# Squarefree values of polynomial discriminants II

Manjul Bhargava, Arul Shankar, and Xiaoheng Wang

###### Abstract

We determine the density of integral binary forms of given degree that have
squarefree discriminant, proving for the first time that the lower density is
positive. Furthermore, we determine the density of integral binary forms that
cut out maximal orders in number fields. The latter proves, in particular, an
“arithmetic Bertini theorem” conjectured by Poonen for
${\mathbb{P}}^{1}_{\mathbb{Z}}$.

Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number
fields of degree $n$ having associated Galois group $S_{n}$ and absolute
discriminant less than $X$, improving the best previously known lower bound of
$\gg X^{1/2+1/n}$.

Finally, our methods correct an error in and thus resurrect earlier
(retracted) results of Nakagawa on lower bounds for the number of totally
unramified $A_{n}$-extensions of quadratic number fields of bounded
discriminant.

##  1 Introduction

In the first article [10] of this two-part series, we proved that when monic
integer polynomials $f(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n}$ of fixed degree $n$
are ordered by ${\rm max}\\{|a_{1}|,\ldots,|a_{n}|^{1/n}\\}$, a positive
proportion have squarefree discriminant. The purpose of this article is to
prove the analogous result for integral binary $n$-ic forms.

Recall that the discriminant $\Delta(f)$ of a binary $n$-ic form over a field
$K$ is a homogeneous polynomial of degree $2n-2$ in the coefficients of $f$,
whose nonvanishing is equivalent to $f$ having $n$ distinct linear factors
over an algebraic closure $\overline{K}$ of $K$. We order integral binary
$n$-ic forms $f(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+\cdots+a_{n}y^{n}$ by their
height $H(f)$ given by $H(f):={\rm max}\\{|a_{0}|,\ldots,|a_{n}|\\},$ i.e.,
the maximum of the absolute values of the coefficients. Then a natural
question is: when ordered by height, what is the density of integral binary
$n$-ic forms whose discriminant is squarefree? For $n=2$, classical methods in
sieve theory yield the answer. For $n=3$ and $n=4$, results of
Davenport–Heilbronn [13] and the first and second authors [9], respectively,
answer the question in the related setting in which we consider ${\rm
GL}_{2}({\mathbb{Z}})$-orbits on binary $n$-ic forms. However, for $n\geq 5$,
it has not previously been known whether this density exists or even whether
the lower density is positive. In this paper, we prove:

###### Theorem 1

Let $n\geq 2$ be an integer. When integral binary $n$-ic forms
$f(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+\cdots+a_{n}y^{n}$ are ordered by $H(f):={\rm
max}\\{|a_{0}|,\ldots,|a_{n}|\\}.$, the density of forms having squarefree
discriminant exists and is equal to

$\begin{array}[]{lll}\displaystyle\mfrac
12\prod_{p>2}\Bigl{(}1-\frac{1}{p}\Bigr{)}\Bigl{(}1+\frac{1}{p}-\frac{1}{p^{3}}\Bigr{)}&\approx\;\;38.97\%&\mbox{
if }n=2;\\\\[12.64746pt] \displaystyle\mfrac
38\prod_{p>2}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{2}\Bigl{(}1+\frac{1}{p}\Bigr{)}^{2}&\approx\;\;24.64\%&\mbox{
if }n=3;\\\\[12.64746pt] \displaystyle\mfrac
38\prod_{p>2}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{2}\Bigl{(}1+\frac{2}{p}-\frac{2}{p^{4}}+\frac{1}{p^{5}}\Bigr{)}&\approx\;\;21.18\%&\mbox{
if }n=4;\\\\[12.64746pt] \displaystyle\mfrac
38\prod_{p>2}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{2}\Bigl{(}1+\frac{1}{p}\Bigr{)}\Bigl{(}1+\frac{1}{p}-\frac{1}{p^{2}}\Bigr{)}&\approx\;\;20.83\%&\mbox{
if }n\geq 5.\end{array}$

To any nonzero integral binary $n$-ic form
$f(x,y)=a_{0}x^{n}+\cdots+a_{n}y^{n}$, we may naturally attach a rank-$n$ ring
$R_{f}$ (see Birch–Merriman [11], Nakagawa [19], and Wood [30]), defined as
follows when $a_{0}\neq 0$. Let $\theta$ denote the image of $x$ in
$K_{f}:={\mathbb{Q}}[x]/(f(x,1))$. Let $R_{f}$ be the free rank-$n$
${\mathbb{Z}}$-submodule of $K_{f}$ generated by
$1,\,a_{0}\theta,\,a_{0}\theta^{2}+a_{1}\theta,\,\ldots,\,a_{0}\theta^{n-1}+\cdots+a_{n-1}\theta$.
Then $R_{f}$ is in fact closed under multiplication and forms a ring whose
discriminant is equal to the discriminant of $f(x)$. Our next result
determines the density of irreducible integral binary forms $f$ for which
$R_{f}$ is the maximal order in its field of fractions.

###### Theorem 2

Let $n\geq 2$ be an integer. When irreducible integral binary $n$-ic forms
$f(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+\cdots+a_{n}y^{n}$ are ordered by $H(f):={\rm
max}\\{|a_{0}|,\ldots,|a_{n}|\\}$, the density of forms $f$ such that $R_{f}$
is the ring of integers in its field of fractions exists and is equal to

$\begin{array}[]{lll}\displaystyle\prod_{p}\Big{(}1-\frac{1}{p^{2}}-\frac{1}{p^{3}}+\frac{1}{p^{4}}\Big{)}&\approx\;\;53.59\%&\mbox{
if }n=2;\\\\[12.46661pt] \zeta(2)^{-1}\zeta(3)^{-1}&\approx\;\;50.57\%&\mbox{
if }n\geq 3.\end{array}$

In particular, Theorem 2 yields the first unconditional Bertini theorem for
arithmetic schemes of dimension $\geq 2$ as conjectured by Poonen [23, §5].
Indeed, for a quasiprojective subscheme $X$ of ${\mathbb{P}}^{n}_{\mathbb{Z}}$
that is regular of dimension $m$, Poonen conjectured that the density of
hyperplane sections of $X$ that are regular of dimension $m-1$ should equal
$\zeta_{X}(m+1)$, where $\zeta_{X}$ denotes the zeta function of $X$. Since
the subscheme of ${\mathbb{P}}^{1}_{\mathbb{Z}}$ cut out by an integral binary
$n$-ic form $f$ is regular if and only if $R_{f}$ is maximal, and the zeta
function of ${\mathbb{P}}^{1}_{\mathbb{Z}}$ is given by
$\zeta_{{\mathbb{P}}^{1}_{\mathbb{Z}}}(s)=\zeta(s)\zeta(s-1)$, we have
$\zeta_{{\mathbb{P}}^{1}_{\mathbb{Z}}}(\dim({\mathbb{P}}^{1}_{\mathbb{Z}})+1)^{-1}=\zeta(2)^{-1}\zeta(3)^{-1}$.
Therefore, Theorem 2 yields an unconditional proof of [23, Theorem 5.1] for
the case $X={\mathbb{P}}^{1}_{\mathbb{Z}}$ with the usual “box ordering” on
the forms defining the hyperplane sections. In fact, we prove the stronger
result that for every fixed $n\geq 3$, the density of regular binary $n$-ic
forms is $\zeta(2)^{-1}\zeta(3)^{-1}$, while arithmetic Bertini only claims
this in the limit as $n\to\infty$.

As a further application of our methods, we obtain the following theorem:

###### Theorem 3

For each $n\geq 3$, the number of isomorphism classes of number fields of
degree $n$ with associated Galois group $S_{n}$ and absolute discriminant less
than $X$ is $\gg X^{1/2+1/(n-1)}$.

Our lower bound in Theorem 3 on the number of degree-$n$ $S_{n}$-number fields
of absolute discriminant less than $X$ improves the previous best-known lower
bound of $X^{1/2+1/n}$ obtained in [10]. We note that the number fields
constructed in Theorem 3 can all be taken to have squarefree discriminant.

Our results also correct an error in, and thus resurrect, all the results of
Nakagawa [19] and [21] that had been subsequently retracted in [20] and [22].
Specifically, the retracted theorems [19, Theorems 3–4] and [21, Theorem 2]
regarding binary forms and $A_{n}$-extensions of quadratic fields can now be
taken to be true. In particular, we obtain:

###### Theorem 4

For $n\geq 3$, the total number of unramified $A_{n}$-extensions of real
$($resp., imaginary$)$ quadratic fields $F$, across all such $F$ such that
$|{\rm Disc}(F)|<X$, is $\gg X^{(n+1)/(2n-2)}$.

Theorem 4 yields the best-known lower bounds on the number of unramified
$A_{n}$-extensions of quadratic fields when $n>5$. For improved bounds in the
cases $n\leq 5$, see [3, Theorem 1.4]. For the best-known bounds on the number
of quadratic fields of bounded discriminant admitting an unramified
$A_{n}$-extension, see Kedlaya [18, Corollary 1.4]. Other related works
include Uchida [29], Yamamoto [32], and Yamamura [33].

The main technical ingredient required to prove all the above results is a
“tail estimate” which shows that not too many discriminants of integral binary
$n$-ic forms $f$ are divisible by $p^{2}$ when $p$ is large relative to the
discriminant of $f$ (here, large means larger than $H(f)$, say). It is these
tail estimates that were missing in Nakagawa’s work. For a prime $p$, and an
integral binary $n$-ic form $f$ such that $p^{2}\mid\Delta(f)$, we say that
$p^{2}$ strongly divides $\Delta(f)$ if $p^{2}\mid\Delta(f+pg)$ for every
integral binary $n$-ic form $g$; otherwise, we say $p^{2}$ weakly divides
$\Delta(f)$. For any squarefree integer $m>0$, let
${\mathcal{W}}_{m}^{\rm{(1)}}$ (resp., ${\mathcal{W}}_{m}^{\rm{(2)}}$) denote
the set of integral binary $n$-ic forms whose discriminants are strongly
divisible (resp., weakly divisible) by $p^{2}$ for every prime factor $p$ of
$m$.

We prove the following tail estimates:

###### Theorem 5

For an integer $n\geq 3$, a positive real number $M$ and any $\epsilon>0$, we
have:

$\begin{array}[]{rlcl}\displaystyle\rm{(a)}&\displaystyle\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{f\in{\mathcal{W}}_{m}^{\rm{(1)}}:H(f)<X\\}&=&\displaystyle
O_{\epsilon}\Bigl{(}\frac{X^{n+1+\epsilon}}{M}+X^{n}\Bigr{)};\\\\[14.45377pt]
\displaystyle\rm{(b)}&\displaystyle\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{f\in{\mathcal{W}}_{m}^{\rm{(2)}}:H(f)<X\\}&=&\displaystyle
O_{\epsilon}\Bigl{(}\frac{X^{n+1+\epsilon}}{M}+X^{n+1-1/(5n)+\epsilon}\Bigr{)},\mbox{
if }2\nmid n;\\\\[14.45377pt]
\displaystyle\rm{(c)}&\displaystyle\\#\bigcup_{\begin{subarray}{c}m>M\\\
m\;\mathrm{squarefree}\end{subarray}}\\{f\in{\mathcal{W}}_{m}^{\rm{(2)}}:H(f)<X\\}&=&\displaystyle
O\Bigl{(}\frac{X^{n+1+1/(88n^{5})}}{\sqrt{M}}+X^{n+1-1/(88n^{6})}\Bigr{)},\mbox{
if }2\mid n.\end{array}$

The estimate in the strongly divisible Case (a) of Theorem 5 follows from
geometric techniques, namely, the quantitative version of the Ekedahl
geometric sieve as developed by the first author [3]. The estimates in the
weakly divisible Cases (b) and (c) of Theorem 5 are considerably more
difficult (particularly (c)), and we describe their proofs in the next
section. Our tail estimate in fact allows us to prove Theorems 1 and 2 with
power-saving error terms:

###### Theorem 6

Let $V_{n}={\rm Sym}^{n}(2)$ denote the space of binary $n$-ic forms. Define
$\eta_{n}$ to be $1/(5n)$ when $n$ is odd and $1/(88n^{6})$ when $n$ is even.
Then

$\begin{array}[]{ccl}\displaystyle\\#\\{f\in V_{n}({\mathbb{Z}}):H(f)<X\mbox{
and $\Delta(f)$
squarefree}\\}&\\!\\!=\\!\\!&\alpha_{n}\cdot(2X)^{n+1}+O_{\epsilon}(X^{n+1-\eta_{n}+\epsilon});\\\\[9.03374pt]
\displaystyle\\#\\{f\in V_{n}({\mathbb{Z}}):H(f)<X\mbox{ and $R_{f}$
maximal}\\}&\\!\\!=\\!\\!&\beta_{n}\cdot(2X)^{n+1}+O_{\epsilon}(X^{n+1-\eta_{n}+\epsilon}).\end{array}$

These power saving bounds have applications towards level-of-distribution
questions when counting integral binary $n$-ic forms $f$ of bounded height
with $\Delta(f)$ squarefree (resp., $R_{f}$ maximal) satisfying splitting
conditions at finitely many primes. Such level-of-distribution results in turn
have applications towards a host of problems in analytic number theory, such
as studying statistics of Artin $L$-functions attached to binary $n$-ic forms
and proving lower bounds on the number of degree-$n$ number fields which are
ramified only at a bounded number of primes, among many others.

We remark that our methods imply that the analogues of all of the above
results also hold when local conditions are imposed at finitely many places
(including at infinity); the orders of magnitudes in these theorems remain the
same, provided that no local conditions are imposed that force the sets being
counted in Theorems 1 and 2 to be empty.

Finally, the methods introduced in [10] and in the current article have
applications beyond just squarefree values of polynomial discriminants. They
have been recently adapted in [8] to determine the density of squarefree
discriminants of elliptic curves over ${\mathbb{Q}}$ having two marked
rational points. Other applications include determining the density of
conductors in some families of elliptic curves [27] and the density of
squarefree values taken by $a^{4}+b^{3}$ ([24]).

##  2 Outline of proof

As mentioned in the introduction, the uniformity estimate in Theorem 5 is the
key to deducing Theorems 1, 2, and 6 via a squarefree sieve. Case (a) of
Theorem 5 follows directly from the results in [3]. Case (b), which pertains
to odd degrees $n$, can be proven using methods similar to those developed in
our previous work [10]. However, these methods fail to work for Case (c),
which pertains to even degrees $n$, and a number of new ideas are required to
handle this case. It is the proof of this case to which the bulk of our paper
is devoted; it requires, in particular, the introduction of a new technique in
the geometry of numbers that we believe may also be useful in other contexts.

In this section, we give a detailed outline of the proof of Case (b)
pertaining to odd $n$. We then explain why this strategy breaks down (quite
spectacularly!) when $n$ is even, and finally we describe the new techniques
required to complete the proof of Theorem 5(c).

