each prime factor $p$ of $m$. Let
$N_{q}({\mathcal{B}};m,B_{0})$ denote the number of $B\in{\mathcal{B}}$ such
that $B\equiv B_{0}\pmod{m}$ and $q(B)=0$.

For each $r\geq 1$, set

$C_{q}(r)=\frac{1}{r^{9}}\sum_{\begin{subarray}{c}0\leq a<r\\\
\gcd(a,r)=1\end{subarray}}\sum_{B\bmod{r}}e\left(\frac{a}{r}q(B)\right)$ (15)

Define the singular series

$\mathfrak{S}(q)=\sum_{r\geq 1}C_{q}(r),$ (16)

and for each prime $p$, the series

$\mathfrak{S}(q;p)=\sum_{\ell\geq 0}C_{q}(p^{\ell}).$ (17)

Define the singular integral

$\mathfrak{S}_{\infty}({\mathcal{B}};q)=\int_{{\mathbb{R}}}\int_{{\mathcal{B}}}e(\theta
q(B))\,dB\;d\theta,$ (18)

where $e(x)=e^{2\pi ix}$. Then,

$N_{q}({\mathcal{B}};m,B_{0})=\frac{1}{m^{8}}\left(\prod_{p\mid
m}\mathfrak{S}(q;p)^{-1}\right)\mathfrak{S}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(\frac{X^{6.85(1+2\delta)}}{m^{5.5}}\log
X\right),$ (19)

with the implied constant being absolute. All the series defined above
converge absolutely and they have positive value.

###  4.1 Proof of Theorem 4.2 using the circle method

Fix an odd squarefree $m$. For any $\alpha\in[0,1]$, let

$S_{\mathcal{B}}(\alpha;m,B_{0})=\sum_{\begin{subarray}{c}B\in{\mathcal{B}}\cap
V({\mathbb{Z}})\\\ B\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{\alpha}{m}q(B)\right).$

Then,

$N_{q}({\mathcal{B}};m,B_{0})=\int_{0}^{1}S_{\mathcal{B}}(\alpha;m,B_{0})d\alpha.$

Let $r_{1}$ and $r_{2}$ be positive real numbers, to be picked later, with
$r_{1}\ll\frac{X^{1-2\delta}}{m}$ and $r_{2}\gg X^{1+2\delta}$. Split the
interval $[0,1]$ into the major arc $\mathfrak{M}$ and the minor arc
$\mathfrak{m}=[0,1]\setminus\mathfrak{M}$, where

$\mathfrak{M}=\left\\{\alpha:\left|\alpha-\frac{a}{r}\right|\leq\frac{1}{rr_{2}},\gcd(a,r)=1,0\leq
a<r\leq r_{1}\right\\}.$

####  4.1.1 Major arc estimate

We estimate first the major arc integral

$\int_{\mathfrak{M}}S_{\mathcal{B}}(\alpha;m,B_{0})d\alpha=\sum_{r\leq
r_{1}}\sum_{\begin{subarray}{c}0\leq a<r\\\
\gcd(a,r)=1\end{subarray}}\int_{|\theta|\leq\frac{1}{rr_{2}}}S_{\mathcal{B}}\left(\frac{a}{r}+\theta;m,B_{0}\right)d\theta.$

Fix some $\alpha=\frac{a}{r}+\theta\in\mathfrak{M}$, where
$|\theta|\leq\frac{1}{rr_{2}}$. We have

$\displaystyle S_{\mathcal{B}}(\alpha;m,B_{0})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}B_{1}\bmod{rm}\\\ B_{1}\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{a}{rm}q(B_{1})\right)\sum_{\begin{subarray}{c}B\in{\mathcal{B}}\cap
V({\mathbb{Z}})\\\ B\equiv
B_{1}\bmod{rm}\end{subarray}}e\left(\frac{\theta}{m}q(B)\right)$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}B_{1}\bmod{rm}\\\
B_{1}\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{a}{rm}q(B_{1})\right)\sum_{B^{\prime}\in{\mathcal{B}}^{\prime}\cap
V({\mathbb{Z}})}e\left(\frac{\theta}{m}q(rmB^{\prime}+B_{1})\right),$

where ${\mathcal{B}}^{\prime}=\\{B^{\prime}\in V({\mathbb{R}})\colon
rmB^{\prime}+B_{1}\in{\mathcal{B}}\\}$ is another box. To compute the
exponential sum over a box, we use the following result from [15, Proposition
8.7].

###### Lemma 4.3

Let $f(x)$ be a real function on an interval $[a,b]$ such that
$|f^{\prime}(x)|\leq\frac{1}{2}$ for all $x\in(a,b)$. Suppose further that
$f^{\prime\prime}(x)\geq 0$ on $(a,b)$ or that $f^{\prime\prime}(x)\leq 0$ on
$(a,b)$. Then,

$\sum_{a<n<b}e(f(n))=\int_{a}^{b}e(f(x))\,dx+O(1),$

with the implied constant being absolute.

We note that [15, Proposition 8.7] requires that $f^{\prime\prime}(x)>0$, but
the same proof applies when $f^{\prime\prime}(x)\geq 0$ or when
$f^{\prime\prime}(x)\leq 0.$ The following multivariable version also follows
immediately.

###### Lemma 4.4

Let $f(x_{1},\ldots,x_{\ell})$ be a real function on a box
${\mathcal{R}}=\prod_{i}[a_{i},b_{i}]$ such that $|\frac{\partial f}{\partial
x_{i}}(x)|\leq\frac{1}{2}$ on ${\mathcal{R}}$ for all $i=1,\ldots,\ell$.
Suppose for any $i=1,\ldots,\ell$ and for any fixed $x_{j}\in(a_{j},b_{j})$
for all $j\neq i$, the second partial derivative
$\frac{\partial^{2}f}{\partial x_{i}^{2}}(x)$ as a function of $x_{i}$ is
either non-negative on $(a_{i},b_{i})$ or non-positive on $(a_{i},b_{i})$.
Then

$\sum_{n\in{\mathcal{R}}\cap{\mathbb{Z}}^{\ell}}e(f(n))=\int_{\mathcal{R}}e(f(x))\,dx+O({\rm
max}\\{{\rm Vol}(\bar{{\mathcal{R}}}),1\\}),$

where ${\rm Vol}(\bar{{\mathcal{R}}})$ denotes the greatest $d$-dimensional
volume of any projection of ${\mathcal{R}}$ onto a coordinate subspace
obtained by equating $\ell-d$ coordinates to zero, where $d$ takes all values
from $1$ to $\ell-1$. The implied constant depends only on $\ell$.

We apply Lemma 4.4 to the box ${\mathcal{B}}^{\prime}$ and the quadratic
polynomial $f(b_{ij})=\frac{\theta}{m}q(rmB^{\prime}+B_{1})$ viewed as a
function in the coordinates of $B^{\prime}$. The partial derivative of $f$
with respect to $b_{ij}$ equals $\theta r\frac{\partial q}{\partial
b_{ij}}(rmB^{\prime}+B_{1})$ which is bounded by $c_{3}c_{1}\theta
rX^{1+2\delta}$ where $c_{3}$ is a constant depending only on $q$ (and equals
$2$ in this case). Hence, we can bound the first order partial derivatives by
$\frac{1}{2}$ by taking $r_{2}\geq 2c_{3}c_{1}X^{1+2\delta}$. The second
partial derivative of $f$ with respect to any $b_{ij}$ is a constant since $f$
is quadratic. Finally, the side lengths of ${\mathcal{B}}^{\prime}$ are of the
form $2X_{ij}/(rm)\gg X^{1-2\delta}/(r_{1}m)\gg 1$ by the assumption on
$r_{1}$. Hence, we have

$\displaystyle\sum_{B^{\prime}\in{\mathcal{B}}^{\prime}\cap
V({\mathbb{Z}})}e\left(\frac{\theta}{p}q(rmB^{\prime}+B_{1})\right)$
$\displaystyle=$
$\displaystyle\int_{{\mathcal{B}}^{\prime}}e\left(\frac{\theta}{m}q(rmB^{\prime}+B_{1})\right)dB^{\prime}+O\left(\left(\frac{X^{1+2\delta}}{rm}\right)^{8}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{r^{9}m^{9}}\int_{{\mathcal{B}}}e\left(\frac{\theta}{m}q(B)\right)dB+O\left(\left(\frac{X^{1+2\delta}}{rm}\right)^{8}\right).$

Summing over the $r^{9}$ possible $B_{1}$’s then gives

$S_{\mathcal{B}}(\alpha;m,B_{0})=c_{q}(a;r,m,B_{0})\int_{{\mathcal{B}}}e\left(\frac{\theta}{m}q(B)\right)dB+O\left(\frac{rX^{8(1+2\delta)}}{m^{8}}\right),$
(20)

where

$c_{q}(a;r,m,B_{0})=\frac{1}{r^{9}m^{9}}\sum_{\begin{subarray}{c}B_{1}\bmod{rm}\\\
B_{1}\equiv B_{0}\bmod{m}\end{subarray}}e\left(\frac{a}{rm}q(B_{1})\right).$

In the light of (15), we define for any integer $r\geq 1$ and any integer $a$
coprime to $r$,

$c_{q}(a;r)=\frac{1}{r^{9}}\sum_{B\bmod{r}}e\left(\frac{a}{r}q(B)\right).$

###### Lemma 4.5

If $\gcd(r,m)=1$, then $\displaystyle
c_{q}(a;r,m,B_{0})=\frac{1}{m^{9}}c_{q}(a;r)$. Otherwise
$c_{q}(a;r,m,B_{0})=0$.