#### Sketch of the proof of the tail estimate for odd $n$

Our proof of Theorem 5(b) makes use of the representation of $G={\rm SL}_{n}$
on the space $W=2\otimes{\rm Sym}_{2}(n)$ of pairs $(A,B)$ of symmetric
$n\times n$ matrices, studied in detail in [31, 4, 6, 7]. The group $G$ acts
on $W$ via $\gamma\cdot(A,B)=(\gamma A\gamma^{t},\gamma B\gamma^{t}$) for
$\gamma\in G$ and $(A,B)\in W$. We define the invariant binary form of an
element $(A,B)\in W$ by

$\smash{f_{A,B}(x,y)=(-1)^{n(n-1)/2}\det(Ax-By).}$

Then $f_{A,B}$ is a binary $n$-ic form satisfying $f_{\gamma(A,B)}=f_{A,B}$.
Moreover, the ring of polynomial invariants for the action of $G$ on $W$ is
freely generated by the coefficients of the invariant binary form (reference?
it is in some classification of coregular spaces). Define the discriminant
$\Delta(A,B)$ and height $H(A,B)$ of an element $(A,B)\in W$ by
$\Delta(A,B)=\Delta(f_{A,B})$ and $H(A,B)=H(f_{A,B})$.

The first step of our proof is the construction, for every squarefree integer
$m>0$, of a map

$\sigma_{m}:{\mathcal{W}}_{m}^{\rm{(2)}}\to W({\mathbb{Z}}),$

such that $f_{\sigma_{m}(f)}(x,y)=f(x,y)$ for every
$f\in{\mathcal{W}}_{m}^{\rm{(2)}}$. In our construction, the image of
$\sigma_{m}$ in fact lies in $W_{0}({\mathbb{Z}})$, where $W_{0}$ is the
subspace of $W$ consisting of pairs of matrices whose top left $g\times g$
blocks are $0$, where $n=2g+1$. The action of the group $G$ does not preserve
$W_{0}$, and we take $G_{0}$ to be the maximal parabolic subgroup of $G$ that
does preserve $W_{0}$. When the discriminant polynomial
$\Delta\in{\mathbb{Z}}[W]$ is restricted to $W_{0}$, it is no longer
irreducible but rather is divisible by the square of a polynomial
$Q\in{\mathbb{Z}}[W_{0}]$. This polynomial $Q$ is a relative invariant for the
action of $G_{0}$ on $W_{0}$. Its significance is that, by construction of
$\sigma_{m}$, every element in the the image of $\sigma_{m}$ has $Q$-invariant
equal to $m$. To prove Part (b) of Theorem 5, it therefore suffices to
estimate the number of $G_{0}({\mathbb{Z}})$-orbits on $W_{0}({\mathbb{Z}})$
having height less than $X$ and $Q$-invariant greater than $M$.

Bounding the number of these orbits is complicated by the fact that $G_{0}$ is
not reductive. We are rescued by using the full action of $G({\mathbb{Z}})$ on
$W({\mathbb{Z}})$. This necessitates expanding the definition of the
$Q$-invariant from $W_{0}({\mathbb{Z}})$ to all “distinguished” elements of
$W({\mathbb{Z}})$. An element $(A,B)\in W({\mathbb{Z}})$ is distinguished if
$A$ and $B$ have a common isotropic $g$-dimensional subspace defined over
${\mathbb{Q}}$. Thus every element in $W_{0}({\mathbb{Z}})$ (and thus every
element in the image of $\sigma_{m}$) is distinguished. The $Q$-invariant,
though defined initially on $W_{0}$, can be extended as a function on the set
of all triples $(A,B,\Lambda)$, where $(A,B)\in W({\mathbb{Z}})$ is
distinguished, and $\Lambda$ is a common isotropic subspace of $A$ and $B$.
For all but a negligible number of distinguished elements $(A,B)\in
W({\mathbb{Z}})$, $A$ and $B$ have exactly one common isotropic subspace
$\Lambda$ defined over ${\mathbb{Q}}$. Thus we may define a
$G({\mathbb{Z}})$-invariant function $Q$ on the set of distinguished pairs
$(A,B)\in W({\mathbb{Z}})$ outside a negligible number of them. It then
suffices to bound the number of $G({\mathbb{Z}})$-orbits on distinguished
elements in $W({\mathbb{Z}})$ having bounded height and large $Q$-invariant.

To obtain such a bound, we construct fundamental domains for the action of
$G({\mathbb{Z}})$ on elements in $W({\mathbb{R}})$ with height less than $X$.
Such a fundamental domain has a natural partition into three parts that we
term the main body, the shallow cusp, and the deep cusp. We have little
control over the $Q$-invariants of elements in the main body and the shallow
cusp. However, it is known [17, Proposition 4.3] that there are a negligible
number of integral elements in the shallow cusp. Meanwhile, distinguished
elements occur rarely in the main body, a fact we prove via the Selberg sieve.

Finally, the deep cusp lies in $W_{0}$, where an upper bound for the
$Q$-invariant can be obtained. Imposing the condition that this upper bound is
greater than $M$, and counting the number of such points in the deep cusp
using the averaging method of [2], gives the desired saving for the number of
elements in the deep cusp having $Q$-invariant larger than $M$. Combining the
estimates for the main body, the shallow cusp, and the deep cusp yields Part
(b) of Theorem 5.

#### Sketch of the proof of the tail estimate for even $n$

With $W$ again denoting the space of pairs of symmetric $n\times n$ matrices,
we may attempt to proceed in the same manner as in the case of odd $n$, by
constructing a map

$\smash{\sigma_{m}:{\mathcal{W}}_{m}^{\rm{(2)}}\to W({\mathbb{Z}})}$

such that $f_{\sigma_{m}(f)}(x,y)=f(x,y)$ for every
$f\in{\mathcal{W}}_{m}^{\rm{(2)}}$. However, such a map does not exist in the
case that $n$ is even! Indeed, there exist integral binary $n$-ic forms
$f(x,y)$ that cannot be expressed as $\det(Ax-By)$—even up to sign—for any
integral $n\times n$ symmetric matrices $A$ and $B$. This phenomenon was
extensively studied in [4, 6, 7]. It is in this sense that the strategy to
prove Theorem 5(b) for odd $n$ fails spectacularly for even $n$—and at the
very first step.

We address this issue by replacing $f(x,y)\in{\mathcal{W}}_{m}^{(2)}$ by
$xf(x,y)$, which is a reducible binary $(n+1)$-ic form whose discriminant, at
least generically, remains weakly divisible by $m^{2}$. For these forms
$xf(x,y)$, we can use the lift $\sigma_{m}$ constructed in the odd case.
However, since $xf(x,y)$ has vanishing $y^{n+1}$ term, the image of
$\sigma_{m}$ lies within the set of pairs $(A,B)$ where $B$ is singular.

The singularity of $B$ introduces additional difficulties with respect to both
the algebraic and the analytic aspects of the proof. On the algebraic side,
the main new problem is that distinguished elements $(A,B)$ with $B$ singular
have at least two values for the $Q$-invariant, since they share at least two
different common isotropic $(g+1)$-dimensional subspaces, where $n=2g+2$. So
it is no longer well-defined to impose the condition that $Q$ is large.
Imposing the condition that the maximum value of $Q$ is large does not yield
sufficient savings to prove an analogue of Theorem 5(b). We thus instead
construct a new invariant, termed $q$, such that for all but a negligible
number of elements $(A,B)$ in the image of our map $\sigma_{m}$, the invariant
$q$ is the minimum value taken by $Q$, and it satisfies
$q(\sigma_{m}(xf(x,y)))=\pm m$.

As in the odd degree case, we once again construct fundamental domains
${\mathcal{F}}_{X}$ for the action of $G({\mathbb{Z}})$ on $W({\mathbb{R}})$
with height less than $X$, and partition such a domain into three parts: the
main body, the shallow cusp, and the deep cusp. However, we must now only
count integer elements $(A,B)$ where $B$ is singular. The beautiful work of
Eskin and Katznelson [15] provides asymptotics for the number of singular
symmetric matrices in homogenously expanding domains. But this work is not
directly applicable to our case since we need to estimate the number of
singular symmetric matrices $B$ in skewed domains. To achieve this, we provide
a simplification of the proof of the upper bounds in [15], at the cost of some
extra $\log$ factors, which gives us a flexible method by which to obtain
upper bounds on the number of singular symmetric matrices in arbitrarily
skewed domains.

Accounting for the singularity of the $B$’s introduces complications in each
region of the fundamental domain. In the main body, the lack of an exact count
with a power-saving error term means we cannot directly apply a Selberg sieve
to bound the number of distinguished elements. Instead, we fiber over the
singular matrices $B$ and apply the Selberg sieve to bound the number of
possible $A$’s. This requires us to prove new density estimates on the number
of distinguished elements $(A,B)$ over ${\mathbb{F}}_{p}$, when $B$ is fixed.

Furthermore, unlike in the odd degree case, we no longer have an automatic
power-saving on the number of pairs $(A,B)\in W({\mathbb{Z}})$ lying in the
shallow cusp of the fundamental domain and where $B$ is singular. As we go
closer to the deep cusp, there are regions in which imposing the condition
that $B$ is singular yields no saving whatsoever. To obtain the required
bounds, we isolate this region of the shallow cusp and prove that integral
elements $(A,B)$ in them either satisfy $\Delta(A,B)=0$ or $|q(A,B)|$ is
small.

Finally, for the deep cusp of ${\mathcal{F}}$, we once again use the condition
that the $q$-invariant is large to obtain a power saving. Unlike the situation
with the $Q$-invariant in the odd-degree case, the invariant $q$ in the even
degree case behaves more wildly and is much harder to control. This is because
$q$ is not a polynomial in the coefficients of $W_{0}$ but rather is a minimum
of the different possible values of $Q$. In fact, there are regions within the
deep cusp where the $q$-invariant of elements $(A,B)$ are not small. However,
we show that these regions correspond to an archimedean condition on the
invariant binary form $f$ of $(A,B)$, namely, that the discriminant of $f$ is
much smaller than is typical for the height bound on $f$. Separately bounding
the number of such binary forms yields the desired result.

#### Organization of the paper

This paper is organized as follows. We begin in §3 by recalling the arithmetic
invariant theory for the representations $W_{n}:=2\times{\rm Sym}_{2}(n)$ of
${\rm SL}_{n}$ and $2\otimes g\otimes(g+1)$ of ${\rm SL}_{2}\times{\rm
SL}_{g}\times{\rm SL}_{g+1}$. In particular, we define the fundamental
invariants $Q$ and $q$. We then construct our maps from
${\mathcal{W}}_{m}^{(2)}$ into $W_{n}({\mathbb{Z}})$ when $n$ is odd and into
$W_{n+1}({\mathbb{Z}})$ when $n$ is even.

The analytic parts of the paper are carried out in §4–6. In §4, we prove the
tail estimates of Theorem 5 for odd degrees $n$ using geometry-of-numbers
techniques. In §5, we carry out the necessary groundwork to count the number
of singular symmetric matrices that lie in skewed domains. Using these
results, we prove the tail estimates for even degrees $n$ in §6, completing
the proof of Theorem 5. In §7, we deduce the main results, Theorems 1–4, from
the tail estimates using a squarefree sieve, although the exact constants
occurring in Theorems 1 and 2 remain conditional upon certain local density
computations. Finally, in the Appendix, we compute the local densities of
integral binary $n$-ic forms whose discriminants are indivisible by $p^{2}$
(resp., whose associated rings are maximal at $p$), thereby completing the
proofs of Theorems 1 and 2.

##  3 Invariant theory on spaces associated to binary $n$-ic forms

Fix a positive integer $n$ and consider the space $V_{n}={\rm Sym}^{n}(2)$ of
binary $n$-ic forms of degree $n$. The group ${\rm SL}_{2}$ acts on $V_{n}$
via linear change of variables: we have $\gamma\cdot
f(x,y):=f((x,y)\cdot\gamma)$ for $\gamma\in{\rm SL}_{2}$ and $f\in V_{n}$.

Let $W_{n}=2\otimes{\rm Sym}_{2}(n)$ denote the space of pairs of $n\times n$
symmetric matrices $(A,B)$. The group ${\rm SL}_{2}\times{\rm SL}_{n}$ acts on
$(A,B)$ via

$(\gamma_{2},\gamma_{n})\cdot(A,B)=(\gamma_{n}A\gamma_{n}^{t},\gamma_{n}B\gamma_{n}^{t})\cdot\gamma_{2}^{t}.$

There is a natural map $W_{n}\to V_{n}$ given by

$\begin{array}[]{rcl}(A,B)&\mapsto&f_{A,B}:=(-1)^{n(n-1)/2}\det(Ax-
By),\end{array}$ (1)

sending an element of $W_{n}$ to its invariant binary $n$-ic form. The ring of
${\rm SL}_{n}({\mathbb{C}})$-invariant polynomials on $W_{n}({\mathbb{C}})$ is
freely generated by the coefficients of the invariant binary $n$-ic form.

###  3.1 Arithmetic invariant theory for the representation $2\otimes{\rm
Sym}_{2}(n)$ of ${\rm SL}_{n}$

First, let $n=2g+1$ be an odd integer with $g\geq 1$. We recall some of the
arithmetic invariant theory of the representation $W:=W_{n}$ of ${\rm SL}_{n}$
and its map (1) to $V:=V_{n};$ see [6] for more details.

Let $k$ be a field of characteristic not $2$. For a binary $n$-ic form
$f(x,y)=a_{0}x^{n}+\cdots+a_{n}y^{n}\in V(k)$ with $\Delta(f)\neq 0$ and
$a_{0}\neq 0$, let $C_{f}$ denote the smooth hyperelliptic curve
$z^{2}=f(x,y)y$ of genus $g$ viewed as a curve in the weighted projective
space ${\mathbb{P}}(1,1,g+1)$. Let $J_{f}$ denote the Jacobian of $C_{f}$.
Then the stabilizer of an element $(A,B)\in W(k)$ with invariant binary form
$f(x,y)$ is isomorphic to $J_{f}[2](k)$. The set of ${\rm SL}_{n}(k)$-orbits
on $W(k)$ with invariant binary form $f(x,y)$ maps injectively into
$H^{1}(k,J_{f}[2])$. An element $(A,B)$ (or an ${\rm SL}_{n}(k)$-orbit) is
distinguished if $\Delta(A,B)\neq 0$ and there exists a $g$-dimensional
subspace defined over $k$ that is isotropic with respect to both $A$ and $B$.
If $(A,B)$ is distinguished, then its ${\rm SL}_{n}(k)$-orbit corresponds to
the identity element of $H^{1}(k,J_{f}[2])$, and the set of these
$g$-dimensional subspaces is in bijection with $J_{f}[2](k)$.