Proof: We consider the case $\gcd(r,m)=1$ first. Let $\bar{m}$ be any integer
such that $m\bar{m}\equiv 1\pmod{r}$. For any integer $n$ divisible by $m$, we
have $\frac{a}{rm}n\equiv\frac{a\bar{m}}{r}n\pmod{1}$. Suppose now
$B_{1},B^{\prime}\in V({\mathbb{Z}})$ with $B_{1}\equiv B_{0}\pmod{m}$ and
$B_{1}\equiv B^{\prime}\pmod{r}$. Then $q(B_{1})\equiv q(B_{0})\equiv
0\pmod{m}$ and $q(B_{1})\equiv q(B^{\prime})\pmod{r}$ and so

$e\left(\frac{a}{rm}q(B_{1})\right)=e\left(\frac{a\bar{m}}{r}q(B_{1})\right)=e\left(\frac{a\bar{m}}{r}q(B^{\prime})\right).$

Since $m$ and $r$ are coprime, we have by the Chinese Remainder Theorem,

$\sum_{\begin{subarray}{c}B_{1}\bmod{rm}\\\ B_{1}\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{a}{rm}q(B_{1})\right)=\sum_{B^{\prime}\bmod{r}}e\left(\frac{a}{r}q(B^{\prime})\right).$

Dividing by $r^{9}m^{9}$ gives us $\displaystyle
c_{q}(a;r,m,B_{0})=\frac{1}{m^{9}}c_{q}(a;r)$.

Now, we consider the case $\gcd(r,m)>1$. Suppose that $p$ is a prime dividing
$\gcd(r,m)$. We rewrite the sum as

$\sum_{\begin{subarray}{c}B_{1}\bmod{rm}\\\ B_{1}\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{a}{rm}q(B_{1})\right)=\sum_{\begin{subarray}{c}B^{\prime}\bmod{rm/p}\\\
B^{\prime}\equiv
B_{0}\bmod{m}\end{subarray}}\sum_{B_{0}^{\prime}\bmod{p}}e\left(\frac{a}{rm}q\left(\frac{rm}{p}B_{0}^{\prime}+B^{\prime}\right)\right).$

Given $v,w\in V$, we write $\langle v,w\rangle=q(v+w)-q(v)-q(w)$ for the
associated bilinear form. Hence, we have

$q\left(\frac{rm}{p}B_{0}^{\prime}+B^{\prime}\right)=q(B^{\prime})+\frac{rm}{p}\langle
B^{\prime},B_{0}^{\prime}\rangle+\frac{r^{2}m^{2}}{p^{2}}q(B_{0}^{\prime}).$

Since $rm\mid\frac{r^{2}m^{2}}{p^{2}}$, the inner sum equals

$\sum_{B_{0}^{\prime}\bmod{p}}e\left(\frac{a}{rm}\left(q(B^{\prime})+\frac{rm}{p}\langle
B^{\prime},B_{0}^{\prime}\rangle\right)\right)=e\left(\frac{a}{rm}q(B^{\prime})\right)\sum_{B_{0}^{\prime}\bmod{p}}e\left(\frac{a}{p}\langle
B^{\prime},B_{0}^{\prime}\rangle\right).$

Since $B^{\prime}\equiv B_{0}$ is nonzero modulo $p$ and $q$ is non-degenerate
modulo $p$ for $p\geq 3$, the linear form $\langle
B^{\prime},*\rangle:V({\mathbb{F}}_{p})\to{\mathbb{F}}_{p}$ is nonzero.
Moreover, $a$ is coprime to $p$ since $p\mid r$ and $a$ is coprime to $r$.
Therefore, the above exponential sum vanishes and as a result,
$c_{q}(a;r,m,B_{0})=0$. $\Box$

Integrating (20) over the arc $|\theta|\leq\frac{1}{rr_{2}}$ and summing over
$a$ and $r$ now give

$\displaystyle\int_{\mathfrak{M}}S(\alpha;m,B_{0})d\alpha$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}r\leq r_{1}\\\
\gcd(r,m)=1\end{subarray}}\sum_{\begin{subarray}{c}0\leq a<r\\\
\gcd(a,r)=1\end{subarray}}\frac{1}{m^{9}}c_{q}(a;r)\int_{|\theta|\leq\frac{1}{rr_{2}}}\int_{{\mathcal{B}}}e\left(\frac{\theta}{m}q(B)\right)dB\;d\theta+O\left(\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}\right)$
(21) $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}r\leq r_{1}\\\
\gcd(r,m)=1\end{subarray}}\frac{1}{m^{8}}C_{q}(r)\int_{|\theta|\leq\frac{1}{mrr_{2}}}\int_{{\mathcal{B}}}e(\theta
q(B))\,dB\;d\theta+O\left(\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}\right),$

Our aim is to replace the above truncated sum by the singular series

$\mathfrak{S}_{m}(q)=\sum_{\gcd(r,m)=1}\frac{1}{m^{8}}C_{q}(r)$ (22)

and the above integral by the singular integral
$\mathfrak{S}_{\infty}({\mathcal{B}};q)$. To this end, we prove the following
bounds:

###### Lemma 4.6

With notations as above, we have:

  1. (a)

for all $r\geq 1$,

$|C_{q}(r)|\leq 4r^{-7/2};$ (23)

  2. (b)

for all $\theta\neq 0$,

$\int_{\mathcal{B}}e(\theta q(B))\,dB\ll{\rm min}\\{X^{9},|\theta|^{-9/2}\\}.$
(24)

  3. (c)

the singular integral

$\mathfrak{S}_{\infty}({\mathcal{B}};q)=\int_{\mathbb{R}}\int_{\mathcal{B}}e(\theta
q(B))\,dBd\theta\ll X^{7}.$ (25)

The above implied constants depend only on $q$ (which is fixed).

Proof: We prove first the bound

$\sum_{B\bmod{r}}e\left(\frac{a}{r}q(B)\right)\leq 8r^{9/2}.$ (26)

Also, we prove the bound with the constant $8$ replaced by $\sqrt{2}$ for $r$
odd. Recall that
$q(b_{ij})=-b_{11}b_{44}-b_{22}b_{33}-2b_{12}b_{34}-2b_{13}b_{24}-2b_{14}^{2}$.
Hence (26) follows from

$\sum_{x,y\bmod{r}}e\left(\frac{a}{r}xy\right)=\gcd(a,r)r,\quad\left|\sum_{x\bmod{r}}e\left(\frac{a}{r}x^{2}\right)\right|\leq(2\gcd(a,r)r)^{1/2},$

where $\gcd(a,r)\mid 2$. Note that the second sum is a standard quadratic
Gauss sum and the bound follows, for example, from [2, §1.3–§1.6]. Thus, for
$r$ odd, $|C_{q}(r)|\leq\sqrt{2}r^{-9/2}\phi(r)\leq\sqrt{2}r^{-7/2}$, where
$\phi(r)$ is the Euler’s totient function. Meanwhile, for $r$ even, we have
$|C_{q}(r)|\leq 8r^{-9/2}\phi(r)\leq 4r^{-7/2}$. This proves (23).