Let $W_{0}\subset W$ be the subspace of pairs of matrices whose top left
$g\times g$ blocks are zero. Then elements $(A,B)$ in $W_{0}(k)$ with nonzero
discriminant are all distinguished since the $g$-dimensional subspace $Y_{g}$
spanned by the first $g$ basis vectors is isotropic with respect to both $A$
and $B$. Moreover, every distinguished element of $W(k)$ is ${\rm
SL}_{n}(k)$-equivalent to some element in $W_{0}(k)$ since ${\rm SL}_{n}(k)$
acts transitively on the set of $g$-dimensional subspaces of
${\mathbb{P}}^{n-1}(k)$. Let $G_{0}$ be the maximal parabolic subgroup of
${\rm SL}_{n}$ consisting of elements $\gamma$ that preserve $Y_{g}$. Elements
of $W_{0}$ have block matrix form

$(A,B)=\left(\Bigl{(}\begin{array}[]{cc}0&A^{\rm top}\\\ (A^{\rm
top})^{t}&A_{1}\end{array}\Bigr{)},\Bigl{(}\begin{array}[]{cc}0&B^{\rm top}\\\
(B^{\rm top})^{t}&B_{1}\end{array}\Bigr{)}\right),$ (2)

where $A^{\rm top}$, $B^{\rm top}$ are $g\times(g+1)$ matrices and $A_{1}$,
$B_{1}$ are $(g+1)\times(g+1)$-symmetric matrices. Meanwhile, elements of
$G_{0}$ have the block matrix form

$\gamma=\Bigl{(}\begin{array}[]{cc}\gamma_{1}&0\\\
n&\gamma_{2}\end{array}\Bigr{)}\in\Bigl{(}\begin{array}[]{cc}{\rm GL}_{g}&0\\\
M_{(g+1)\times g}&{\rm GL}_{g+1}\end{array}\Bigr{)}.$ (3)

An element $\gamma\in G_{0}$ acts on the top right $g\times(g+1)$ block of
elements of $W_{0}$ by

$\gamma(A^{\rm top},B^{\rm top})=(\gamma_{1}A^{\rm
top}\gamma_{2}^{t},\gamma_{1}B^{\rm top}\gamma_{2}^{t})$

where we use the superscript “top” to denote the top right $g\times(g+1)$
block of an $n\times n$ symmetric matrix. The action of $G_{0}$ on $W_{0}$
restricts to an action on the space $U_{g}:=2\otimes g\otimes(g+1)$ of pairs
of $g\times(g+1)$-matrices, Moreover, the unipotent radical $M_{(g+1)\times
g}$ of $G_{0}$ acts trivially on $U_{g}$. We study the invariant theory for
this action more closely in the next subsection.

We will also need some results in the case when $n=2g+2$ is even in Section 6
(specifically in the proof of Lemma 6.7). Let
$f(x,y)=a_{0}x^{n}+\cdots+a_{n}y^{n}\in V(k)$ with $\Delta(f)\neq 0$ and
$a_{0}\neq 0$. Let $L=k[x]/(f(x))$. Let $V_{f}(k)$ denote the set of $(A,B)\in
W_{n}(k)$ with $f_{A,B}=f(x,y)$. Then $V_{f}(k)$ is nonempty if and only if
$f\in k^{\times 2}N_{L/k}(L^{\times})$. Note in particular that if $f(x,y)\in
V({\mathbb{R}})$ is negative definite, so that
$L={\mathbb{R}}[x]/(f(x))\simeq{\mathbb{C}}^{n/2}$ and $a_{0}<0$, then
$V_{f}({\mathbb{R}})$ is empty. On the other hand, if $k$ is a finite field of
characteristic not $2$, then $V_{f}(k)$ is always nonempty and the number of
${\rm SL}_{n}(k)$-orbits equals the number of even degree factorizations of
$f(x,y)$ over $k$.

###  3.2 The representation $2\otimes g\otimes(g+1)$ of ${\rm
SL}_{2}\times{\rm SL}_{g}\times{\rm SL}_{g+1}$ and the $Q$-invariant

In this section, we collect some algebraic facts about the representation
$U_{g};=2\otimes g\otimes(g+1)$ of the group $H_{g}:={\rm SL}_{2}\times{\rm
SL}_{g}\times{\rm SL}_{g+1}$. We start with the following proposition.

###### Proposition 3.1

The representation $U_{g}$ of ${\mathbb{G}}_{m}\times H_{g}$ is
prehomogeneous, i.e., the action of ${\mathbb{G}}_{m}\times H_{g}$ on $U_{g}$
has a single Zariski open orbit. Furthermore, the stabilizer in
$H_{g}({\mathbb{C}})$ of an element in the open orbit of $U_{g}({\mathbb{C}})$
is isomorphic to ${\rm SL}_{2}({\mathbb{C}})$.

Proof: We prove this by induction on $g$. The assertion is clear for $g=1$,
where the representation is that of ${\mathbb{G}}_{m}\times{\rm
SL}_{2}\times{\rm SL}_{2}$ on $2\times 2$ matrices; the single relative
invariant in this case is the determinant, and the open orbit consists of
nonsingular matrices. For higher $g$, we note that $U_{g}$ is a castling
transform of $U_{g-1}$ in the sense of Sato and Kimura [25]. As a result, the
orbits of ${\mathbb{G}}_{m}\times{\rm SL}_{2}\times{\rm SL}_{g}\times{\rm
SL}_{g-1}$ on $2\otimes g\otimes(g-1)$ are in natural one-to-one
correspondence with the orbits of ${\mathbb{G}}_{m}\times{\rm
SL}_{2}\times{\rm SL}_{g}\times{\rm SL}_{g+1}$ on $2\otimes
g\otimes(2g-(g-1))=2\otimes g\otimes(g+1)$, and under this correspondence, the
open orbit in $U_{g-1}$ maps to an open orbit in $U_{g}$ (cf. [25]). Thus all
the representations $U_{g}$ for the action of ${\mathbb{G}}_{m}\times H_{g}$
are prehomogeneous.

Note that castling transforms preserve stabilizers over ${\mathbb{C}}$. Since
the generic stabilizer for the action of $H_{1}({\mathbb{C}})$ on
$U_{1}({\mathbb{C}})$ is clearly isomorphic to ${\rm SL}_{2}({\mathbb{C}})$,
it follows that this remains the generic stabilizer for the action of
$H_{g}({\mathbb{C}})$ on $U_{g}({\mathbb{C}})$ for all $g\geq 1$. $\Box$

Since castling transforms also preserve polynomial invariants and their
irreducibility ([25, Proposition 18]), it follows that the ring of polynomial
invariants for this action of $H_{g}$ on $U_{g}$ is generated by an
irreducible polynomial. We now give an explicit description of this invariant.

Write an element in $U_{g}=2\times g\times(g+1)$ as a pair $(A^{\rm
top},B^{\rm top})$ of $g\times(g+1)$ matrices. For $1\leq i\leq g+1$, let
$A_{i}$ and $B_{i}$ denote the $g\times g$-matrices obtained from $A^{\rm
top}$ and $B^{\rm top}$, respectively, by deleting the $i$th column. Define
the binary $(g+1)$-ic form $f_{i}(x,y)$ to be $(-1)^{i+1}\det(A_{i}x-B_{i}y)$.
Consider the $(g+1)\times(g+1)$ matrix $C$ whose $(i,j)$-entry is the $j$th-
coefficient of $f_{i}(x,y)$. Taking the determinant of $C$ yields a polynomial
$Q=Q(A^{\rm top},B^{\rm top})$ in the coordinates of $U_{g}$. The polynomial
$Q$ is the hyperdeterminant of the $2\times g\times(g+1)$ matrix $(A^{\rm
top},B^{\rm top})$ (cf. [16, Theorem 3.18]). As a consequence, it is
irreducible and invariant under the action of $H_{g}$ on $U_{g}$ and thus
generates the ring of polynomials for the action of $H_{g}$ on $U_{g}$.

Let $n=2g+1$ again be an odd integer. We return to the representation $W_{0}$
of $G_{0}$. Given an element $(A,B)\in W_{0}$, recall that we obtain an
element $(A^{\rm top},B^{\rm top})\in U_{g}$ by taking the top right
$g\times(g+1)$ blocks of $A$ and $B$. We define the $Q$-invariant of $(A,B)\in
W_{0}$ as the $Q$-invariant of $(A^{\rm top},B^{\rm top})$:

$Q(A,B):=Q(A^{\rm top},B^{\rm top}).$ (4)

Then the $Q$-invariant is a relative invariant for $G_{0}$. More precisely,
for any $\gamma\in G_{0}$ in the block matrix form (3), we have

$Q(\gamma\cdot(A,B))=\det(\gamma_{1})^{g+1}\det(\gamma_{2})^{g}Q(A,B)=\det(\gamma_{1})Q(A,B),$
(5)

since $\det(\gamma_{1})\det(\gamma_{2})=1$. If $\gamma\in
G_{0}({\mathbb{Z}})$, then we have $\det(\gamma_{1})=\det(\gamma_{2})=\pm 1$.
Hence the absolute value $|Q|$ of $Q$ is an invariant for the action of
$G_{0}({\mathbb{Z}})$ on $W_{0}({\mathbb{Z}})$.

###  3.3 Divisibility properties of $\Delta$ when restricted to $W_{0}$

Let $n=2g+1$ be an odd integer. Write the coordinates on $W_{0}$ as
$a_{ij},b_{ij}$ with $i,j$ in the appropriate ranges. Let $R$ denote the ring
of regular functions of $W_{0}$ over ${\mathbb{Z}}$, i.e.,
$R={\mathbb{Z}}[W_{0}]={\mathbb{Z}}[a_{ij},b_{ij}]$. Consider the discriminant
polynomial $\Delta\in R$ given by $\Delta(A,B):=\Delta(f_{A,B})$. In this
section, we prove that $Q^{2}\mid\Delta$ as polynomials in $R$, along with
another useful divisibility result.

Let $Z$ be the closed subvariety of $W_{0}$ consisting of elements $(A,B)$
with $\Delta(A,B)=0$, and let $Y\subset Z$ denote the closed subvariety of
$W_{0}$ consisting of elements $(A,B)$ such that $f_{A,B}$ is either divisible
by the cube of a binary form with degree $\geq 1$ or the square of a binary
form with degree $\geq 2$. Both of these varieties $Y$ and $Z$ are defined
over ${\mathbb{Z}}$ and are clearly ${\rm SL}_{2}\times G_{0}$-invariant.

Our first result states that the variety in $W_{0}$ cut out by $Q=0$ does not
lie in $Y$.

###### Proposition 3.2

Let $(A,B)=((a_{ij})_{ij},(b_{ij})_{ij})\in W_{0}(R)$ be the generic element.
Then

$(A,B){\rm{\;mod\;}}Q\not\in Y(R/(Q)).$

Proof: Fix an odd prime $p$. Let $f(x,y)$ be an element of $V({\mathbb{Z}})$,
such that the reduction of $f(x,y)$ modulo $p$ factors as $x^{2}h(x,y)$, where
$h$ is irreducible. In particular, $f(x,y)$ mod $p$ is not divisible by either
the cube of a binary form with degree $\geq 1$, or the square of a binary form
with degree $\geq 2$. Let $(A_{f},B_{f})\in W_{0}({\mathbb{Z}})$ be an element
with invariant binary $n$-ic form equal to $f$ and $Q(A_{f},B_{f})=p$. Such an
element $(A_{f},B_{f})$ is constructed in the next subsection.

Let $\pi:R\rightarrow{\mathbb{Z}}$ denote the specialization map assigning
integer values to $a_{ij},b_{ij}$ such that

$\pi(A,B)=(A_{f},B_{f}).$

Then $\pi(Q)=p$ and so $\pi$ induces a map $R/(Q)\rightarrow{\mathbb{F}}_{p}$.
Since $(A_{f},B_{f})\mbox{ mod }p\notin Y({\mathbb{F}}_{p})$, we see that
$(A,B)\mbox{ mod }Q\notin Y(R/(Q))$. $\Box$

The next lemma, which follows from a direct computation, gives the
$Q$-invariant for elements in $W_{0}$ having a specific form.

###### Lemma 3.3

Let $k$ be a field and let $(A,B)\in W_{0}(k)$ be an element such that the top
right $g\times(g+1)$ blocks of $(A,B)$ are of the following form:

$(A^{\rm top},B^{\rm
top})=\left(\left(\begin{array}[]{cccccc}0&0&\cdots&0&0&a_{1}\\\
0&&&&a_{2}&*\\\ 0&&&a_{3}&*&*\\\
\vdots&&\reflectbox{$\ddots$}&&\vdots&\vdots\\\
0&a_{g}&\cdots&*&*&*\end{array}\right),\left(\begin{array}[]{cccccc}0&\cdots&0&0&b_{1}&0\\\
&&&b_{2}&0&0\\\ &&b_{3}&&0&0\\\ &\reflectbox{$\ddots$}&&&\vdots&\vdots\\\
b_{g}&&&&0&0\end{array}\right)\right).$ (6)

Then

$Q(A,B)=\pm a_{1}^{g}a_{2}^{g-1}\cdots a_{g}b_{1}b_{2}^{2}\cdots b_{g}^{g}.$

Next, we have the following proposition that gives a normal form for elements
$(A,B)\not\in Y$ whose $Q$-invariant is $0$.

###### Proposition 3.4

Let $k$ be a field. Let $(A,B)$ be an element of $W_{0}(k)\backslash Y(k)$
such that $Q(A,B)=0$. Then $(A,B)$ is ${\rm SL}_{2}(k)\times
G_{0}(k)$-equivalent to an element of the form $(A^{\prime},B^{\prime})$ where
the top right $g\times(g+1)$ blocks of $A^{\prime}$ and $B^{\prime}$ are given
by

$(A^{\prime{\rm top}},B^{\prime{\rm
top}})=\left(\left(\begin{array}[]{cccccc}0&0&\cdots&0&0&a_{1}\\\
0&&&&a_{2}&*\\\ 0&&&a_{3}&*&*\\\
\vdots&&\reflectbox{$\ddots$}&&\vdots&\vdots\\\
0&a_{g}&\cdots&*&*&*\end{array}\right),\left(\begin{array}[]{cccccc}0&\cdots&0&0&0&0\\\
&&&b_{2}&0&0\\\ &&b_{3}&&0&0\\\ &\reflectbox{$\ddots$}&&&\vdots&\vdots\\\
b_{g}&&&&0&0\end{array}\right)\right),$ (7)

where $a_{1},\ldots,a_{g},b_{2},\ldots,b_{g}\in k^{\times}.$ In the displayed
matrices above, any empty entry is $0$.

Proof: The action of $G_{0}(k)$ allows us to perform simultaneous row
operations and simultaneous column operations on $(A^{\rm top},B^{\rm top})$.
As a first step, we perform column operations to ensure that the rightmost
column of $B^{\rm top}$ is $0$. Next, recall that the $Q$-invariant of $(A,B)$
is the determinant of the $(g+1)\times(g+1)$ matrix $C$, whose rows come from
the coefficients of the $g\times g$ minors of $A^{\rm top}x-B^{\rm top}y$. It
follows that row operations on $(A^{\rm top},B^{\rm top})$ leave $C$
unchanged, while adding $\alpha$ times the $i$-th columns of $A^{\rm
top},B^{\rm top}$ to the $j$-th column has the effect of adding $\alpha$ times
the $j$-th row of $C$ to the $i$-th row of $C$ and leaving the rest unchanged.
Since $\det(C)=Q(A^{\rm top},B^{\rm top})=0$, it follows that by adding
multiples of the last columns of $A^{\rm top},B^{\rm top}$ to the other
columns, we may assume that the last row of $C$ is $0$. Denoting the $g\times
g$ matrices obtained by removing the last columns of $A^{\rm top}$ and $B^{\rm
top}$ by $M$ and $N$, respectively, we have $\det(Mx-Ny)=0$.