Next we prove the bound (24). The $X^{9}$ bound is trivial since ${\rm
Vol}({\mathcal{B}})\ll X^{9}$. We may also assume that $\theta>0$ as the case
$\theta<0$ follows by complex conjugation. Setting $B^{\prime}=\theta^{1/2}B$,
we see that it suffices to prove the following general statement: for any box
${\mathcal{B}}^{\prime}$ centered at the origin,

$\int_{{\mathcal{B}}^{\prime}}e(q(B^{\prime}))\,dB^{\prime}\ll 1.$

Again, using the explicit formula of $q$, it reduces to proving that for any
$X,Y>0$,

$\int_{-X}^{X}\int_{-Y}^{Y}e(xy)\,dydx\ll 1,\quad\int_{-X}^{X}e(x^{2})dx\ll
1.$

The first integral can be computed as follows:

$\int_{-X}^{X}\int_{-Y}^{Y}e(xy)dx\;dy=\int_{-X}^{X}\frac{\sin(2\pi xY)}{\pi
x}dx=\int_{-XY}^{XY}\frac{\sin(2\pi x)}{\pi x}dx\ll 1.$

Now, we bound the second integral. Since $e(x^{2})$ is an even function, we
can write

$\int_{-X}^{X}e(x^{2})dx=2\int_{0}^{X}e(x^{2})dx.$

For $0<X<1$, we can use the trivial estimate. For $X\geq 1$, we use the
trivial estimate for $x\in[0,1]$ and partial integration for $x\in[1,X]$:

$\int_{0}^{X}e(x^{2})dx\ll
1+\left[\frac{e(x^{2})}{2x}\right]_{1}^{X}+\int_{1}^{X}\frac{e(x^{2})}{2x^{2}}dx\ll
1+1+\left[\frac{1}{2x}\right]_{1}^{X}\ll 1.$

Finally, by using (24), we have

$\mathfrak{S}_{\infty}({\mathcal{B}};q)\ll\int_{|\theta|\leq
X^{-2}}X^{9}d\theta+\int_{|\theta|\geq X^{-2}}|\theta|^{-9/2}d\theta\ll
X^{7},$

which is the desired bound (25). $\Box$

Note the bound (23) on $C_{q}(r)$ implies that

$|\mathfrak{S}(q)-1|\leq 4(\zeta(7/2)-1)<1$

and that

$|\mathfrak{S}(q;p)-1|\leq 4\sum_{\ell\geq
1}p^{-(7/2)\ell}=\frac{4}{p^{7/2}-1}<1.$

Hence, the series defined by (16) and (17) have positive values.

Combining the bounds (23), (24) and (25) with (21), we have

$\displaystyle\int_{\mathfrak{M}}S(\alpha;p,B_{0})d\alpha$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}r\leq r_{1}\\\
\gcd(r,m)=1\end{subarray}}\left(\frac{1}{m^{8}}C_{q}(r)(\mathfrak{S}_{\infty}({\mathcal{B}};q)+O((prr_{2})^{7/2})\right)+O\left(\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}\right)$
(27) $\displaystyle=$
$\displaystyle\left(\mathfrak{S}_{m}(q)+\sum_{r>r_{1}}O(r^{-7/2}m^{-8})\right)\mathfrak{S}_{\infty}({\mathcal{B}};q)+\sum_{r\leq
r_{1}}O(r^{-7/2}m^{-8}(mrr_{2})^{7/2})+O\left(\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}\right)$
$\displaystyle=$
$\displaystyle\mathfrak{S}_{m}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(\frac{X^{7}}{r_{1}^{5/2}m^{8}}+\frac{r_{1}r_{2}^{7/2}}{m^{9/2}}+\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}\right),$

where $\mathfrak{S}_{m}(q)$ is defined in (22).

####  4.1.2 Minor arc estimate

We now estimate the minor arc integral. Fix some
$\alpha=\frac{a}{r}+\theta\in\mathfrak{m}$, where $r_{1}<r\leq r_{2}$ and
$|\theta|\leq\frac{1}{rr_{2}}$. Then

$\displaystyle|S_{\mathcal{B}}(\alpha;m,B_{0})|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}B^{\prime},B^{\prime\prime}\in{\mathcal{B}}\cap
V({\mathbb{Z}})\\\ B^{\prime},B^{\prime\prime}\equiv
B_{0}\bmod{m}\end{subarray}}e\left(\frac{\alpha}{m}(q(B^{\prime\prime})-q(B^{\prime}))\right)$
$\displaystyle=$ $\displaystyle\sum_{B\in
V({\mathbb{Z}})}\sum_{\begin{subarray}{c}B^{\prime}\in{\mathcal{B}}\cap
V({\mathbb{Z}})\\\ B^{\prime}\equiv B_{0}\bmod{m}\\\
B^{\prime}+mB\in{\mathcal{B}}\cap
V({\mathbb{Z}})\end{subarray}}e\left(\frac{\alpha}{m}(m^{2}q(B)+m\langle
B^{\prime},B\rangle)\right),$

where the second equality follows by setting $B^{\prime\prime}=B^{\prime}+mB.$
The set of $B\in V({\mathbb{Z}})$ for which the inner sum is non-empty is
contained in the box
${\mathcal{B}}^{\prime\prime}=\frac{1}{m}({\mathcal{B}}-{\mathcal{B}})=\\{\frac{1}{m}(B^{\prime\prime}-B^{\prime})\colon
B^{\prime},B^{\prime\prime}\in{\mathcal{B}}\\}.$ Taking absolute values now
give

$|S_{\mathcal{B}}(\alpha;m,B_{0})|^{2}\leq\sum_{B\in{\mathcal{B}}^{\prime\prime}\cap
V({\mathbb{Z}})}\left|\sum_{\begin{subarray}{c}B^{\prime}\in{\mathcal{B}}\cap
V({\mathbb{Z}})\\\ B^{\prime}\equiv B_{0}\bmod{m}\\\
B^{\prime}+mB\in{\mathcal{B}}\cap
V({\mathbb{Z}})\end{subarray}}e\left(\alpha\langle
B^{\prime},B\rangle\right)\right|=\sum_{B\in{\mathcal{B}}^{\prime\prime}\cap
V({\mathbb{Z}})}\left|\sum_{\begin{subarray}{c}B_{1}\in V({\mathbb{Z}})\\\
B_{0}+mB_{1}\in{\mathcal{B}}\\\
B_{0}+mB_{1}+mB\in{\mathcal{B}}\end{subarray}}e\left(\alpha m\langle
B_{1},B\rangle\right)\right|.$ (28)

Let $b_{ij}$ denote the entries of $B$ and let $x_{ij}$ denote the entries of
$B_{1}$, then from the explicit formula for $q$, we have

$\langle
B_{1},B\rangle=-b_{11}x_{44}-b_{44}x_{11}-b_{22}x_{33}-b_{33}x_{22}-2b_{12}x_{34}-2b_{34}x_{12}-2b_{13}x_{24}-2b_{24}x_{13}-4b_{14}x_{14}.$
(29)

Each $x_{ij}$ takes all integer values within an interval, depending only on
$b_{ij}$, of length at most $2X_{ij}/m$. Hence, the inner sum in (28) factors
into a product of geometric sums. For each $(i,j)$, let $\alpha_{ij}$ denote
the integer coefficient in front of each $b_{ij}$ in (29) and let $I_{ij}$
denote the closed interval $[-2X_{ij}/m,2X_{ij}/m]$. Then we have

$|S_{\mathcal{B}}(\alpha;m,B_{0})|^{2}\ll\prod_{(i,j)}\,\sum_{b_{ij}\in
I_{ij}\cap{\mathbb{Z}}}{\rm
min}\left\\{\frac{X_{ij}}{m},||\alpha\alpha_{ij}mb_{ij}||^{-1}\right\\},$

where $||\cdot||$ is the distance to the nearest integer function.

Note we have

$|\theta\alpha_{ij}mb_{ij}|\leq\frac{4mb_{ij}}{rr_{2}}\leq\frac{8X_{ij}}{rr_{2}}\leq\frac{8c_{1}X^{1+2\delta}}{rr_{2}}\leq\frac{1}{2r}$

by taking $r_{2}\geq 16c_{1}X^{1+2\delta}$. So we have the lower bound

$||\alpha\alpha_{ij}mb_{ij}||\geq\frac{1}{2}||\frac{a}{r}m\alpha_{ij}b_{ij}||.$

Write $r_{ij}=r/\gcd(r,m\alpha_{ij})\geq r/(4m)$ and
$a_{ij}=am\alpha_{ij}/\gcd(r,m\alpha_{ij}).$ We have

$|S_{\mathcal{B}}(\alpha;m,B_{0})|^{2}\ll\prod_{(i,j)}\sum_{b_{ij}\in
I_{ij}\cap{\mathbb{Z}}}{\rm
min}\left\\{\frac{X_{ij}}{m},||\frac{a_{ij}}{r_{ij}}b_{ij}||^{-1}\right\\}.$

Now if $r_{ij}>4X_{ij}/m$, then

$\sum_{b_{ij}\in I_{ij}\cap{\mathbb{Z}}}{\rm
min}\left\\{\frac{X_{ij}}{m},||\frac{a_{ij}}{r_{ij}}b_{ij}||^{-1}\right\\}\leq\frac{X_{ij}}{m}+2\sum_{\ell=1}^{\lceil
2X_{ij}/m\rceil}\frac{r_{ij}}{\ell}\ll\frac{X_{ij}}{m}+r_{ij}\log X_{ij}.$
(30)

If $r_{ij}\leq 4X_{ij}/m$, then

$\sum_{b_{ij}\in I_{ij}\cap{\mathbb{Z}}}{\rm
min}\left\\{\frac{X_{ij}}{m},||\frac{a_{ij}}{r_{ij}}b_{ij}||^{-1}\right\\}\ll\frac{X_{ij}}{m}\frac{4X_{ij}/m}{r_{ij}}+\frac{4X_{ij}/m}{r_{ij}}\sum_{\ell=1}^{\lfloor
r_{ij}/2\rfloor}\frac{r_{ij}}{\ell}\ll\frac{X_{ij}^{2}}{rm}+\frac{X_{ij}}{m}\log
X_{ij}.$ (31)