We next claim that by performing simultaneous row and column operations on
$(M,N)$, we may bring $M$ and $N$ in the form of the first $g$ columns of
$A^{\prime\,{\rm top}}$ and $B^{\prime\,{\rm top}}$, respectively, for
$(A^{\prime\,{\rm top}},B^{\prime\,{\rm top}})$ as given in (7) with
$b_{i}\neq 0$ for all $2\leq i\leq g$. Since $\det(M)=0$, after appropriate
column operations we may assume that the first column of $M$ is $0$. Now the
first column of $N$ cannot be identically $0$ for otherwise, the invariant
binary form of $(A,B)$ has a factor of the form $h(x,y)^{2}$ with $\deg h=g$,
contradicting $(A,B)\notin Y(k)$. By applying row operations, we may ensure
that the bottom left entry of $N$ is $b_{g}\neq 0$ and the rest of the first
column of $N$ is $0$. We then use this nonzero cofficient $b_{g}$ to clear out
the rest of the bottom row of $N$ (without changing $M$).

Let $M_{1}$ and $N_{1}$ denote the top right $(g-1)\times(g-1)$ block of $M$
and $N$. Then $\det(Mx-Ny)=(-1)^{g}b_{g}y\det(M_{1}x-N_{1}y).$ Hence
$\det(M_{1}x-N_{1}y)=0$ and the first column of $M_{1}$ can be made $0$. As in
the previous case, all the coefficients of the first column of $N_{1}$ can be
made $0$ except for the bottom left entry, which is $b_{g-1}\neq 0$. We then
clear out the bottom row of $N_{1}$ as before. Proceeding in this way, we
transform the first $g-1$ columns of $M$ and $N$ to be in the required form.
Since the $b_{i}$’s are nonzero for $2\leq i\leq g$, and since $\det(Mx-
Ny)=0$, it follows that the top right coefficients of $M$ and $N$ are $0$,
completing the proof of the claim.

Note that this transformation of $M$ and $N$ did not change the last column of
$B^{\rm top}$, which remains $0$. Thus to complete the proof of Proposition
3.4, it remains to show that $a_{i}\neq 0$ for $1\leq i\leq g$. Since the
first row and column of $B^{\prime}$ are $0$, we see that $x^{2}a_{1}^{2}\mid
f_{A^{\prime},B^{\prime}}$. Hence $a_{1}\neq 0$. Suppose for contradiction
that $i=2,\ldots,g$ is the smallest index such that $a_{i}=0$. Then we may
clear out the $i$-th row of $A^{\prime}$ using the second up to the $(i-1)$-th
rows of $A^{\prime}$. That is, $(A^{\prime},B^{\prime})$ is ${\rm
SL}_{n}(k)$-equivalent to some $(A^{\prime\prime},B^{\prime\prime})$ where the
only nonzero entries in the $i$-th row and the $i$-th column of
$A^{\prime\prime}$ appear in the last entry. This allows us to factor out an
extra factor of $y^{2}$ in $\det(A^{\prime\prime}x-B^{\prime\prime}y)=\pm
f_{A,B}$, contradicting the assumption that $(A,B)\notin Y(k)$ since we
already had $x^{2}\mid f_{A,B}$. $\Box$

We are now ready to prove that $Q^{2}\mid\Delta$:

###### Theorem 3.5

We have $Q^{2}\mid\Delta$ in ${\mathbb{Z}}[W_{0}]$.

Proof: Let $(A,B)\in W_{0}(R)$ be the generic element. We begin by proving
that $(A,B)\in Z(R/(Q))$, or equivalently that $Q\mid\Delta$ in $R$. Let
$(\bar{A},\bar{B})\in W_{0}(R/(Q))$ denote the reduction of $(A,B)$ mod $Q$
and let $F$ denote the field of fractions of $R/(Q)$. By Proposition 3.2, we
know $(\bar{A},\bar{B})\notin Y(F)$. Since $Q(\bar{A},\bar{B})=0$, by
Proposition 3.4, there exists $\gamma\in{\rm SL}_{2}(F)\times G_{0}(F)$ such
that $\gamma(\bar{A},\bar{B})=(A^{\prime},B^{\prime})$ where $(A^{\prime\,{\rm
top}},B^{\prime\,{\rm top}})$ is of the form (7). The invariant binary form of
$(A^{\prime},B^{\prime})$ has a factor of $x^{2}$, and so
$(A^{\prime},B^{\prime})\in Z(F)$. Since $Z$ is ${\rm SL}_{2}\times
G_{0}$-invariant, we see that $(\bar{A},\bar{B})\in Z(F)$.

Since $Q\mid\Delta$ in $R$, there exists an element $\delta\in R$ such that
$\Delta=Q\delta$. Let $Z_{1}$ denote the closed subvariety of $W_{0}$ cut out
by $\delta$. It now suffices to prove that $Q\mid\delta$ or, equivalently,
that the generic element $(A,B)$ belongs to $Z_{1}(R/Q)$. We claim that for
any field $k$, and every element $(A,B)\in W_{0}(k)$ such that $(A^{\rm
top},B^{\rm top})$ has the form (7), we have $\delta(A,B)=0$. Indeed, let
$(A,B)$ be such an element. Let $(A^{(\epsilon)},B^{(\epsilon)})\in
W_{0}(k[\epsilon])$ be such that $A^{(\epsilon)}=A$, the $(1,n-1)$-entry and
the $(n-1,1)$-entry of $B^{(\epsilon)}$ equal $\epsilon$, and the other
coefficients of $B^{(\epsilon)}$ are the same as those of $B$. By Lemma 3.3,
we have

$Q(A^{(\epsilon)},B^{(\epsilon)})=\pm\epsilon\,a_{1}^{g}a_{2}^{g-1}\cdots
a_{g}b_{2}^{2}\cdots b_{g}^{g}.$

Moreover, $\epsilon^{2}$ divides the $y^{n}$-coefficient of
$f_{A^{(\epsilon)},B^{(\epsilon)}}$ and $\epsilon$ divides the
$xy^{n-1}$-coefficient of $f_{A^{(\epsilon)},B^{(\epsilon)}}$. Hence
$\epsilon^{2}\mid\Delta(A^{(\epsilon)},B^{(\epsilon)})$, which implies (since
$\epsilon^{2}\nmid Q(A^{(\epsilon)},B^{(\epsilon)})$) that
$\epsilon\mid\delta(A^{(\epsilon)},B^{(\epsilon)})$. Since $(A,B)$ is obtained
from $(A^{(\epsilon)},B^{(\epsilon)})$ by setting $\epsilon=0$, we have
$\delta(A,B)=0$. We have proven that the generic element $(A,B)\in W_{0}(R)$
belongs to $Z_{1}(R/(Q))$. Therefore, $Q\mid\delta$. $\Box$

We end this section with another divisibility result for $\Delta$, which will
be used in §6.

###### Proposition 3.6

We have $\det(A^{\rm top}(A^{\rm top})^{t})\det(B^{\rm top}(B^{\rm
top})^{t})\mid\Delta$ as elements in ${\mathbb{Z}}[W_{0}]$.

Proof: It suffices to prove that $\det(B^{\rm top}(B^{\rm top})^{t})$ divides
$\Delta$ in ${\mathbb{Z}}[W_{0}]$. Suppose $(A,B)\in W_{0}({\mathbb{C}})$ with
$\det(B^{\rm top}(B^{\rm top})^{t})=0$. Then $B^{\rm top}$ does not have full
rank. Hence there exists some nonzero
$v\in\text{Span}_{\mathbb{C}}\\{e_{1},\ldots,e_{g}\\}$ such that $Bv=0$.
However, any such $v$ is isotropic with respect to $A$. As a result,
$\Delta(A,B)=0$. Thus, by the Nullstellensatz, $\det(B^{\rm top}(B^{\rm
top})^{t})\mid c\Delta^{d}$ in ${\mathbb{Z}}[W_{0}]$ for some nonzero integer
$c$ and positive integer $d$.

Define $P_{g}\in{\mathbb{Z}}[M_{g\times(g+1)}]$ by $P_{g}(M)=\det(MM^{t})$.
For the purpose of proving Proposition 3.6, it suffices to prove that $P_{g}$
is squarefree in ${\mathbb{Z}}[M_{g\times(g+1)}]$. We proceed by induction on
$g$. Denote the $(i,j)$-entry of any $M\in M_{g\times(g+1)}$ by $u_{ij}$. When
$g=1$, we have $P_{1}=u_{11}^{2}+u_{12}^{2}$, which is squarefree in
${\mathbb{Z}}[u_{11},u_{12}]$. For general $g\geq 2$, consider

$M=\begin{pmatrix}u_{11}&\cdots&u_{1\,g-1}&u_{1\,g}&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots\\\
u_{g-1\,1}&\cdots&u_{g-1\,g-1}&u_{g-1\,g}&0\\\
0&\cdots&0&\alpha&\beta\end{pmatrix}.$

Then

$\det(MM^{t})=\beta^{2}P_{g-1}+\alpha^{2}D_{g-1}^{2}$

where $D_{g-1}$ is the determinant of the top left $(g-1)\times(g-1)$ block of
$M$. Any square factor of $\det(MM^{t})$ must be a common square factor of
$P_{g-1}$ and $D_{g-1}^{2}$, which can only be $\pm 1$ since $P_{g-1}$ is
squarefree by induction. We have shown that $P_{g}$ is squarefree even after
setting certain variables to $0$. Therefore, $P_{g}$ is squarefree in
${\mathbb{Z}}[M_{g\times(g+1)}]$. $\Box$

###  3.4 Embedding ${\mathcal{W}}_{m,n}^{\rm{(2)}}$ into
$W_{n}({\mathbb{Z}})$, for $n$ odd

Let $n=2g+1$ be an odd integer, and set $W=W_{n}$. For an odd squarefree
integer $m>0$, let
${\mathcal{W}}_{m}^{\rm{(2)}}={\mathcal{W}}_{m,n}^{\rm{(2)}}$ denote the set
of integer binary forms whose discriminants are weakly divisible by $p^{2}$
for every prime factor $p$ of $m$. Fix an element
$f(x,y)\in{\mathcal{W}}_{m}^{\rm{(2)}}$. Then just as shown in [10, §3.2],
there exists an ${\rm SL}_{2}({\mathbb{Z}})$-change of variable such that
$f((x,y)\gamma)$ has the form

$f((x,y)\gamma)=m^{2}b_{0}x^{n}+mb_{1}x^{n-1}y+\cdots+b_{n}y^{n}$ (8)

for some integers $b_{0},\ldots,b_{n}$ and where $m$ and $b_{0}$ are coprime.

Consider the following pair of matrices:

$A=\left(\begin{array}[]{ccccccc}&&&&&&1\\\ &&&&&\iddots&\\\ &&&&m&&\\\
&&&c_{0}&&&\\\ &&m&&c_{2}&&\\\ &\iddots&&&&\ddots&\\\
1&&&&&&c_{n-1}\end{array}\right),\;\;B=\left(\begin{array}[]{ccccccc}&&&&&1&0\\\
&&&&\iddots&\iddots&\\\ &&&1&r&&\\\ &&1&c_{1}&&&\\\ &\iddots&r&&c_{3}&&\\\
1&\iddots&&&&\ddots&\\\ 0&&&&&&c_{n}\end{array}\right).$

Here the dots on the antidiagonal of $A$ are all $1$. We claim that $c_{i},r$
can be chosen to be integers so that $(-1)^{g}\det(xA-yB)=f((x,y)\gamma)$. It
is clear that $c_{0}=b_{0}$ and $2mrc_{0}+m^{2}c_{1}=-mb_{1}$. Choose
$r\in{\mathbb{Z}}$ such that $m\mid 2rc_{0}+b_{1}$; this then determines
$c_{1}$. It is then not hard to check that the coefficient of $x^{n-i}y^{i}$
in $(-1)^{g}\det(xA-yB)$ is of the form
$(-1)^{i}c_{i}+L(c_{0},\ldots,c_{i-1})$ where $L$ is a linear form with
coefficients in ${\mathbb{Z}}[r]$. The existence of integers
$c_{2},\ldots,c_{n}$ now follows by induction.

Set $\sigma_{m}(f)=\sigma_{m,n}(f)$ to be the element $(A_{f},B_{f})$ such
that

$\begin{pmatrix}A_{f}\\\ -B_{f}\end{pmatrix}=\gamma^{-1}\begin{pmatrix}A\\\
-B\end{pmatrix}.$

Then $f_{\sigma_{m}(f)}=f.$ Next, we note that $(A,B)$ and thus
$(A_{f},B_{f})$ are in $W_{0}({\mathbb{Z}})$ and from Lemma 3.3, we obtain
that $|Q|(A,B)=m$. Since $Q$ is ${\rm SL}_{2}$-invariant, we conclude that

$|Q|({\sigma_{m}(f)})=m.$

We have proven the following theorem.

###### Theorem 3.7

Let $m>0$ be a squarefree integer. There exists a map
$\sigma_{m}:{\mathcal{W}}_{m}^{\rm{(2)}}\to W_{0}({\mathbb{Z}})$ such that

$f_{\sigma_{m}(f)}=f,\qquad|Q|(\sigma_{m}(f))=m$

for every $f\in{\mathcal{W}}_{m}^{\rm{(2)}}$.

We will later use the image of $\sigma_{1}$ as a fundamental set for the
action of ${\rm SL}_{n}({\mathbb{R}})$ on the set of distinguished elements of
$W({\mathbb{R}})$. We now extend the function $|Q|$ to the set of
distinguished elements of $W({\mathbb{Z}})$ having irreducible invariant
binary form. Suppose that $(A,B)$ is a distinguished element of
$W({\mathbb{Z}})$. Then there is a $g$-dimensional subspace $X$ isotropic with
respect to $A$ and $B$. Let $\Lambda=X\cap{\mathbb{Z}}^{n}$ be the primitive
lattice in $X$. There exists an element $\gamma$ in ${\rm
SL}_{n}({\mathbb{Z}})$, unique up to left multiplication by an element in
$G_{0}({\mathbb{Z}})$, such that
$\Lambda=\gamma^{t}\cdot\,\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g}\\}$,
where $e_{1},\ldots,e_{n}$ is the standard basis of ${\mathbb{Z}}^{n}$. Then
$\gamma\cdot(A,B)\in W_{0}({\mathbb{Z}})$, and we can thus define the
$|Q|$-invariant on the triple $(A,B,\Lambda)$ by

$|Q|(A,B,\Lambda):=|Q|(\gamma\cdot(A,B)).$

That is, we complete an integral basis of $\Lambda$ to an integral basis of
${\mathbb{Z}}^{n}$ with respect to which the pair $(A^{\prime},B^{\prime})$ of
Gram matrices for the quadratic forms defined by $A$ and $B$ lies inside
$W_{0}({\mathbb{Z}})$, and we define $|Q|(A,B,\Lambda)$ to be
$|Q|(A^{\prime},B^{\prime}).$

We end with the following result that will be crucial in Section 4.

###### Proposition 3.8

Let $n=2g+1$ be an odd integer with $n\geq 3$. Let $m$ be an odd positive
squarefree integer. Let $f(x,y)\in{\mathcal{W}}_{m}^{(2)}$ be an irreducible
integral binary $n$-ic form. Let $(A,B)$ be any element in ${\rm
SL}_{n}({\mathbb{Z}})\cdot\sigma_{m}(f)$. Then there is a unique primitive
$g$-dimensional lattice $\Lambda$ that is isotropic with respect to both $A$
and $B$. Moreover, $|Q|(A,B):=|Q|(A,B,\Lambda)=m$. In particular, if
$f(x,y)\in{\mathcal{W}}_{m}^{(2)}\cap{\mathcal{W}}_{m^{\prime}}^{(2)}$ is
irreducible where $m$ and $m^{\prime}$ are distinct odd positive squarefree
integers, then $\sigma_{m}(f(x,y))$ and $\sigma_{m^{\prime}}(f(x,y))$ are not
${\rm SL}_{n}({\mathbb{Z}})$-equivalent.

Proof: Let $C_{f}$ denote the smooth hyperelliptic curve $z^{2}=f(x,y)y$ of
genus $g$ viewed as a curve in the weighted projective space
${\mathbb{P}}(1,1,g+1)$, and let $J_{f}$ denote its Jacobian. Since $(A,B)$ is
${\rm SL}_{n}({\mathbb{Z}})$-equivalent to $\sigma_{m}(f)$, it follows that
$(A,B)$ is distinguished. Thus the set of common isotropic $g$-dimensional
subspaces of $A$ and $B$ over ${\mathbb{Q}}$ is in bijection with
$J_{f}[2]({\mathbb{Q}})$. Since $f$ is irreducible, we have
$J_{f}[2]({\mathbb{Q}})=1$. Therefore, there is a unique primitive
$g$-dimensional lattice $\Lambda$ which is isotropic with respect to both $A$
and $B$.

Let $\gamma\in{\rm SL}_{n}({\mathbb{Z}})$ be an element such that
$\gamma(A,B)=\sigma_{m}(f)=:(A_{f},B_{f})\in W_{0}({\mathbb{Z}})$. Since we
know that $\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g}\\}$ is a primitive
$g$-dimensional lattice isotropic with respect to $A_{f}$ and $B_{f}$, we see
that $\gamma^{t}\cdot\,\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g}\\}$ is a
primitive $g$-dimensional lattice isotropic with respect to $A$ and $B$. By
uniqueness, it follows that
$\Lambda=\gamma^{t}\cdot\,\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g}\\}$,
and so by definition, $|Q|(A,B,\Lambda)=|Q|(\sigma_{m}(f))=m$, where the final
equality is Theorem 3.7. $\Box$

###  3.5 Embedding ${\mathcal{W}}_{m,n}^{(2),\,{\rm gen}}$ into
$W_{n+1}({\mathbb{Z}})$, for $n$ even

Suppose now that $n=2g+2$ is even with $g\geq 1$. For an odd squarefree
integer $m>0$, let ${\mathcal{W}}_{m,n}^{\rm{(2)}}$ denote the set of integer
binary forms having discriminant weakly divisible by $p^{2}$ for every prime
factor $p$ of $m$. Let ${\mathcal{W}}_{m,n}^{(2),\,{\rm
gen}}\subset{\mathcal{W}}_{m,n}^{(2)}$ consist of those $f(x,y)$ with $f(0,1)$
coprime to $m$. Since $\Delta(xf(x,y))=\Delta(f(x,y))f(0,1)^{2}$, we see that
if $f(x,y)\in{\mathcal{W}}_{m,n}^{(2),\,{\rm gen}}$, then
$xf(x,y)\in{\mathcal{W}}_{m,n+1}^{(2)}$. We define
$\sigma_{m,n}:{\mathcal{W}}_{m,n}^{(2),\,{\rm gen}}\rightarrow
W_{n+1}({\mathbb{Z}})$ via $\sigma_{m,n}(f)=\sigma_{m,n+1}(xf)$. For the rest
of this subsection, we drop subscripts and denote $\sigma_{m,n}$ by
$\sigma_{m}$, ${\mathcal{W}}_{m,n}^{{\rm{(2)}},\,{\rm gen}}$ by
${\mathcal{W}}_{m}^{{\rm{(2)}}\,{\rm gen}}$, and $W_{n+1}$ by $W$.

We now define the finer $q$-invariant. Let $f\in{\mathcal{W}}_{m}^{(2),\,{\rm
gen}}$ and suppose $(A,B)=\sigma_{m}(f)$. Then $B$ is singular since $xf(x,y)$
has vanishing $y^{n}$-term. Moreover, since $\Delta(xf)\neq 0$, the kernel of
$B$ has dimension exactly $1$ and is not isotropic with respect to $A$ (see
Lemma 6.4). Fix an integral domain $D$. Let $W_{1}(D)$ be the subset of $W(D)$
consisting of pairs $(A,B)$ of symmetric $(n+1)\times(n+1)$ matrices
satisfying the following conditions:

  * (a)

The top left $(g+1)\times(g+1)$ block of $A$ is $0$.

  * (b)

The top left $(g+2)\times(g+2)$ block of $B$ is $0$ (implying that $B$ is
singular).

  * (c)

The kernel of $B$ has dimension exactly $1$ (over the fraction field of $D$)
and is not isotropic with respect to $A$.

Take any $(A,B)\in W_{1}(D)$. Conditions (a) and (c) imply that the first
$g+1$ columns of $B$ are linearly independent over the fraction field of $D$.
Let $B^{\prime}$ denote the top right $(g+1)\times(g+1)$ block of $B$. Since
$B$ is symmetric, we see that $B^{\prime}$ is nonsingular. It is now easy to
see from the definition of the $Q$-invariant that as polynomials in the
coordinates of $W_{1}(D)$, we have

$\det(B^{\prime})\mid Q(A^{\rm top},B^{\rm top}).$

We define the quotient to be the $q$-invariant of $(A,B)$:

$q(A,B):=Q(A^{\rm top},B^{\rm top})/\det(B^{\prime}).$ (9)

Let $G_{1}(D)$ denote the subgroup of ${\rm SL}_{n+1}(D)$ preserving
$W_{1}(D)$. Then elements of $G_{1}(D)$ have the following block matrix form

$\gamma=\left(\begin{array}[]{ccc}\gamma_{1}&0&0\\\ n_{1}&\gamma_{2}&0\\\
n_{2}&n_{3}&\gamma_{3}\end{array}\right)\in\left(\begin{array}[]{ccc}{\rm
GL}_{g+1}&&\\\ M_{1\times(g+1)}&{\rm GL}_{1}&\\\
M_{(g+1)\times(g+1)}&M_{(g+1)\times 1}&{\rm GL}_{g+1}\end{array}\right).$ (10)

It is easy to check that for any $(A,B)\in W_{1}(D)$,

$q(\gamma(A,B))=\det(\gamma_{1})(\det(\gamma_{1})\det(\gamma_{3}))^{-1}q(A,B)=\det(\gamma_{1})\gamma_{2}\,q(A,B).$
(11)

We now consider the situation over ${\mathbb{Z}}$. Let $(A,B)\in
W({\mathbb{Z}})$ be a distinguished element having nonzero discriminant such
that $B$ is singular. Let $X$ denote a common isotropic $(g+1)$-dimiensional
subspace of $A$ and $B$. We know that the kernel $\langle v\rangle$ of $B$ has
trivial intersection with $X$. Denote the span of $X$ and $v$ by $X^{\prime}$,
which is a $(g+2)$-dimensional subspace containing $X$ that is isotropic with
respect to $B$. Let $\Lambda=X\cap{\mathbb{Z}}^{n+1}$ and
$\Lambda^{\prime}=X^{\prime}\cap{\mathbb{Z}}^{n+1}$ be the primitive lattices
in $X$ and $X^{\prime}$, respectively. There exists an element $\gamma$ in
${\rm SL}_{n+1}({\mathbb{Z}})$, unique up to left multiplication by an element
in $G_{1}({\mathbb{Z}})$, such that
$\Lambda=\gamma^{t}.\,\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g+1}\\}$ and
$\Lambda^{\prime}=\gamma^{t}.\,\text{Span}_{\mathbb{Z}}\\{e_{1},\ldots,e_{g+2}\\}$.
Then $\gamma(A,B)\in W_{1}({\mathbb{Z}})$, and we can thus define the
$|q|$-invariant for the quadruple $(A,B,\Lambda,\Lambda^{\prime})$ by

$|q|(A,B,\Lambda,\Lambda^{\prime}):=|q(\gamma(A,B))|.$

In other words, we complete an integral basis $\\{v_{1},\ldots,v_{g+1}\\}$ of
$\Lambda$ to an integral basis $\\{v_{1},\ldots,v_{n+1}\\}$ of
${\mathbb{Z}}^{n+1}$ such that $\\{v_{1},\ldots,v_{g+2}\\}$ forms an integral
basis of $\Lambda^{\prime}$. When expressed in this basis, the pair
$(A^{\prime},B^{\prime})$ of Gram matrices for the quadratic forms defined by
$A$ and $B$ lies in $W_{1}({\mathbb{Z}})$ and we define
$|q|(A,B,\Lambda,\Lambda^{\prime}):=|q|(A^{\prime},B^{\prime}).$

Finally, we compute the $|Q|$\- and $|q|$-invariants of $\sigma_{m}(f(x,y))$,
where $f(x,y)\in{\mathcal{W}}_{m}^{(2),\,{\rm gen}}$ is irreducible.

###### Proposition 3.9

Let $n=2g+2$ with $g\geq 1$. Let $m$ be an odd positive squarefree integer.
Let $f(x,y)\in{\mathcal{W}}_{m}^{(2),\,{\rm gen}}$ be irreducible. Let $(A,B)$
be any element in ${\rm SL}_{n+1}({\mathbb{Z}})\cdot\sigma_{m}(f(x,y))$. Let
$\Lambda$ be a $(g+1)$-dimensional primitive lattice contained in a
$(g+2)$-dimensional primitive lattice $\Lambda^{\prime}$ such that $\Lambda$
is isotropic with respect to $A$ and $\Lambda^{\prime}$ is isotropic with
respect to $B$. Then $|Q|(A,B,\Lambda)$ is either $m$ or $|f(0,1)|m$, and
$|q|(A,B):=|q|(A,B,\Lambda,\Lambda^{\prime})=m$, independent of
$(\Lambda,\Lambda^{\prime})$. In particular, if
$f(x,y)\in{\mathcal{W}}_{m}^{(2),\,{\rm
gen}}\cap{\mathcal{W}}_{m^{\prime}}^{(2),\,{\rm gen}}$ is irreducible where
$m$ and $m^{\prime}$ are distinct odd positive squarefree integers, then
$\sigma_{m}(f(x,y))$ and $\sigma_{m^{\prime}}(f(x,y))$ are not ${\rm
SL}_{n+1}({\mathbb{Z}})$-equivalent.

Proof: The size of $J_{f}[2]({\mathbb{Q}})$ is $2$ since $xf(x,y)$ has a
unique even degree factor (namely, $f(x,y)$) over ${\mathbb{Q}}$. Therefore,
the pair $(A,B)$ has two $(g+1)$-dimensional common isotropic subspaces
$X_{1}$ and $X_{2}$ over ${\mathbb{Q}}$. Let $\Lambda_{1}$ and $\Lambda_{2}$
denote the corresponding primitive lattices contained in $X_{1}$ and $X_{2}$.
The unique $(g+2)$-dimensional subspace $X_{1}^{\prime}$ (resp.,
$X_{2}^{\prime}$) isotropic with respect to $B$ and containing $X_{1}$ (resp.,
$X_{2}$) is the span of $X_{1}$ (resp., $X_{2}$) with the kernel of $B$. Let
$\Lambda^{\prime}_{1}$ and $\Lambda_{2}^{\prime}$ denote the primitive
lattices contained in $X_{1}^{\prime}$ and $X_{2}^{\prime}$. We compute the
$|Q|$\- and $|q|$-invariants associated to these lattices.

We may assume that $(A,B)=\sigma_{m}(f)$ since the action of ${\rm
SL}_{n+1}({\mathbb{Z}})$ does not change the $|Q|$\- or $|q|$ invariants.
Since $|Q|$ is ${\rm SL}_{2}$-invariant, and $|q|$ remains unchanged when we
add a multiple of $B$ to $A$, we may also assume that

$xf(x,y)=m^{2}b_{0}x^{n+1}+mb_{1}x^{n}y+\cdots+b_{n}xy^{n}$

and so

$A=\left(\begin{array}[]{ccccccc}&&&&&&1\\\ &&&&&\iddots&\\\ &&&&m&&\\\
&&&c_{0}&&&\\\ &&m&&c_{2}&&\\\ &\iddots&&&&\ddots&\\\
1&&&&&&c_{n}\end{array}\right),\;\;B=\left(\begin{array}[]{ccccccc}&&&&&1&0\\\
&&&&\iddots&\iddots&\\\ &&&1&r&&\\\ &&1&c_{1}&&&\\\ &\iddots&r&&c_{3}&&\\\
1&\iddots&&&&\ddots&\\\ 0&&&&&&c_{n+1}\end{array}\right).$

Comparing the $y^{n+1}$\- and the $xy^{n}$-coefficients, we have $c_{n+1}=0$
and $c_{n}=b_{n}$.

Let $\langle\,,\rangle_{A}$ and $\langle\,,\rangle_{B}$ denote the quadratic
forms corresponding to $A$ and $B$. Let $e_{1},\ldots,e_{n+1}$ be the standard
basis on ${\mathbb{Z}}^{n+1}$. Since $c_{n+1}=0$, the vector $e_{n+1}$ spans
the kernel of $B$. We may take the subspace spanned by $e_{1},\ldots,e_{g+1}$
as $X_{1}$. Then by construction $|Q|(A,B,\Lambda_{1})=m$. When expressed in
terms of the ordered integral basis
$\\{e_{1},\ldots,e_{g+1},e_{n},e_{g+2},\ldots,e_{n-1}\\}$, the top right
$(g+1)\times(g+1)$ block of $B$ has $1$’s on the antidiagonal and $0$’s above
the antidiagonal, and so has determinant $\pm 1$. Hence
$|q|(A,B,\Lambda_{1},\Lambda_{1}^{\prime})=m$.

The second common isotropic $(g+1)$-dimensional subspace $X_{2}$ is the
reflection of $X_{1}$ in the hyperplane perpendicular to $e_{n+1}$ with
respect to $\langle\,,\rangle_{A}$. That is,

$\displaystyle X_{2}$ $\displaystyle=$ $\displaystyle\displaystyle{\rm
Span}_{\mathbb{Q}}\Bigl{\\{}e_{1}-\frac{2\langle
e_{1},e_{n+1}\rangle_{A}}{\langle
e_{n+1},e_{n+1}\rangle_{A}}e_{n+1},\ldots,e_{g+1}-\frac{2\langle
e_{g+1},e_{n+1}\rangle_{A}}{\langle
e_{n+1},e_{n+1}\rangle_{A}}e_{n+1}\Bigr{\\}}$ $\displaystyle=$
$\displaystyle\displaystyle{\rm
Span}_{\mathbb{Q}}\Bigl{\\{}e_{1}-\frac{2}{b_{n}}e_{n+1},e_{2},e_{3},\ldots,e_{g+1}\Bigr{\\}}.$

Suppose first that $b_{n}$ is odd. Then we have the following integral basis
for ${\mathbb{Z}}^{n+1}$:

$\Big{\\{}b_{n}e_{1}-2e_{n+1},e_{2},e_{3},\ldots,e_{g+1},\frac{b_{n}+1}{2}e_{1}-e_{n+1},e_{g+2},\ldots,e_{n}\Big{\\}}.$

In terms of this basis, the top right $(g+1)\times(g+2)$ blocks of $A$ and $B$
have the following form:

$A^{\rm top}=\left(\begin{array}[]{cccccc}-1&0&0&\cdots&\cdots&0\\\
0&0&0&&\iddots&1\\\ \vdots&\vdots&\vdots&\iddots&\iddots&\vdots\\\
\vdots&\vdots&0&1&&\vdots\\\
0&0&m&\cdots&\cdots&0\end{array}\right),\;\;B^{\rm
top}=\left(\begin{array}[]{cccccc}0&&&&&b_{n}\\\ 0&&&&1&\\\
\vdots&&&\iddots&&\\\ \vdots&&\iddots&&&\\\ 0&1&&&&\end{array}\right).$ (12)

It is then easy to check that $|Q|(A,B,\Lambda_{2})=|b_{n}|m$ and
$|q|(A,B,\Lambda_{2},\Lambda_{2}^{\prime})=m$.