Combining (30) and (31) then gives

$\sum_{b_{ij}\in I_{ij}\cap{\mathbb{Z}}}{\rm
min}\left\\{\frac{X_{ij}}{m},||\frac{a_{ij}}{r_{ij}}b_{ij}||^{-1}\right\\}\ll\frac{X^{2(1+2\delta)}}{rm}+r_{2}\,\log
X.$

Raising it to the power $9$ and taking square root give

$S_{\mathcal{B}}(\alpha;m,B_{0})\ll\frac{X^{9(1+2\delta)}}{r^{9/2}m^{9/2}}+r_{2}^{9/2}\,\log^{9/2}X.$

Finally, integrating over the minor arc gives

$\displaystyle\int_{\mathfrak{m}}S_{\mathcal{B}}(\alpha;m,B_{0})$
$\displaystyle\ll$ $\displaystyle\sum_{r_{1}<r\leq
r_{2}}\sum_{\begin{subarray}{c}0\leq a<r\\\
(a,r)=1\end{subarray}}\int_{|\theta|\leq\frac{1}{rr_{2}}}\left(\frac{X^{9(1+2\delta)}}{r^{9/2}m^{9/2}}+r_{2}^{9/2}\,\log^{9/2}X\right)d\theta$
(32) $\displaystyle\ll$ $\displaystyle\sum_{r_{1}<r\leq
r_{2}}\left(\frac{X^{9(1+2\delta)}}{r_{2}r^{9/2}m^{9/2}}+r_{2}^{7/2}\log^{9/2}X\right)$
$\displaystyle\ll$
$\displaystyle\frac{X^{9(1+2\delta)}}{r_{2}r_{1}^{7/2}m^{9/2}}+r_{2}^{9/2}\log^{9/2}X,$

where the last bound follows from $r_{2}\gg X^{1+2\delta}.$

####  4.1.3 Proof of Theorem 4.2

We are ready to prove Theorem 4.2. By (27) and (32), we have

$N_{q}({\mathcal{B}};m,B_{0})=\mathfrak{S}_{m}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(\frac{X^{7}}{r_{1}^{5/2}m^{8}}+\frac{r_{1}r_{2}^{7/2}}{m^{9/2}}+\frac{r_{1}^{2}X^{8(1+2\delta)}}{r_{2}m^{8}}+\frac{X^{9(1+2\delta)}}{r_{2}r_{1}^{7/2}m^{9/2}}+r_{2}^{9/2}\log^{9/2}X\right),$

where $\mathfrak{S}_{m}(q)$ is defined in (22). Since $X\gg m^{3}$ and
$\delta<0.01$, we may pick

$r_{1}=X^{\frac{2}{11}(1+2\delta)}m^{\frac{7}{11}},\qquad
r_{2}=\frac{X^{1.522(1+2\delta)}}{m^{1.223}\log X}$

so that $r_{1}m\ll X^{1-2\delta}$ and $X^{1+2\delta}\ll r_{2}\ll X^{2}$. With
this choice of $r_{1}$ and $r_{2}$, we have

$N_{q}({\mathcal{B}};m,B_{0})=\mathfrak{S}_{m}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(\frac{X^{6.85(1+2\delta)}}{m^{5.5}}\log
X\right).$

Finally, since $C_{q}$ is multiplicative (as easily verified), the singular
series $\mathfrak{S}_{m}(q)$ defined in (22) equals

$\mathfrak{S}_{m}(q)=\frac{1}{m^{8}}\left(\prod_{p\mid
m}\mathfrak{S}(q;p)^{-1}\right)\mathfrak{S}(q).$

This completes the proof of Theorem 4.2.

###  4.2 Proof of Theorem 4.1 using the Selberg sieve

For any prime $p$, we say an element $B\in V({\mathbb{F}}_{p})$ (or
$W({\mathbb{F}}_{p})$) is ${\mathbb{F}}_{p}$-_reducible_ if either
$\Delta(B)=0\in{\mathbb{F}}_{p}$ or $\Delta(B)\neq 0$ and $B$ is
${\mathbb{F}}_{p}$-distinguished in the sense of Section 2.1. Then any element
$B\in V({\mathbb{Z}})$ that is ${\mathbb{Q}}$-distinguished is
${\mathbb{F}}_{p}$-reducible for all primes $p$.

We now apply the Selberg sieve to prove Theorem 4.1. We follow the setup as in
[19, §3]. Let $z$ be a number less than $X^{1/3}$. Let $P$ be the product of
all primes $p$ with $N\leq p<z$ where $N$ is some large enough absolute
constant. For each $m\mid P$, let $a_{m}$ be the number of elements
$B\in{\mathcal{B}}\cap V({\mathbb{Z}})$ such that:

  * •

$q(B)=0$;

  * •

for any prime $p\mid\frac{P}{m}$, $B$ is ${\mathbb{F}}_{p}$-reducible;

  * •

for any prime $p\mid m$, $B$ is not ${\mathbb{F}}_{p}$-reducible.

For $m\nmid P$, we set $a_{m}=0$. Then, applying the Selberg sieve will give
us the count for

$a_{1}=\sum_{\gcd(n,P)=1}a_{n},$

which is the number of elements $B\in{\mathcal{B}}\cap V({\mathbb{Z}})$ with
$q(B)=0$ and is ${\mathbb{F}}_{p}$-reducible for all primes $p\mid P$.

For any squarefree $m\mid P$, the expression

$\sum_{n\equiv 0\bmod{m}}a_{n}$

counts the number of elements $B\in{\mathcal{B}}\cap V({\mathbb{Z}})$ such
that $q(B)=0$ and $B$ is not ${\mathbb{F}}_{p}$-reducible for any $p\mid m$.
Recall that for any prime $p$, we defined $d_{p}$ in Proposition 2.3 for the
number of $B_{0}\in W({\mathbb{F}}_{p})$ with $f_{B_{0}}\in
U({\mathbb{F}}_{p})$ and are not ${\mathbb{F}}_{p}$-distinguished, which is
the same as the number of $B_{0}\in V({\mathbb{F}}_{p})$ with $q(B_{0})=0$ and
are not ${\mathbb{F}}_{p}$-reducible. The condition that $\Delta(B_{0})\neq 0$
in ${\mathbb{F}}_{p}$ also implies that $B_{0}$ is nonzero modulo $p$. Thus,
by Proposition 2.3 and Theorem 4.2, we have

$\displaystyle\sum_{n\equiv 0\bmod{m}}a_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{8}}\prod_{p\mid
m}\Big{(}d_{p}\mathfrak{S}(q;p)^{-1}\Big{)}\mathfrak{S}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(X^{6.85(1+2\delta)}m^{2.5}\log
X\right)$ $\displaystyle=$ $\displaystyle\left(\prod_{p\mid
m}\frac{d_{p}\mathfrak{S}(q;p)^{-1}}{p^{8}}\right)\mathfrak{S}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+O\left(X^{6.85(1+2\delta)}m^{2.5}\log
X\right).$

We set $\displaystyle g(m)=\prod_{p\mid m}g(p)$ and
$u_{m}=O\left(X^{6.85(1+2\delta)}m^{2.5}\log X\right)$ for each squarefree
$m\mid P$ where

$g(p)=\frac{d_{p}\mathfrak{S}(q;p)^{-1}}{p^{8}}$

for each prime $p\mid P$. By (23), we have

$\mathfrak{S}(q;p)=1+O\Big{(}\sum_{\ell\geq
1}p^{-7\ell/2}\Big{)}=1+O(p^{-7/2}).$

Recall from Proposition 2.3, we have the bound

$\frac{1}{16}+O(p^{-1})\,\leq\,\frac{d_{p}}{p^{8}}\,\leq\,\frac{3}{4}+O(p^{-1}).$

Hence, by taking $N$ large enough, we have the bound
$\displaystyle\frac{1}{32}\leq g(p)\leq\frac{7}{8}$ for $p\geq N$.

Now, set $\displaystyle h(m)=\prod_{p\mid m}\frac{g(p)}{1-g(p)}$ for all
squarefree $m\mid P$. Let $D>1$ with $D<z$ be a real number to be picked later
and set

$H=\sum_{\begin{subarray}{c}m<\sqrt{D}\\\ m\mid P\end{subarray}}h(m).$

Then, by [15, Theorem 6.4], we have

$a_{1}=\sum_{\gcd(n,P)=1}a_{n}\leq
H^{-1}\mathfrak{S}(q)\mathfrak{S}_{\infty}({\mathcal{B}};q)+R,$

where

$|R|\leq\sum_{\begin{subarray}{c}m<\sqrt{D}\\\ m\mid
P\end{subarray}}\tau_{3}(m)u_{m}\ll_{\epsilon}X^{6.85(1+2\delta)}\log
X\sum_{m<\sqrt{D}}m^{2.5+\epsilon}\ll_{\epsilon}X^{6.85(1+2\delta)}D^{1.75+\epsilon}\log
X$

for any $\epsilon>0$.