When $b_{n}$ is even, we have the following integral basis

$\Big{\\{}\frac{b_{n}}{2}e_{1}-e_{n},e_{2},e_{3},\ldots,e_{g+1},(b_{n}+1)e_{1}-2e_{n+1},e_{g+2},\ldots,e_{n}\Big{\\}}.$

In terms of this basis, the top right $(g+1)\times(g+2)$ blocks of $A$ and $B$
have the same form as in (12). Hence the $|Q|$\- and $|q|$-invariants are as
stated in the proposition. $\Box$

##  4 A uniformity estimate for odd degree polynomials

Throughout this section, we fix an odd integer $n=2g+1$ with $g\geq 1$. Our
goal is to prove Theorem 5(b) by obtaining a bound on the number of integral
binary $n$-ic forms having bounded height and discriminant weakly divisible by
the square of a large squarefree integer.

Let $m>0$ be an odd squarefree integer. Recall that we defined a map
$\sigma_{m}:{\mathcal{W}}^{(2)}_{m}\to W_{0}({\mathbb{Z}})$ in Theorem 3.7
with the following two properties: $f_{\sigma_{m}(f)}=f$ for every
$f\in{\mathcal{W}}^{(2)}_{m}$, and $|Q|(\sigma_{m}(f))=m$. Moreover, in
Proposition 3.8, we proved that when $f\in{\mathcal{W}}_{m}^{(2)}$ is
irreducible, it is possible to naturally extend the definition of the
$|Q|$-invariant to the set ${\rm SL}_{n}({\mathbb{Z}})\cdot\sigma_{m}(f)$.

Let $W({\mathbb{Z}})^{\rm dist}$ denote the set of distinguished elements in
$W({\mathbb{Z}})$, and for any set $L\subset W({\mathbb{Z}})$, let $L^{\rm
irr}$ denote the set of elements $w\in L$ such that $f_{w}$ is irreducible.
There is a natural extension of the $|Q|$-invariant to the set
$W({\mathbb{Z}})^{{\rm dist},{\rm irr}}$. For a positive real number $M$ and
any set $S\subset W({\mathbb{Z}})^{{\rm dist},{\rm irr}}$, let $S_{|Q|>M}$
denote the set of elements $w\in S$ with $|Q(w)|>M$. By [5], the number of
reducible elements $f\in V({\mathbb{Z}})$ with $H(f)<X$ is $O(X^{n})$. Hence
we have the bound

$\\#\bigcup_{\begin{subarray}{c}m>M\\\ {\rm
squarefree}\end{subarray}}\bigl{\\{}f\in{\mathcal{W}}_{m}^{(2)}:H(f)<X\bigr{\\}}\ll\\#\bigl{(}{\rm
SL}_{n}({\mathbb{Z}})\backslash\bigl{\\{}w\in W({\mathbb{Z}})^{{\rm dist},{\rm
irr}}_{|Q|>M}:H(w)<X\\}\bigr{)}+O(X^{n}).$ (13)

In this section, we obtain an upper bound on the number of ${\rm
SL}_{n}({\mathbb{Z}})$-orbits on $W({\mathbb{Z}})^{{\rm dist},{\rm
irr}}_{|Q|>M}$ with height bounded by $X$. First, in §4.1, we lay out the
reduction theory necessary to express the number of such orbits in terms of
the counts of lattice points in certain bounded regions. Then in §4.2, we
partition these regions into three parts, the main body, the shallow cusp, and
the deep cusp. We prove the desired estimate for each of these parts, thereby
obtaining Theorem 5(b).

###  4.1 Reduction theory and averaging over fundamental domains

Recall that the Iwasawa decomposition of ${\rm SL}_{n}({\mathbb{R}})$ is given
by ${\rm SL}_{n}({\mathbb{R}})=NTK,$ where $N$ is the group of unipotent lower
triangular matrices in ${\rm SL}_{n}({\mathbb{R}})$, $K={\rm SO}(n)$ is a
maximal compact subgroup of ${\rm SL}_{n}({\mathbb{R}})$, and $T$ is the split
torus of ${\rm SL}_{n}({\mathbb{R}})$ consisting of $n\times n$ diagonal
matrices with positive diagonal entries and determinant $1$. We denote
elements in $T$ by $s={\rm diag}(t_{1}^{-1},t_{2}^{-1},\ldots,t_{n}^{-1})$,
where $t_{i}>0$ for $1\leq 1\leq n$ and $t_{1}t_{2}\cdots t_{n}=1$. It will be
convenient to make the following change of variables. For $1\leq i\leq n-1$,
set $s_{i}$ to be

$s_{i}=(t_{i}/t_{i+1})^{1/n},\mbox{ which implies
}\;t_{i}=\prod_{k=1}^{i-1}s_{k}^{-k}\prod_{k=i}^{n-1}s_{k}^{n-k}$

for $1\leq i<n$. The Haar measure of $G({\mathbb{R}})$ in these coordinates is
then given by

$dg=dn\delta(s)d^{\times}sdk,\;\;\;\;\;\mbox{where}\;\;\;\;\;\delta(s)=\prod_{1\leq
i<j\leq n}\frac{t_{j}}{t_{i}}=\prod_{i=1}^{n-1}s_{k}^{-nk(n-k)},$

$dn$ and $dk$ are Haar measures on $N$ and $K$, respectively, and
$d^{\times}s=\prod_{i=1}^{n-1}s_{i}^{-1}ds_{i}$.

We denote the coordinates on $W$ by $a_{ij},b_{ij}$ for $1\leq i\leq j\leq n$.
These coordinates are eigenvectors for the action of $T$ on the dual $W^{*}$
of $W$. Denote the $T$-weight of a coordinate $\alpha$ on $W$, or more
generally a product $\alpha$ of powers of such coordinates, by $w(\alpha)$.
Then $w(a_{ij})=w(b_{ij})=t_{i}^{-1}t_{j}^{-1}.$ It will be useful in what
follows to compute the weight of the $Q$-invariant, which is a homogeneous
polynomial of degree $g(g+1)$ in the coordinates of $W_{0}$. We view the torus
$T$ as sitting inside $G_{0}$. Then by (5), we have

$w(Q)=\prod_{k=1}^{g}t_{k}^{-1}.$ (14)

Let ${\mathcal{F}}$ be a fundamental set for the action of ${\rm
SL}_{n}({\mathbb{Z}})$ on ${\rm SL}_{n}({\mathbb{R}})$ that is contained in a
Siegel set, i.e., contained in $N^{\prime}T^{\prime}K$, where $N^{\prime}$ is
a set consisting of elements in $N$ whose coefficients are absolutely bounded
and $T^{\prime}\subset T$ consists of elements in $s\in T$ with $s_{i}\geq c$
for some positive constant $c$. Let ${\mathcal{W}}(1)$ denote the subset of
real binary $n$-ic forms of height bounded by $1$ and let
$R^{\prime}=\sigma_{1}({\mathcal{W}}(1))$, where $\sigma_{1}$ is as in §3.4.
Set $R:={\mathbb{R}}_{>0}\cdot R^{\prime}$, and note that every distinguished
element of $W({\mathbb{R}})$ is ${\rm SL}_{n}({\mathbb{R}})$-equivalent to
some element in $R$.

Let $H_{0}$ be a nonempty open bounded left $K$-invariant set in ${\rm
SL}_{n}({\mathbb{R}})$. Denote the set $H_{0}\cdot R^{\prime}$ by
${\mathcal{B}}_{1}$. Then ${\mathcal{B}}_{1}$ is an absolutely bounded set in
$W({\mathbb{R}})$. Let ${\mathcal{L}}$ be any ${\rm
SL}_{n}({\mathbb{Z}})$-invariant subset of $W({\mathbb{Z}})$ consisting of
elements that are distinguished over ${\mathbb{R}}$, and denote the set of
elements in ${\mathcal{L}}$ with height less than $X$ by ${\mathcal{L}}_{X}$.
Throughout this section, let $Y=X^{1/n}$. Then the averaging method as
described in [9, §2.3] yields the bound

$\displaystyle\\#\bigl{(}{\rm
SL}_{n}({\mathbb{Z}})\backslash{\mathcal{L}}_{X}\bigr{)}\;\ll\;\displaystyle\int_{\gamma\in{\mathcal{F}}}\\#\bigl{(}\gamma(Y{\mathcal{B}}_{1})\cap{\mathcal{L}}\bigr{)}d\gamma\;\ll\;\displaystyle\int_{\begin{subarray}{c}s=(s_{i})_{i}\\\
s_{i}\geq
c\end{subarray}}\\#\bigl{(}s(Y{\mathcal{B}})\cap{\mathcal{L}}\bigr{)}\delta(s)d^{\times}s$
(15)

for some absolutely bounded open set ${\mathcal{B}}$ containing
${\mathcal{B}}_{1}$.

We denote the second integral on the right hand side of (15) by
${\mathcal{I}}_{X}({\mathcal{L}})$, and break it up into an integral over the
main body, the shallow cusp, and the deep cusp. We define the main body to be
the range of the integral where $|a_{11}|\geq 1$ for some element in
$s(Y{\mathcal{B}})$, and denote the main-body portion of
${\mathcal{I}}_{X}({\mathcal{L}})$ by ${\mathcal{I}}_{X}^{\rm
main}({\mathcal{L}})$. We define the shallow cusp to be the range of the
integral where $|a_{11}|<1$ for all elements in $s(Y{\mathcal{B}})$ but
$|a_{ij}|\geq 1$ for some $i,j\leq g$, and denote the shallow-cusp portion of
${\mathcal{I}}_{X}({\mathcal{L}})$ by ${\mathcal{I}}_{X}^{\rm
scusp}({\mathcal{L}})$. We define the deep cusp to be the range of the
integral where $|a_{ij}|<1$ for all $i,j\leq g$ and all elements in
$s(Y{\mathcal{B}})$, and denote the deep-cusp portion of
${\mathcal{I}}_{X}({\mathcal{L}})$ by ${\mathcal{I}}_{X}^{\rm
dcusp}({\mathcal{L}})$. Then

${\mathcal{I}}_{X}({\mathcal{L}})={\mathcal{I}}_{X}^{\rm
main}({\mathcal{L}})+{\mathcal{I}}_{X}^{\rm
scusp}({\mathcal{L}})+{\mathcal{I}}_{X}^{\rm dcusp}({\mathcal{L}}).$ (16)

In the next subsection, we prove bounds for the main body, the shallow cusp,
and the deep cusp when ${\mathcal{L}}=W({\mathbb{Z}})_{|Q|>M}^{{\rm dist},{\rm
irr}}$.

###  4.2 The number of orbits of distinguished elements with large
$Q$-invariant

In this subsection, we obtain the following upper bound on
${\mathcal{I}}_{X}(W({\mathbb{Z}})^{{\rm dist},{\rm irr}}_{|Q|>M})$, thus
yielding the same bound on the quantity $\\#\bigl{(}{\rm
SL}_{n}({\mathbb{Z}})\backslash\\{w\in W({\mathbb{Z}})^{{\rm dist},{\rm
irr}}_{|Q|>M}:H(w)<X\\}\bigr{)}$ by (15).

###### Theorem 4.1

We have ${\mathcal{I}}_{X}(W({\mathbb{Z}})^{{\rm dist},{\rm
irr}}_{|Q|>M})\ll_{\epsilon}X^{n+1-1/{5n}+\epsilon}+{X^{n+1+\epsilon}}/{M}.$

Note that (13), (15), and Theorem 4.1 immediately imply Part (b) of Theorem 5.

We bound ${\mathcal{I}}_{X}(W({\mathbb{Z}})^{{\rm dist},{\rm irr}}_{|Q|>M})$
by obtaining bounds for the main body, the shallow cusp, and the deep cusp. We
consider first the main body.

###### Proposition 4.2

We have ${\mathcal{I}}_{X}^{\rm main}(W({\mathbb{Z}})^{\rm
dist})\ll_{\epsilon}X^{n+1-1/{(5n)}+\epsilon}.$

Proof: In [17, Proposition 4.6], an upper bound of $o(X^{n+1})$ is obtained on
${\mathcal{I}}_{X}^{\rm main}(W({\mathbb{Z}})^{\rm dist})$. This is proved
using the following two ingredients: estimates with a power saving error tern
on ${\mathcal{I}}_{X}^{\rm main}({\mathcal{L}})$ for lattices
${\mathcal{L}}\subset W({\mathbb{Z}})$, and a proof that the density of
elements in $W({\mathbb{F}}_{p})$ that are not
${\mathbb{F}}_{p}$-distinguished is bounded away from $1$, independent of $p$.
Proposition 4.2 follows from these two ingredients along with an application
of the Selberg sieve used exactly as in [28]. $\Box$

Next, a bound on the shallow cusp follows directly from the proof of [17,
Proposition 4.3]:

###### Proposition 4.3

We have ${\mathcal{I}}_{X}^{\rm scusp}(W({\mathbb{Z}}))\ll X^{n+1-1/{n}}.$

In [17, Proposition 4.3], the shallow and deep cusps were treated
simultaneously, but the points in the deep cusp were ruled out since only
nondistinguished elements were counted there. Hence the proof of [17,
Proposition 4.3] yields the claimed bound in Proposition 4.3.

Finally, to treat the deep cusp, let $U=\\{a_{ij},b_{ij}:1\leq i\leq j\leq
n\\}$ denote the set of coordinates on $W$, and let
$U_{0}=\\{a_{ij},b_{ij}\mid i\leq j,\>j\geq g+1\\}$ denote the set of
coordinates on $W_{0}$. We define a partial order $\lesssim$ on $U$ by setting
$\alpha\lesssim\beta$ if all the powers of $s_{i}$ in $w(\alpha)^{-1}w(\beta)$
are nonnegative. Explicitly, $a_{ij}\leq a_{i^{\prime}j^{\prime}}$ if and only
if $i\leq i^{\prime}$ and $j\leq j^{\prime}$ (and similarly for $b_{ij}$, as
$a_{ij}$ and $b_{ij}$ have the same weight). A subset ${\mathcal{Z}}$ of
$U_{0}$ is saturated if for any $\beta\in{\mathcal{Z}}$ and any $\alpha\in
U_{0}$ with $\alpha\lesssim\beta$, the coordinate $\alpha$ also lies in
${\mathcal{Z}}$. We pick positive constants $c_{ij}$ for $1\leq i\leq j\leq n$
such that:

  1. (a)

If $|Yw(a_{ij})|<c_{ij}$, then $|a_{ij}|<1$ and $|b_{ij}|<1$ for every
$(A,B)\in s(Y{\mathcal{B}})$.

  2. (b)

For all $s\in T^{\prime}$ and $a_{ij}\lesssim a_{i^{\prime}j^{\prime}}$, we
have $w(a_{ij})/c_{ij}\leq
w(a_{i^{\prime}j^{\prime}})/c_{i^{\prime}j^{\prime}}$.

More explicitly, we may choose $c_{nn}$ to be sufficiently small and take

$c_{ij}=\Bigl{(}\sup_{s\in
T^{\prime}}\frac{w(a_{ij})}{w(a_{nn})}\Bigr{)}c_{nn},\quad\mbox{for }i\leq
j\leq n.$

The significance of these constants $c_{ij}$ is the following: for every
$Y>1$, first, if $Yw(a_{ij})<c_{ij}$, then every integral element in
$s(Y{\mathcal{B}})$ has $a_{ij}$\- and $b_{ij}$-coordinates equal to $0$; and
second, if $a_{ij}\lesssim a_{i^{\prime}j^{\prime}}$, then
$Yw(a_{i^{\prime}j^{\prime}})<c_{i^{\prime}j^{\prime}}$ implies
$Yw(a_{ij})<c_{ij}$.

We have the following result due to Davenport which we use to estimate the
number of lattice points in bounded regions.

######  Proposition 4.4 ([12])

Let ${\mathcal{R}}$ be a bounded, semi-algebraic multiset in
${\mathbb{R}}^{n}$ having maximum multiplicity $m$ that is defined by at most
$k$ polynomial inequalities, each having degree at most $\ell$. Let
${\mathcal{R}}^{\prime}$ denote the image of ${\mathcal{R}}$ under any
$($upper or lower$)$ triangular, unipotent transformation of
${\mathbb{R}}^{n}$. Then the number of lattice points $($counted with
multiplicity$)$ contained in the region ${\mathcal{R}}^{\prime}$ is given by

${\rm Vol}({\mathcal{R}})+O({\rm max}\\{\overline{{\rm
Vol}}(\overline{{\mathcal{R}}}),1\\}),$

where $\overline{\rm Vol}(\overline{{\mathcal{R}}})$ denotes the greatest
$d$-dimensional volume of any projection of ${\mathcal{R}}$ onto a coordinate
subspace obtained by equating $n-d$ coordinates to zero, as $d$ ranges over
all values in $\\{1,\dots,n-1\\}$. The implied constant in the second summand
depends only on $n$, $m$, $k$, and $\ell$.

The following lemma gives conditions that ensure an element in
$W({\mathbb{R}})$ has discriminant $0$.

###### Lemma 4.5

Suppose that $(A,B)\in W({\mathbb{R}})$ satisfies $a_{ij}=b_{ij}=0$ for all
$i\leq k$ and $j\leq n-k$ for some $k\in\\{1,\ldots,g\\}$. Then the
discriminant of $(A,B)$ is $0$.

Proof: One checks that $f_{A,B}$ has a square factor of degree $k$ and so has
discriminant $0$. $\Box$

The next lemma states that when ${\mathcal{L}}\subset W({\mathbb{Z}})$
consists of elements with nonzero discriminant, the integral defining
${\mathcal{I}}_{X}({\mathcal{L}})$ can be cut off by conditions of the form
$s_{i}\ll X^{\Theta}$ for some absolute constant $\Theta$ depending only on
$n$.

###### Lemma 4.6

There exists an absolute constant $\Theta$ depending only on $n$ such that if
$s\in T^{\prime}$ with $s_{i}\gg X^{\Theta}$ for some $i$, then
$s(Y{\mathcal{B}})\cap W({\mathbb{Z}})$ contains only points with discriminant
$0$.

Proof: Let $s={\rm diag}(t_{1}^{-1},\cdots,t_{n}^{-1})\in T^{\prime}$; then
$t_{1}\gg t_{2}\gg\cdots\gg t_{n}$ and $t_{1}t_{2}\cdots t_{n}=1$. Because of
the relation between the $t_{j}$’s and the $s_{i}$’s, it suffices to prove
that if $s(Y{\mathcal{B}})$ contains an integral element with nonzero
discriminant, then $t_{1}$ is bounded from above by some power of $X$ or,
equivalently, $t_{n}$ is bounded from below by some power of $X$. By Lemma
4.5, for $s(Y{\mathcal{B}})$ to contain an integral element with nonzero
discriminant, we must have $Yw(a_{k,n-k})\gg 1$ for every
$k\in\\{1,\ldots,g\\}$. That is, $t_{k}t_{n-k}\ll Y$ for every
$k\in\\{1,\ldots,g\\}$. Multiplying these conditions together, we obtain
$t_{n}\gg Y^{-g}$. The lemma follows. $\Box$

We now estimate the contribution to
${\mathcal{I}}_{X}(W({\mathbb{Z}})_{|Q|>M}^{{\rm dist},{\rm irr}})$ coming
from the deep cusp.

###### Proposition 4.7

We have ${\mathcal{I}}_{X}^{\rm dcusp}(W({\mathbb{Z}})_{|Q|>M}^{\rm
irr})\ll_{\epsilon}{X^{n+1+\epsilon}}/{M}.$

Proof: For a subset ${\mathcal{Z}}$ of $U_{0}$, let $T^{\prime}_{\mathcal{Z}}$
denote the subset of $s\in T^{\prime}$ with $Y^{g(g+1)}w(Q)\gg M$, and
$|Yw(a_{ij})|<c_{ij}$ precisely for those $(i,j)$ where
$a_{ij}\in{\mathcal{Z}}$ or $b_{ij}\in{\mathcal{Z}}$. Note that
$T^{\prime}_{\mathcal{Z}}$ is empty if ${\mathcal{Z}}$ is not saturated.
Define

$\begin{array}[]{rcl}N({\mathcal{Z}},X)&:=&\displaystyle\int_{s\in
T^{\prime}_{\mathcal{Z}}}\\#\bigl{(}s(Y{\mathcal{B}})\cap
W_{0}({\mathbb{Z}})\bigr{)}\,\delta(s)d^{\times}s\\\\[7.22743pt]
&\ll&\displaystyle\int_{s\in T^{\prime}_{\mathcal{Z}}}\big{(}\prod_{\alpha\in
U_{0}\backslash{\mathcal{Z}}}Yw(\alpha)\big{)}\,\delta(s)d^{\times}s\\\\[7.22743pt]
&=&\displaystyle\int_{s\in T^{\prime}_{\mathcal{Z}}}\big{(}\prod_{\alpha\in
U_{0}}Yw(\alpha)\big{)}\big{(}\prod_{\alpha\in{\mathcal{Z}}}Y^{-1}w(\alpha)^{-1}\big{)}\,\delta(s)d^{\times}s,\end{array}$

where the bound on the second line follows from Proposition 4.4. Let
$U^{\prime}:=\\{a_{ij},b_{ij}\mid i+j<n\\}$. If ${\mathcal{Z}}$ is saturated
and not contained in $U^{\prime}$, then ${\mathcal{Z}}$ contains $a_{k,n-k}$
for some $k=1,\ldots,g$. Hence, for any $s\in T^{\prime}_{\mathcal{Z}}$, every
integral element in $s(Y{\mathcal{B}})\cap W_{0}({\mathbb{Z}})$ satisfies
$a_{ij}=b_{ij}=0$, for $i\leq k$ and $j\leq n-k$, and so has zero discriminant
by Lemma 4.5. Therefore,

${\mathcal{I}}_{X}^{\rm dcusp}(W({\mathbb{Z}})^{\rm
irr}_{|Q|>M})\ll\sum_{\mathcal{Z}}N({\mathcal{Z}},X)$

where the sum is over saturated subsets ${\mathcal{Z}}$ of $U_{0}$ contained
in $U^{\prime}$.

Now

$\displaystyle\prod_{\alpha\in U_{0}}Yw(\alpha)$ $\displaystyle=$
$\displaystyle Y^{n(n+1)-g(g+1)}(t_{1}\cdots t_{g+1})^{2g+4}$ (17)
$\displaystyle=$ $\displaystyle\frac{Y^{n(n+1)}}{Y^{g(g+1)}w(Q)}(t_{1}\cdots
t_{g+1})^{g+2}(t_{g+2}\cdots t_{n})^{-(g+1)}$ $\displaystyle=$
$\displaystyle\frac{Y^{n(n+1)}}{Y^{g(g+1)}w(Q)}\prod_{i=1}^{g+1}\prod_{j=g+2}^{n}\frac{t_{i}}{t_{j}}.$

Fix a saturated subset ${\mathcal{Z}}$ of $U_{0}$ contained in $U^{\prime}$.
We define a map $\pi:{\mathcal{Z}}\rightarrow U_{0}\backslash U^{\prime}$ by

$\pi(a_{ij})=a_{i,n-i},\qquad\pi(b_{ij})=b_{n-j,j}.$

Note that for any $\alpha\in{\mathcal{Z}}$, we have $\pi(\alpha)\notin
U^{\prime}$ and so $Yw(\pi(\alpha))\gg 1$. Furthermore, for every $\alpha\in
U^{\prime}$, we have $\alpha\lesssim\pi(\alpha)$ and so
$w(\pi(\alpha))/w(\alpha)\gg 1$. Hence for any $s\in
T^{\prime}_{\mathcal{Z}}$,

$\prod_{\alpha\in{\mathcal{Z}}}(Yw(\alpha))^{-1}\ll\prod_{\alpha\in{\mathcal{Z}}}\frac{Yw(\pi(\alpha))}{Yw(\alpha)}\ll\prod_{\alpha\in
U^{\prime}}\frac{Yw(\pi(\alpha))}{Yw(\alpha)}=\Big{(}\prod_{g+2\leq i<j\leq
n}\frac{t_{i}}{t_{j}}\Big{)}\Big{(}\prod_{1\leq i<j\leq
g+1}\frac{t_{i}}{t_{j}}\Big{)}.$ (18)

Here the first product on the right hand side is the contribution from all
$a_{ij}\in U_{1}$, and the second product is the contribution from all
$b_{ij}\in U_{1}$. Combining (17) and (18) gives, for any $s\in
T^{\prime}_{\mathcal{Z}}$,

$\prod_{\alpha\in
U_{0}\backslash{\mathcal{Z}}}Yw(\alpha)\ll\frac{Y^{n(n+1)}}{Y^{g(g+1)}w(Q)}\prod_{1\leq
i<j\leq
n}\frac{t_{i}}{t_{j}}=\frac{Y^{n(n+1)}}{Y^{g(g+1)}w(Q)}\,\delta(s)^{-1}\ll\frac{Y^{n(n+1)}}{M}\,\delta(s)^{-1}.$

Since each $s_{i}$ is bounded below by an absolute constant and bounded above
by a power of $X$ by Lemma 4.6, we obtain

$N({\mathcal{Z}},X)=O_{\epsilon}\Big{(}\frac{X^{n+1+\epsilon}}{M}\Big{)}.$

The proof is completed by summing over all saturated subsets ${\mathcal{Z}}$
contained in $U_{1}$. $\Box$

Theorem 4.1 now follows from Propositions 4.2, 4.3 and 4.7.

##  5 A bound on the number of singular symmetric matrices in skewed boxes

Let $n\geq 2$ be a positive integer and let $S={\rm Sym}_{2}(n)$ denote the
space of symmetric $n\times n$ matrices. Let $|\cdot|$ denote Euclidean length
on $S({\mathbb{R}})$ obtained by identifying $S({\mathbb{R}})$ with
${\mathbb{R}}^{\dim S}={\mathbb{R}}^{n(n+1)/2}$. Let ${\mathcal{D}}\subset
S({\mathbb{R}})$ be a bounded open set. For an integer $r$ with $1\leq r<n$,
let $S({\mathbb{Z}})_{(r)}$ denote the set of elements in $S({\mathbb{Z}})$
having rank $r$. In [15], Eskin and Katznelson obtained asymptotics for the
number of elements in $Y{\mathcal{D}}\cap S({\mathbb{Z}})_{(r)}$ for
$r\in\\{1,\ldots,n-1\\}$.

In this paper, we will not need exact asymptotics; upper bounds will suffice.
In this section, our goal is to obtain upper bounds on the number of elements
of $S({\mathbb{Z}})_{(r)}$ in skew balls.

The group ${\rm SL}_{n}$ acts on $S$ via $\gamma(A)=\gamma A\gamma^{t}$ for
$\gamma\in{\rm SL}_{n}$ and $A\in S$. Let $T\subset{\rm SL}_{n}({\mathbb{R}})$
denote the subgroup of diagonal matrices with positive coefficients. We denote
elements in $T$ by $s={\rm diag}(t_{1}^{-1},\ldots,t_{n}^{-1})$. We are
interested in studying the skew ball $s(Y{\mathcal{D}})$. By symmetry, we may
assume that $s\in T^{\prime}$, i.e., we have $t_{1}\gg t_{2}\gg\ldots\gg
t_{n}$. Moreover, in light of Lemma 4.6, we will assume that $t_{1}\ll
Y^{\Theta}$ and $t_{n}\gg Y^{-\Theta}$ for some absolute constant $\Theta$
depending only on $n$.

For $s\in T$ and $r\in\\{1,\ldots,n-1\\}$, we define the constants $C(r,s)$ by

$C(r,s)=\prod_{i=1}^{r}\prod_{j=1}^{n-i}\frac{t_{j}}{t_{n-i+1}}=\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ j>n-r\end{subarray}}\frac{t_{i}}{t_{j}}.$ (19)

When $s\in T^{\prime}$, these constants satisfy

$C(r,s)\,\ll\,C(n-1,s)=\prod_{1\leq i<j\leq
n}\frac{t_{i}}{t_{j}}=\delta(s)^{-1},$

where as before $\delta(s)$ is the character of the torus appearing in the
Haar measure of ${\rm SL}_{n}({\mathbb{R}})$.

Finally, for $1\leq r<n$, a positive real number $Y$, and $s\in T$, let
$N_{r}(Y,s)$ denote the number of elements in $s(Y{\mathcal{D}})\cap
S({\mathbb{Z}})_{(r)}$. We prove the following result.

###### Theorem 5.1

Let $n\geq 2$ and $1\leq r<n$ be positive integers. Let $\Theta>0$ be a real
number. Let $Y>1$ be a real number, and let $s\in T^{\prime}$ with $t_{1}\ll
Y^{\Theta}$ and $t_{n}\gg Y^{-\Theta}$. Then

$N_{r}(Y,s)=O\bigl{(}C(r,s)Y^{nr/2}\log^{r}Y\bigr{)}$

where the implied constants depend only on $n$ and $\Theta$.

The case $s=1$ of Theorem 5.1 follows from the work of Eskin-Katznelson [15].
Their strategy is to express the set of singular symmetric matrices of rank
$r$ as a union of lattices, each of which consists of elements having a fixed
row span. They count the number of elements in each such lattice having
bounded norm, and then sum over all possible row spans. We follow this
strategy, explaining the modifications necessary to bound integer points in
skew balls.