Meanwhile, for $p$ prime, we have $\frac{1}{31}\leq h(p)\leq 8$ and so for any
$\epsilon>0$,

$H\gg\pi(\sqrt{D})\gg_{\epsilon}D^{0.5-\epsilon}.$

Thus, we get

$N_{q}^{\rm dist}({\mathcal{B}})\leq
a_{1}\ll_{\epsilon}X^{7}D^{-0.5+\epsilon}+X^{6.85(1+2\delta)}D^{1.75+\epsilon}\log
X.$

Taking $D=X^{(1/15)-(54.8/9)\delta}$ gives the desired bound (14).

##  5 Proof of Theorem 1.1 and Theorem 1.3

We prove Theorem 1.3 first. By the paragraph following Theorem 1.3, it is
enough to consider the case $\alpha=256$ and $\beta=-27$. We note that for
$(a,b)\in{\mathbb{Z}}^{2}$ with $H(a,b)<X$, there are at most $X^{\epsilon}$
integers $m$ whose square divides $\Delta(a,b)$. Hence, it is enough to prove
the bound (1) for

$\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{(a,b)\in{\mathbb{Z}}^{2}\colon
H(a,b)<X,m^{2}\mid\Delta(a,b)\\}.$

Moreover, if $m^{2}\mid\Delta(a,b)$, then we can factor $m=m_{1}m_{2}$ where
$m_{1}$ is the product of all prime factors $p$ of $m$ such that $p^{2}$
strongly divides $\Delta(a,b)$, and $m_{2}$ is the product of all prime
factors $p$ of $m$ such that $p^{2}$ weakly divides $\Delta(a,b)$. Since at
least one of $m_{1}$ or $m_{2}$ is at least $m^{\prime}$ for some squarefree
integer $m^{\prime}\geq\sqrt{m}$, we have

$\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{(a,b)\in{\mathbb{Z}}^{2}\colon
H(a,b)<X,m^{2}\mid\Delta(a,b)\\}\quad\subset\bigcup_{\begin{subarray}{c}m^{\prime}>\sqrt{M}\\\
m^{\prime}\;\mathrm{squarefree}\end{subarray}}{\mathcal{W}}_{m^{\prime}}^{(1)}\quad\cup\bigcup_{\begin{subarray}{c}m^{\prime}>\sqrt{M}\\\
m^{\prime}\;\mathrm{squarefree}\end{subarray}}{\mathcal{W}}_{m^{\prime}}^{(2)}.$

Theorem 1.3 now follows from Theorem 1.4.

Finally, we prove Theorem 1.1 using an inclusion-exclusion sieve. We have

$N(X;\alpha,\beta)=\sum_{m}\mu(m)N_{m}(X;\alpha,\beta),$

and the following individual count

$N_{m}(X;\alpha,\beta)=X^{7}m^{-4}\rho_{\alpha,\beta}(m^{2})+O(X^{4}m^{-2}\rho_{\alpha,\beta}(m^{2}))+O(\rho_{\alpha,\beta}(m^{2})).$

Since $\rho_{\alpha,\beta}(m^{2})=O(m^{2})$, we sum over $m<X^{\eta}$ for some
$\eta>0$ to get

$\displaystyle\sum_{m\leq X^{\eta}}\mu(m)N_{m}(X;\alpha,\beta)$
$\displaystyle=$ $\displaystyle X^{7}\sum_{m\leq
X^{\eta}}\frac{\rho_{\alpha,\beta}(m^{2})}{m^{4}}+O(X^{4+\eta})+O(X^{1+3\eta})$
(33) $\displaystyle=$ $\displaystyle
C(\alpha,\beta)X^{7}+O(X^{7-\eta})+O(X^{4+\eta})+O(X^{1+3\eta}).$

We take $\eta=0.1$ and apply Theorem 1.3 with $M=X^{0.1}$ to get

$N(X;\alpha,\beta)=C(\alpha,\beta)X^{7}+O(X^{6.9})+O_{\epsilon}(X^{6.9+\epsilon}+X^{6.992+\epsilon}).$

The proof of Theorem 1.1 is now complete.

### Acknowledgments

It is a pleasure to thank Manjul Bhargava for many helpful comments. The first
named author is supported by the University of Waterloo through an MURA
project. The second named author is supported by an NSERC Discovery Grant.

## References

  * [1] T. Apostal, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag (1976).
  * [2] B. Berndt, R. Evans, K. Williams, Gauss and Jacobi Sums, John Wiley and Sons, (1998)
  * [3] M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181 (2015), 191-–242.
  * [4] M. Bhargava, A. Shankar, and X. Wang, Squarefree values of polynomial discriminants I. To appear in Invent. Math.
  * [5] M. Bhargava, A. Shankar, and X. Wang, Squarefree values of polynomial discriminants II, preprint.
  * [6] J. Friedlander and H. Iwaniec, The polynomial $X^{2}+Y^{4}$ captures its primes. Ann. of Math. (2) 148 (1998), no. 3, 945–-1040.
  * [7] A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices (1998), 19, 991–1009.
  * [8] G. Greaves, Power-free values of binary forms, Quart. J. Math. Oxford Ser. (2) 43 (1992), 45–65.
  * [9] D. Heath-Brown, Power-free values of polynomials, Quart. J. Math.  64 (2013), 177–-188.
  * [10] D. Heath-Brown, Primes represented by $x^{3}+2y^{3}$, Acta Math. , 186 (2001), 1–84.
  * [11] D. Heath-Brown and X. Li, Prime values of $a^{2}+p^{4}$, Invent. Math.  208 (2017), 441-–499.
  * [12] D. Heath-Brown and B. Moroz, On the representation of primes by cubic polynomials in two variables, Proc. London Math. Soc. (3) 88 (2004), 289–312.
  * [13] C. On the power-free values of polynomials in two variables: II, J. Number Theory 129 (2009), no. 6, 1443–1455.
  * [14] C. Hooley, On the square-free values of cubic polynomials, J. Reine Angew. Math.  229 (1968), 147–154.
  * [15] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Providence, RI (2004).
  * [16] J. Kowalski, On the proportion of squarefree numbers among sums of cubic polynomials. Ramanujan J.  54 (2021), no. 2, 343–354.
  * [17] R. Murty and H. Paston, Counting squarefree values of polynomials with error term. Int. J. Number Theory 10 (2014), no. 7, 1743–-1760.
  * [18] B. Poonen, Squarefree values of multivariable polynomials, Duke Math. J.  118 no. 2, 353–373.
  * [19] A. Shankar and J. Tsimerman, Counting $S_{5}$-fields with a power saving error term, Forum of Mathematics, Sigma 2, e13 (2014).
  * [20] A. Shankar and X. Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compositio Mathematica 154 (2018). no. 1, 188–222.


# Non-universal tunneling resistance at the quantum critical point of
mesoscopic SQUIDs array

Sujit Sarkar PoornaPrajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.

###### Abstract

We calculate the tunnelling resistance at the quantum critical point of a
mesoscopic SQUIDs array in the presence of magnetic flux. We find the
analytical relation between the magnetic flux induced dissipation strength and
the Luttinger liquid parameter of the system. While the experimental finding
for the system is around $40-50$ mK, we find the behavior of the system even
at lower temperatures through the analysis of renormalization group. Apart
from the length scale dependent superconductor-insulator transition, we also
predict the evidence of length scale independent metallic state. This study
also emphasizes the importance of Co-tunelling effect.

Introduction: Josephson junction arrays have attracted considerable interest
in the recent years owing to their interesting physical properties. Currently
such arrays can be fabricated in restricted geometries both in one and two
dimensions sondhi ; suj1 ; suj2 ; granto1 ; geer ; zant ; kuo ; havi2 ; havi3
; lar . There are a few experimental findings in the field of mesoscopic
SQUIDs array. The authors of Ref. (havi3  ) and Ref. (kuo ) have fabricated
the arrays of SQUIDs junctions of different numbers in a single chip. They
have studied the current voltage characteristics of the mesoscopic SQUIDs
array in the presence of a magnetic field. They have used an external magnetic
field $(B)$ that tunes the effective Josephson coupling (${E_{J}}$) between
the nearest neighbor superconductors by the following relations
${E_{J}}={E_{J0}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$, where $\Phi$ is the
external flux and ${\Phi}_{0}$ is the flux quantum. They have found magnetic
field induced superconductor-insulator (SI) quantum phase transition at around
$40-50$ mK. When the applied magnetic flux is less than the critical value the
system shows constant saturated resistance that may arise from the source of
quantum phase slip centers (QPS) suj2 . When the magnetic flux exceeds the
critical value the system turns into the insulating phase due to the flux
induced Coulomb blockade of Cooper pair tunneling. In this letter, we study
the behavior of the system at very low temperature where there is no
experimental findings, i.e., around few mili Kelvin havi3 ; kuo . The
experimental result shows length scale dependent superconductor-insulator (SI)
transition at the quantum resistance but this is not the whole picture, we
find the length scale independent metallic phase. We calculate the tunneling
resistance at the quantum critical point and also show explicitly the
importance of Co-tunneling effect to get the correct physical behavior of the
system.
Analytical relation between the flux induced dissipative strength and the
Luttinger liquid parameter: Here we derive analytical expression of magnetic
flux induced dissipative strength ($\alpha$) in terms of the interactions of
the system. At first we derive the dissipative action/partition function of a
quantum impurity system. We will see that the analytical structure of this
dissipative action is identical with the mesoscopic SQUIDs array.
Here we consider that the impurity is present at the origin where the fermions
scatter from the left to the right and vice versa. The Hamiltonian describing
this process is ${H_{1}}={V_{0}}({R^{\dagger}}(0){L}(0)+h.c)={V_{0}}\int
dx{\delta}(x)cos{\theta}(x).$ The total Hamiltonian of the system
$H={H_{0}}~{}+~{}{V_{0}}\int dx{\delta}(x)cos{\theta}(x,\tau)$
${H_{0}}=\frac{1}{2\pi}\int{uK{({{\partial}_{x}}{\theta}(x,\tau))}^{2}~{}+~{}\frac{u}{K}{({{\partial}_{x}}{\phi}(x,\tau))}^{2}},$
corresponding Lagrangian of the system is

$\displaystyle{L}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
K}\int{\frac{1}{u}{({{\partial}_{\tau}}{\phi}(x,\tau))}^{2}~{}+~{}{u}{({\partial}{\phi}(x,\tau))}^{2}}$
(1) $\displaystyle+{V_{0}}\int dx\delta(x)cos(\theta(x,\tau))={L_{0}}+{L_{1}}$

where ${L_{0}}$ and ${L_{1}}$ are the non-interacting and the interacting part
of the Lagrangian and $K$ is the Luttinger liquid parameter of the system. The
only non-linear term in this Lagrangian is expressed by the field
$\theta(x=0)$. We would like to express the action of the system as an
effective action by integrating the field $\theta(x\neq 0)$. Therefore one may
consider $\theta(x\neq 0)$ as a heat bath, which yields the source of
dissipation in the system. The constraint condition for the integration is
${\theta}(\tau)={\theta}(x=0,\tau)$. We can write the partition function.

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int
D\theta(x,\tau)e^{-\int_{0}^{\beta}Ld{\tau}}$ (2) $\displaystyle=\int
D\theta(x,\tau)D\theta(\tau)\delta(\theta(\tau)-{\theta}(0,\tau))e^{-\int_{0}^{\beta}Ld{\tau}}$

Here we use the standard trick of introducing the Lagrange multiplier with
auxiliary field $\lambda(\tau)$.
$Z=\int{D\theta(x,\tau)}\int{D\theta(\tau)}\int
D{\lambda(\tau)}e^{-\int_{0}^{\beta}({L_{0}+L_{1}})d{\tau}}$
$e^{i\int_{0}^{\beta}d{\tau}{\lambda(\tau)}~{}(\theta(0,\tau)-{\theta}(\tau))}$
$Z=\int{D\theta(\tau)}e^{-\int_{0}^{\beta}{L_{1}}d{\tau}}\int
D{\lambda(\tau)}e^{-i{\lambda(\tau)}{\theta}(\tau)}$
$\int{D\theta(x,\tau)}e^{\int_{0}^{\beta}(-{L_{0}}+i{\lambda}({\tau}){\theta}(0,\tau)){d\tau}}$
The Fourier transform of the first term of Eq.(2) is
${L_{0}}~{}=~{}\sum_{q}\sum_{i{\omega}_{n}}\frac{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}{2\pi
Kv}{\theta}(q,i{{\omega}_{n}}){\theta}(-q,-i{{\omega}_{n}})$
At first we would like to calculate the integral:
$\int_{0}^{\beta}d{\tau}[{L_{0}}-i{\lambda(\tau)}{\theta(0,\tau)}]$, we can
write this term as
$\sum_{q}\sum_{i{\omega}_{n}}\frac{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}{2\pi
Kv}{\theta}(q,i{{\omega}_{n}}){\theta}(-q,-i{{\omega}_{n}})$
$-\frac{1}{2\sqrt{L}}({\lambda}(i{\omega}_{n}){\theta}(-q,-i{{\omega}_{n}})+{\lambda}(-i{\omega}_{n}){\theta}(q,i{{\omega}_{n}})$.
This integral appears in the integral $\theta(x,\tau)$. This integral is
quadratic in $\theta$. Now we would like to perform the Gaussian integration
by completing the square. We can write the result as
$\frac{-1}{2L}\sum_{i{{\omega}_{n}},q}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}$.
In the infinite length limit one can write, $\frac{1}{2L}\sum_{q}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}~{}=~{}\int\frac{dq}{2\pi}\frac{\pi
Kv}{{{\omega}_{n}}^{2}~{}+~{}{v}^{2}{q}^{2}}=\frac{\pi K}{4{{\omega}_{n}}}$.
Now we would like to append this result of integration in the second integral
of $Z$, i.e., the integral over ${\lambda}$. One can write the integrand as
$\sum_{i{{\omega}_{n}}}(-\frac{\pi
K}{4{{\omega}_{n}}}{\lambda}(i{\omega}_{n}){\lambda}(-i{\omega}_{n})$
$+\frac{i}{2}({\lambda}(i{\omega}_{n}){\theta}(-q,-i{{\omega}_{n}})+{\lambda}(-i{\omega}_{n}){\theta}(q,i{{\omega}_{n}}).$
This integral is again the quadratic integral of $\lambda$, therefore the
Gaussian integral can be performed by completing the square. We get after the
integration $\sum_{i{{\omega}_{n}}}\frac{{\omega}_{n}}{\pi
K}\theta(i{\omega}_{n})\theta(-i{\omega}_{n})$. From these analytical
expression, we obtain the effect of bath on ${\theta}(\tau)$. The appearance
of the factor ${{\omega}_{n}}$ signifies the dissipation. Therefore the
effective action reduces to

$S~{}=~{}\sum_{i{{\omega}_{n}}}\frac{{\omega}_{n}}{\pi
K}\theta(i{\omega}_{n})\theta(-i{\omega}_{n})+\int dx{V_{0}}cos{\theta}(\tau)$
(3)

The above action implies that a single particle moving in the potential
${V_{0}}cos{\theta}(\tau)$ subject to dissipation with friction constant ,
$\frac{1}{\pi K}$.

Now we calculate the dissipative action of mesoscopic SQUIDs array. We have
already proved in Ref. suj2  that the strong coupling phase of the system is
consistent with the experimental findings. Here we calculate the effective
partition function of our system in the strong coupling phase. Our starting
point is the Calderia-Legget cal  formalism. Following reference we write the
action as

$S_{1}~{}=~{}S_{0}~{}+~{}\frac{{\alpha}^{\prime}}{4\pi
T}~{}\sum_{m}{{\omega}_{m}}{|{\theta}_{m}|}^{2}.$ (4)

Here, $S_{1}$ is the standard action for the system with tiled wash-board
potential sch1 ; su1 ; schon  to describe the dissipative physics for low
dimensional superconducting tunnel junctions, $S_{0}$ is the action for non-
dissipative part,
${\alpha}^{\prime}~{}=~{}\frac{R_{Q}}{R_{s}}cos|\frac{\pi\phi}{{\phi}_{0}}|$
(the extra cosine factor which we consider in ${\alpha}^{\prime}$ is entirely
new in the literature to probe the effect of an external magnetic flux and is
also consistent physically), the Matsubara frequency
${\omega}_{m}~{}=~{}\frac{2\pi}{\beta}m$ and $R_{Q}$ ($=6.45k\Omega$) is the
quantum resistance and $R_{s}$ is the tunnel junction resistance, $\beta$ is
the inverse temperature. In the strong potential, tunneling between the minima
of the potential is very small. In the imaginary time path integral formalism,
tunneling effect in the strong coupling limit can be described in terms of
instanton physics. In this formalism, it is convenient to characterize the
profile of $\theta$ in terms of its time derivative,

$\frac{d\theta{(\tau)}}{d{\tau}}~{}=~{}\sum_{i}e_{i}h(\tau-{\tau}_{i}),$ (5)

where $h(\tau-{\tau}_{i})$ is the time derivative at time $\tau$ of one
instanton configuration. ${\tau}_{i}$ is the location of the i-th instanton,
$e_{i}=1$ and $-1$ is the topological charge of instanton and anti-instanton
respectively. Integrating the function over $h$ from $-\infty$ to $\infty$,
$\int_{-\infty}^{\infty}d\tau
h(\tau)={\theta}(\infty)-{\theta}({-\infty})=2\pi.$ It is well known that the
instanton (anti-instanton) is almost universally constant except for a very
small region of time variation. In the QPS process the amplitude of the
superconducting order parameter is zero only in a very small region of space
as a function of time and the phase changes by $\pm 2\pi$. So our system
reduce to a neutral system consisting of equal number of instanton and anti-
instanton. One can find the expression for ${\theta}(\omega)$, after the
Fourier transform to the both sides of Eq. 5 which yields
${\theta}(\omega)=\frac{i}{\omega}\sum_{i}e_{i}h(i\omega)e^{i{\omega}{\tau}_{1}}$
. Now we substitute this expression for ${\theta}(\omega)$ in the second term
of Eq.4 and finally we get this term as
$\sum_{ij}~{}F({\tau}_{i}-{\tau}_{j}){e_{i}}{e_{j}}$, where
$F({\tau}_{i}-{\tau}_{j})=\frac{\pi\alpha}{\beta}\sum_{m}~{}\frac{1}{{|{\omega}_{m}}|}e^{i{\omega}({\tau}_{i}-{\tau}_{j})}$
$\simeq ln({\tau}_{i}-{\tau}_{j})$. We obtain this expression for very small
values of ${\omega}$ ( $\rightarrow 0$). So $F({\tau}_{i}-{\tau}_{j})$
effectively represents the Coulomb interaction between the instanton and anti-
instanton. This term is the main source of dissipation physics of the system.
Following the standard prescription of imaginary time path integral formalism,
we can write the partition function of the system as suj2 ; lar ; gia1 ; kane
; zai1 ; furu .

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}{z}^{n}\sum_{e_{i}}\int_{0}^{\beta}d{{\tau}_{n}}\int_{0}^{{\tau}_{n-1}}d{\tau}_{n-1}...$
(6)
$\displaystyle\int_{0}^{{\tau}_{2}}d{{\tau}_{1}}e^{-F({\tau}_{i}-{\tau}_{j}){e_{i}}{e_{j}}}.$

We would like to express the partion function in terms of integration over
auxiliary field, $q{(\tau)}$. After some extensive analytical calculations, we
get

$Z=\int
Dq(\tau)e^{(-\sum_{i{\omega}_{n}}\frac{|{{\omega}_{n}}|}{4\pi\alpha}q(i{\omega}_{n})q(-i{\omega}_{n})){+(2z\int_{0}^{\beta}d{\tau}cosq(\tau))}}.$
(7)

Thus by comparing the first term of the action of Eq. (3) and the first term
of exponential of Eq. (17), we conclude that the dissipative strength $\alpha$
and the Luttinger liquid parameter of the system are related by the relation,
$K=4\alpha$.
Quantum field theoretical study of model Hamiltonian of the system and
explicit derivation of dissipative strength:
In our previous study, we have shown explicitly that the mesoscopic SQUIDs
array is equivalent to the array of superconducting quantum dots (SQD) with
modulated Josephson coupling. We first write the model Hamiltonian of SQD with
nearest neighbor (NN) Josephson coupling and also with the presence of the on-
site and NN charging energy between SQD,

$H~{}=~{}H_{J1}~{}+~{}H_{EC0}~{}+~{}H_{EC1}.$ (8)

Now we would like to recast our basic Hamiltonians in the spin language. This
is valid when ${E_{C0}}>>{E_{J1}}$. It is also observed from the experiments
that the quantum critical point exists for larger values of the magnetic
field, when the magnetic field induced Coulomb blockade phase is more
prominent than the $E_{J}$ induced SC phase. Thus our theoretical model is
consistent with the experimental findings. During this mapping process we
follow Ref. (lar ; suj1 ; suj2 ).
$H_{J1}~{}=~{}-2~{}E_{J1}\sum_{i}({S_{i}}^{\dagger}{S_{i+1}}^{-}+h.c)$,
$H_{EC0}~{}=~{}{E_{C0}}\sum_{i}{S_{i}}^{Z}.$
$H_{EC1}~{}=~{}4E_{Z1}\sum_{i}{S_{i}}^{Z}~{}{S_{i+1}}^{Z},$
${E_{J1}}={E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$ . At the Coulomb blocked
regime, the higher order expansion leads to the virtual state with energies
exceeding $E_{C0}$. In this second order process, the effective Hamiltonian
reduces to the subspace of charges $0$ and $2$, and takes the form lar ; suj1
; suj2 ,

$H_{C}~{}=~{}-\frac{3{E_{J1}}^{2}}{4E_{C0}}\sum_{i}{{S_{i}}^{Z}}{{S_{i+1}}^{Z}}~{}-~{}\frac{{E_{J1}}^{2}}{E_{C0}}\sum_{i}({S_{i+2}}^{\dagger}{S_{i}}^{-}+h.c).$
(9)

With this corrections $H_{C1}$ become
$H_{EC1}~{}\simeq~{}(4E_{Z1}~{}-~{}\frac{3{E_{J1}}^{2}}{4E_{C0}})~{}\sum_{i}{{S_{i}}^{Z}}{{S_{i+1}}^{Z}}$
One can express spin chain systems to as spinless fermions systems through the
application of Jordan-Wigner transformation. We have transformed all
Hamiltonians in spinless fermions which we have not present in this letter. In
order to study the continuum field theory of these Hamiltonians, we recast the
spinless fermions operators in terms of field operators by a relation gia1 .
${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields
of right- and the left-moving fermions respectively. We would like to express
the fermionic fields in terms of bosonic field by the relation
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))}$,
$r$ is denoting the chirality of the fermionic fields, right (1) or left
movers (-1). The operators $U_{r}$ preserve the anti-commutivity of fermionic
fields. $\phi$ field corresponds to the quantum fluctuations (bosonic) of spin
and $\theta$ is the dual field of $\phi$. They are related by the relations
${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$. The total Hamiltonian is

$\displaystyle H$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{4E_{Z12}}{{(2\pi\alpha)}^{2}}\int~{}dx:cos(4\sqrt{K}\phi(x)):$
(10)
$\displaystyle+\frac{E_{C0}}{\pi\alpha}\int({{\partial}_{x}}\phi(x))~{}dx$

$H_{0}$ is the non-interacting part of the Hamiltonian, The dissipative
strength (${\alpha}_{1}$) of this system in the absence of Co-tunneling effect
is

${{\alpha}_{1}}~{}=~{}\frac{1}{4}\sqrt{\frac{2E_{J1}}{2{E_{J1}}~{}+~{}\frac{16E_{Z1}}{\pi}}}$
(11)

If we consider the total effect of Co-tunneling process, the system reduces to
Heisenberg spin chain with NN and NNN interactions. In this limit the
dissipative strength (${{\alpha}_{2}}$) of the system is

${{\alpha}_{2}}~{}=~{}\frac{1}{4}\sqrt{\frac{2E_{J1}}{E_{J1}~{}+~{}\frac{4}{\pi}(4E_{Z1}~{}-~{}\frac{3{E_{J1}}^{2}}{4E_{C0}})}}$
(12)

We calculate the dissipation strength by calculating K for both cases and then
we use the relation $K=4{\alpha}$. This is the first analytical derivation of
flux induced dissipation strength in terms of the interactions of the system.
We consider these two processes to emphasis the importance of Co-tunneling
effect for this system.

Physical Analysis of Renormalization Group Equation and Calculation of
Tunneling Resistance at The Quantum Critical Point:
The RG equation based on the Eq. (7) is,

$\frac{dz}{dlnb}~{}=~{}(1-{\alpha}^{\prime})z$ (13)

Following Ref.suj2 , we can write fugacity depends on length scale and
temperature as, $z(L)~{}\propto L^{1-{\alpha}^{\prime}}$,
${z}(T)~{}\propto{T}^{{{\alpha}^{\prime}}~{}-~{}1}$. The physical explanations
based on this RG equation are in order.
In our study, the resistance is evolving due to dissipation effect at very low
temperature (few mili Kelvin, less than the superconducting Coulomb blocked
temperature). According to our calculations, for large dissipation
(${\alpha}^{\prime}>1$), $R(T)\propto R_{Q}{T^{{\beta}_{1}}}$ and
${{\beta}_{1}}>0$. Therefore at very low temperature, the system shows stable
SC behaviour (region B of Fig. 1). and the system shows no more saturated
resistance behaviour (region D of Fig. 1). When ${\alpha}^{\prime}<1$, the
resistance of system $R(T)\propto R_{Q}{T^{{-\beta}_{2}}}$ and
${{\beta}_{2}}>0$. So at very low temperature, the resistance of the system
shows Kondo-like divergence behavior (region A of Fig. 1). These stable SC and
Kondo behaviour are unseen in the experimental findings because they have
measured the zero bias resistance at around $40-50$ mK. According to our
calculations, for large dissipation (${\alpha}^{\prime}>1$), $R(T)\propto
R_{Q}{L^{-{\gamma}_{1}}}$ and ${{\gamma}_{1}}>0$. Therefore the longer array
system shows the less resistive state than shorter array in the
superconducting phase. When ${\alpha}^{\prime}<1$, the resistance of system
$R(T)\propto R_{Q}{L^{{\gamma}_{2}}}$, where ${{\gamma}_{2}}>0$
(${{\beta}_{1}},{{\beta}_{2}},{{\gamma}_{1}}$ and ${{\gamma}_{2}}$ are
independent numbers). So the resistance in the insulating state is larger for
longer array system than shorter one. We find the dual behavior of the
resistance for lower and higher values of magnetic field.
From the knowledge of LL physics, we know that for a particular range of $K$,
there is a metallic state in between the insulating phase and the
superconducting phase of the system. The range of $K$ depends on the nature of
the system. Therefore we conclude that there is also a metallic state for a
particular range of magnetic flux induced dissipation strength. This
prediction is unnoticed in the experimental findings. At very low temperature
around $\alpha\sim 1$ (region E of Fig. 1), the quantum phase slip centres
poliferate and as a result a screening appears in the system. A detailed
explanation of QPS for this type of system is discussed in Ref. suj2 .
Physical explanation is as follows.
Here we consider two QPS with co-ordinates (${x_{1}},{\tau}_{1}$) and
(${x_{2}},{\tau}_{2}$), we assume that the cores of the QPS centres donot
overlap, i.e., $|{x_{2}}-{x_{1}}|>{x_{0}}(=\xi)$ and
$|{\tau}_{2}-{\tau}_{1}|>{{\tau}_{0}}(=\frac{1}{\Delta})$. Where $\xi$ and
$\Delta$ are the coherence length and SC orderparameter of the system. We have
already proven that the topological charges interact with each other
logarithmically. Therefore we can write the action of the system as
$S_{QPS}=2S_{core}-{\mu}{{e}_{1}}{{e}_{2}}ln|{x_{1}}-{x_{2}}|$ . QPS with
opposite topological charge attract each other and same charge repel with each
other. Suppose we consider a gas of $n$ QPS and assume that QPS cores do not
overlap. When a current is passing through the system, we can write the total
action of the system as ${S_{QPS}^{n}}=nS_{core}+{S_{int}}$ zai1 . Where,

${S_{int}}=-{\mu}\sum_{i\neq
j}{e_{i}}{e_{j}}ln(\frac{{\rho}_{ij}}{x_{0}})+\sum_{i}\frac{{\Phi}_{0}I}{c}{e_{i}}{{\tau}_{i}}$
(14)

Where ${\rho}_{ij}$ is the distance between the two QPS at the site $i$ and
$j$ and $I$ is the current passing through the system. The grand partition
function of the system is represented as a sum of all topological charges and
the analytical form is equivalent to Eq.6, $z\sim e^{-S_{core}}$. In the
absence of current, Eq. (14) define the model for a 2-dimensional Coulomb gas
interacting logarithmically. We consider the very low temperature limit
($T\rightarrow 0$). The RG equations of this 2-dimensional Coulomb gas are the
following $\frac{\partial\mu}{\partial l}=-4{{\pi}^{2}}{{\mu}^{2}}z^{2}$,
$\frac{\partial z}{\partial l}=(2-{\mu})z$. In our case we can write the
second equation as $\frac{\partial z}{\partial
log(\frac{\Delta}{T})}=(2-{\mu})z$. Finally it yields
$z(T)=z(\Delta){(\frac{\Delta}{T})}^{2-\mu}$. When the interaction coefficient
is less than 2, then the RG equation diverges at the scale ${T^{*}}$ (=
${z}^{\frac{1}{2-\mu}}{\Delta}$). At the scale larger than $\xi$ the
interaction between the QPS is screened and cease to be logarithmic, it
becomes an exponentially decaying function ($\sim
e^{-2{\mu}{K_{0}}(\sqrt{x^{2}+{\tau}^{2}})}$), ${K_{0}}$ is the modified
Bessel function.

Figure 1:  Schematic phase diagram showing the variation of resistance with
temperature based on the renormalization group study. Region A is the Kondo
like behaviour of the system, region B is superconducting phase, region D is
the saturated finite resistance and region E is the metallic phase.
${L_{2}}>{L_{1}}$

When ${\alpha}^{\prime}=1$, i.e.,
${\Phi}~{}=({{\Phi}_{0}}/{\pi})cos^{-1}(\frac{R_{s}}{R_{Q}})$, the system has
no length scale dependence SI transition at very low temperature. This is the
critical behavior of system for a specific value of magnetic field. The
analytical expression for tunneling resistance at the quantum critical point
can be calculated by comparing the expression of ${\alpha}_{1}$ and
${\alpha}_{2}$ with the expression of ${\alpha}^{{}^{\prime}}$.
${R_{S}^{(1)}}=R_{Q}(\frac{16E_{Z1}}{30\pi E_{J1}})$ and
${R_{S}^{(2)}}=R_{Q}(\frac{5\pi
E_{C0}}{4E_{J1}})\sqrt{1+\frac{2.77E_{Z1}}{E_{C0}}}$ are the tunneling
resistance at the quantum critical point in the absence and presence of Co-
tunneling effect respectively. It is clear from our analytical derivation that
the tunneling resistance at quantum critical point is not $R_{Q}$ but it is
rather non-universal. $R_{S}^{(2)}$ is proportional to the $E_{C0}$ as one
would expect from the physical criteria of the system.
Here we discuss the importance of Co-tunelling effect explicitly. In the SC
phase when ${\alpha}>1$, from the analysis of ${\alpha}_{1}$, we get the
condition
$\frac{-16E_{Z1}}{\pi}>30{E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|$. This
condition is unphysical, because the all coupling constants are repulsive. In
presence of Co-tunneling process, from the analysis ${\alpha}_{2}$, we achieve
the condition of SC
${E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|(\frac{192}{4\pi{E_{C0}}}{E_{J10}}|cos(\frac{\pi\Phi}{{\Phi}_{0}})|-30)>\frac{256}{\pi}$.
This condition is physically reliable . Similarly one can do the analysis for
the insulating phase for both absence and presence of co-tunneling effect.
Conclusions: we have found the non-universal tunneling resistance at the
quantum critical point of mesoscopic SQUIDs array. We have found stable SC and
Kondo phase at very low temperature, which is still unseen experimentally. We
have found the length scale independent metallic phase in the system. The
importance of Co-tunneling effect has been studied explicitly.
Acknowledgement: The author would like to acknowledge CCMT of the Physics
Department of IISC for extended facility and Dr. S. Vidyadhiraja for several
important comments on this work and finally Dr. R. Srikanth and Dr. B. Murthy
for reading the manuscript very critically.

## References

  * (1) S. L. Sondhi $et~{}al.$, Rev. Mod. Phys. 69, 315 (1997).
  * (2) L. I. Glazmann $et~{}al.$, Phys. Rev. Lett. 39, (1997) 3786.
  * (3) S. Sarkar, Phys. Rev. B 75, 014528 (2007); Euro. Phys. Lett, 71 (2005) 980.
  * (4) S. Sarkar, Eur. Phys. J. B 67, 559 (2009).
  * (5) L. J. Geerligs $et~{}al.$, Phys. Rev. Lett. 63, 326 (1989).
  * (6) H. S. J. van der Zant $et~{}al.$, Euro. Phys. Lett. 119, 541 (1992).
  * (7) E. Chow $et~{}al$, Phys. Rev. Lett. 81, 204 (1998).
  * (8) W. Kuo $et~{}al$, Phys. Rev. Lett. 87, 186804 (2001).
  * (9) E. Granto $et~{}al$, Phys. Rev. Lett. 65, 1267 (1990).
  * (10) C. D. Chen $et~{}al.$, Phys. Rev. B 51, 15645 (1995).
  * (11) A. O. Calderia, A. J. Leggett, Phys. Rev. Lett 46, 211 (1981).
  * (12) A. Schmid, Phys. Rev. Lett. 51, 1506 (1983).
  * (13) S. Chakravarty $et~{}al.$, Phys. Rev. B 37, 3238 (1988).
  * (14) S. Schon and A. D. Zaikin, Phys. Reports 198, 237 (1990).
  * (15) C. Kane and M. P. A Fisher, Phys. Rev. B 46, 15233 (1992).
  * (16) A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 (1993) ; ibid Phys. Rev. B 47, 3827 (1993).
  * (17) D. S. Golubev and A. D. Zaikin, Phys. Rev. B, 64 014504 (2001); A. D. Zaikin $et~{}al.$, Phys. Rev. Lett., 78 1552 (1997).
  * (18) T. Giamarchi, in Quantum Physics in One Dimension (Claredon Press, Oxford 2004).