Fix positive integers $k$ and $m$ with $k\leq m$, and a lattice $\Lambda$ in
${\mathbb{R}}^{n}$ of rank $r$. A basis $\\{\ell_{1},\ldots,\ell_{k}\\}$ of
$\Lambda$ is reduced if the product $|\ell_{1}||\ell_{2}|\cdots|\ell_{k}|$ is
minimal among all integral bases of $\Lambda$. It is almost reduced if

$|\ell_{1}||\ell_{2}|\cdots|\ell_{k}|\ll d(\Lambda),$

where $d(\Lambda)$ denotes the covolume of $\Lambda$ in
$\Lambda\otimes{\mathbb{R}}$, and the implied constant in the inequality
depends only on $n$. If we order an almost reduced basis
$\\{\ell_{1},\ldots,\ell_{k}\\}$ by length, then the $i$-th successive minimum
of $\Lambda$ is within a constant multiple (depending only on $n$) of
$|\ell_{i}|$ for every $i=1,\ldots,k$. To bound the number elements of
$\Lambda$ in a ball, we use the following result of Schmidt [26].

###### Proposition 5.2

Let $\Lambda$ be a rank $k$ lattice in ${\mathbb{R}}^{m}$ and ${\mathcal{D}}$
a bounded open domain in ${\mathbb{R}}^{m}$. Let $\mu_{1},\ldots,\mu_{k}$ be
the successive minima of $\Lambda$. Then for $Y>0$, we have

$\\#(Y{\mathcal{D}}\cap\Lambda)=O\Big{(}\underset{1\leq j\leq k}{{\rm
max}}\frac{Y^{j}}{\mu_{1}\cdots\mu_{j}}\Big{)}.$ (20)

We use the notation of Theorem 5.1. Given a lattice
$\Lambda\subset{\mathbb{Z}}^{n}$ of rank $r$, let $S(\Lambda)$ denote the
lattice of symmetric matrices $B\in S({\mathbb{Z}})$ such that the row space
(equivalently, the column space) of $B$ is contained in
$\Lambda\otimes{\mathbb{R}}$. For two vectors $v_{1}$ and $v_{2}$ in
${\mathbb{R}}^{n}$, we define

$v_{1}\ast v_{2}:=\left\\{\begin{array}[]{ll}v_{1}\cdot v_{2}^{t}+v_{2}\cdot
v_{1}^{t}&\;{\rm if}\;v_{1}\;{\rm and}\;v_{2}\mbox{ are linearly
independent};\\\ v_{1}\cdot v_{2}^{t}&\;{\rm otherwise}.\end{array}\right.$
(21)

Then $v_{1}\ast v_{2}=v_{2}\ast v_{1}\in S(\text{Span}\\{v_{1},v_{2}\\})$, and

$|v||w|\leq|v\ast w|\leq 2|v||w|.$ (22)

Fix $\gamma\in{\rm SL}_{n}({\mathbb{R}})$. (For our applications, we will take
$\gamma\in T$.) Let $\Lambda\in{\mathbb{Z}}^{n}$ be a primitive lattice of
rank $r$. We bound the number of elements in $\gamma^{-1}(Y{\mathcal{D}})\cap
S(\Lambda)$ using the bijection

$\begin{array}[]{rcl}\gamma^{-1}(Y{\mathcal{D}})\cap
S(\Lambda)&\to&Y{\mathcal{D}}\cap\gamma(S(\Lambda))\\\\[3.61371pt]
A&\mapsto&\gamma A\gamma^{-1}\end{array}$

and instead bounding the number of elements in
$Y{\mathcal{D}}\cap\gamma(S(\Lambda))$. We thus study the lattice
$\gamma(S(\Lambda))\subset S({\mathbb{R}})$. The next result, which gives an
almost reduced basis for $\gamma(S(\Lambda))$ in terms of an almost reduced
basis of $\gamma\Lambda$, follows from the proofs of [15, Proposition 3.3] and
[15, Lemma 3.5].

###### Theorem 5.3

Fix $\gamma\in{\rm SL}_{n}({\mathbb{R}})$. Let
$\Lambda\subset{\mathbb{Z}}^{n}$ be a primitive lattice of rank $r$, and let
$\\{\ell_{1},\ldots,\ell_{r}\\}$ be a basis for $\gamma\Lambda$. Then
$\\{\ell_{i}\ast\ell_{j}\colon 1\leq i\leq j\leq r\\}$ is a basis for
$\gamma(S(\Lambda))$. Furthermore,

$d(\gamma(S(\Lambda)))=2^{r(r-1)/4}d(\gamma\Lambda)^{r+1}=2^{r(r-1)/4}d(\Lambda)^{r+1}.$

In particular, if $\\{\ell_{1},\ldots,\ell_{r}\\}$ is almost reduced, then so
is $\\{\ell_{i}\ast\ell_{j}\colon 1\leq i\leq j\leq r\\}$.

Next, by the proof of [15, Lemma 4.1], we have the following result giving a
necessary condition for the set $Y{\mathcal{D}}\cap\gamma(S(\Lambda))$ to be
nonempty.

###### Proposition 5.4

Let $\gamma\in{\rm SL}_{n}({\mathbb{R}})$ and let
$\Lambda\subset{\mathbb{Z}}^{n}$ be a primitive lattice of rank $r$ such that
the successive minima of $\gamma\Lambda$ are $\mu_{1}\leq\ldots\leq\mu_{r}$.
If $\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))>0$, then $\mu_{i}\mu_{j}\leq
c_{1}Y$ for every pair $(i,j)$ with $i+j\leq r+1$, for some constant $c_{1}$
depending only on $n$.

We now prove an upper bound on $\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))$.

###### Proposition 5.5

Let $\gamma\in{\rm SL}_{n}({\mathbb{R}})$ and let
$\Lambda\subset{\mathbb{Z}}^{n}$ be a primitive lattice of rank $r$ such that
the successive minima of $\gamma\Lambda$ are $\mu_{1}\leq\ldots\leq\mu_{r}$.
Then

$\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))=O\Big{(}\frac{Y^{r(r+1)/2}}{d(\Lambda)^{r+1}}\prod_{\begin{subarray}{c}1\leq
i<j\leq r\\\ i+j\leq r+1\end{subarray}}\frac{\mu_{j}}{\mu_{i}}\Big{)}.$ (23)

Proof: Let $U(r)$ denote the set of pairs $(i,j)$ of positive integers such
that $i\leq j\leq r$ and $i+j>r+1$. In other words, elements in $U(r)$
correspond to the successive minima of the lattice $\gamma(S(\Lambda))$ that
are $\gg Y$. By Proposition 5.2, Theorem 5.3, and (22), we have

$\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))\ll\frac{Y^{r(r+1)/2}}{d(\Lambda)^{r+1}}\prod_{(i,j)\in
U(r)}\Big{(}\frac{Y}{\mu_{i}\mu_{j}}\Big{)}^{-1}.$

Assume that $\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))>0$. Then
$\mu_{r+1-j}\mu_{j}\ll Y$ for all $1\leq j\leq r$ by Proposition 5.4. Thus

$\begin{array}[]{rcl}\\#(Y{\mathcal{D}}\cap\gamma(S(\Lambda))&\ll&\displaystyle\frac{Y^{r(r+1)/2}}{d(\Lambda)^{r+1}}\prod_{(i,j)\in
U(r)}\Big{(}\frac{Y}{\mu_{i}\mu_{j}}\Big{)}^{-1}\frac{Y}{\mu_{r+1-j}\mu_{j}}\\\\[14.45377pt]
&\ll&\displaystyle\frac{Y^{r(r+1)/2}}{d(\Lambda)^{r+1}}\prod_{(i,j)\in
U(r)}\frac{\mu_{i}}{\mu_{r+1-j}}.\end{array}$ (24)

Since $i\leq j$ and $i+j>r+1$ for $(i,j)\in U(r)$, we have the following
injection:

$\begin{array}[]{rcl}U(r)&\to&\\{(k,\ell):1\leq k<\ell\leq r:k+\ell\leq
r+1\\}\\\\[3.61371pt] (i,j)&\mapsto&(r+1-j,i).\end{array}$ (25)

Since $\frac{\mu_{j}}{\mu_{i}}\geq 1$ for $j>i$, the injection (25) implies
that the product of the ratios $\mu_{j}/\mu_{i}$ in (23) is at least as large
as the product of the ratios $\mu_{i}/\mu_{r+1-j}$ in (24). The result
follows. $\Box$

We now sum over the appropriate lattices $\Lambda\subset{\mathbb{Z}}^{n}$
having rank $r$. To this end, we fix an element $s={\rm
diag}(t_{1}^{-1},t_{2}^{-1},\dots,t_{n}^{-1})\in T^{\prime}$. We will apply
the previous results with $\gamma=s^{-1}$. Set $L=(L_{1},\ldots,L_{r})$ with
$0<L_{1}\leq L_{2}\leq\cdots\leq L_{r}$. Let $\Sigma(L,s)$ denote the set of
primitive lattices $\Lambda\subset{\mathbb{Z}}^{n}$ of rank $r$ whose
successive minima $\mu_{1},\ldots,\mu_{r}$ of $s^{-1}\Lambda$ satisfy
$L_{i}\leq\mu_{i}<2L_{i}$ for each $i$.

###### Lemma 5.6

Let $L=(L_{1},\ldots,L_{r})$ and $s={\rm
diag}(t_{1}^{-1},\ldots,t_{n}^{-1})\in T^{\prime}$. Then there is a constant
$c^{\prime}>0$ depending only on $n$ such that if $\\#\Sigma(L,s)>0$, then
$L_{i}t_{j}^{-1}>c^{\prime}$ for all $(i,j)$ with $i+j\geq n+1$.

Proof: Since $\\#\Sigma(L,s)>0$, there exists an integral lattice
$\Lambda\subset{\mathbb{Z}}^{n}$ of rank $r$ with basis
$\\{\ell_{1},\ldots,\ell_{r}\\}$ such that $|s^{-1}\ell_{i}|<2L_{i}$ for
$i\in\\{1,\ldots,r\\}$. For $1\leq j\leq n$, let $u_{ij}$ denote the
(integral) $j$-th entry of $\ell_{i}$. Then $|u_{ij}|\leq 2L_{i}t_{j}^{-1}$
for every $1\leq i\leq r$ and $1\leq j\leq n$. The assumption that $s\in
T^{\prime}$ implies that $L_{i}t_{j}^{-1}\ll
L_{i^{\prime}}t_{j^{\prime}}^{-1}$ whenever $i\leq i^{\prime}$ and $j\leq
j^{\prime}$.

Suppose that there is an integer $k$ with $1\leq k\leq r$ such that
$L_{k}t_{n+1-k}^{-1}<c^{\prime\prime}$ for some sufficiently small constant
$c^{\prime\prime}>0$. Then $|u_{ij}|<1$, and thus $u_{ij}=0$ for all $(i,j)$
with $1\leq i\leq k$ and $1\leq j\leq n+1-k$. However, this implies that the
vectors $\ell_{1},\ldots,\ell_{k}$ are not linearly independent, a
contradiction. Hence such a $k$ does not exist and $L_{k}t_{n+1-k}^{-1}\gg 1$
for all $k$, implying the result. $\Box$

We now determine an upper bound for $\\#\Sigma(L,s)$.

###### Proposition 5.7

Let $L=(L_{1},\ldots,L_{r})$ and $s={\rm
diag}(t_{1}^{-1},\ldots,t_{n}^{-1})\in T^{\prime}$. Then

$\\#\Sigma(L,s)=O\Bigl{(}(L_{1}\cdots L_{r})^{n}\Bigl{(}\prod_{1\leq i<j\leq
r}\frac{L_{i}}{L_{j}}\Bigr{)}C(r,s)\Bigr{)}$ (26)

where $C(r,s)$ is defined as in (19).

Proof: We count lattices $\Lambda$ by counting $r$-tuples of vectors
$(\ell_{1},\ldots,\ell_{r})$ such that each $\ell_{i}\in
s^{-1}{\mathbb{Z}}^{n}$ satisfies $L_{i}\leq|\ell_{i}|<2L_{i}$ and such that
$\\{\ell_{1},\ldots,\ell_{r}\\}$ is a reduced basis of the lattice it
generates. For each $i=1,\ldots,r$, let $\alpha(i)$ be the largest integer
such that $L_{i}t_{\alpha(i)}^{-1}\leq c^{\prime}$, where $c^{\prime}$ is as
in Lemma 5.6, or let $\alpha(i)=0$ if no such integer exists. By Proposition
5.2, the number of possibilities for $\ell_{i}$ is

$\ll\;\prod_{j=1}^{n}{\rm
max}\bigl{(}L_{i}t_{j}^{-1},1\bigr{)}\;\ll\;L_{i}^{n}\prod_{j=1}^{\alpha(i)}\bigl{(}L_{i}^{-1}t_{j}\bigr{)}.$

However, once $\ell_{1}$ is fixed, and given a vector $\ell_{2}$, at most two
of $\ell_{2}-k\ell_{1}$ can be part of a reduced basis for $k\in{\mathbb{Z}}$.
Since $\gg L_{2}/L_{1}$ vectors $\ell_{2}-k\ell_{1}$ satisfy the same size
bound as $\ell_{2}$ (namely, those with $k\ll L_{2}/L_{1}$), the number of
choices for the pair $(\ell_{1},\ell_{2})$ that are part of a reduced basis is

$\frac{L_{1}}{L_{2}}L_{1}^{n}L_{2}^{n}\prod_{j=1}^{\alpha(1)}\bigl{(}L_{1}^{-1}t_{j}\bigr{)}\prod_{j=1}^{\alpha(2)}\bigl{(}L_{2}^{-1}t_{j}\bigr{)}.$

Continuing in this way, we obtain the bound

$\\#\Sigma(L,s)\ll(L_{1}L_{2}\cdots L_{r})^{n}\Bigl{(}\prod_{1\leq i<j\leq
r}\frac{L_{i}}{L_{j}}\Bigr{)}\Bigl{(}\prod_{i=1}^{r}\prod_{j=1}^{\alpha(i)}L_{i}^{-1}t_{j}\Bigr{)}.$
(27)

By Lemma 5.6, we have $\alpha(i)\leq n-i$ for $i\in\\{1,\ldots,r\\}$.
Therefore,

$\prod_{i=1}^{r}\prod_{j=1}^{\alpha(i)}L_{i}^{-1}t_{j}\ll\displaystyle\prod_{i=1}^{r}\prod_{j=1}^{\alpha(i)}L_{i}^{-1}t_{j}L_{i}t_{\alpha(i)+1}^{-1}\ll\displaystyle\prod_{i=1}^{r}\prod_{j=1}^{\alpha(i)}\frac{t_{j}}{t_{n-i+1}}\ll\displaystyle\prod_{i=1}^{r}\prod_{j=1}^{n-i}\frac{t_{j}}{t_{n-i+1}}=\displaystyle
C(r,s).$ (28)

Equations (27) and (28) yield the desired result. $\Box$

We are now ready to prove the main result of this section.

Proof of Theorem 5.1: Let $L=(L_{1},\ldots,L_{r})$ be a tuple such that
$0<L_{1}\leq L_{2}\leq\cdots\leq L_{r}$. Then, by Lemma 5.6, Proposition 5.4
and the definition of $T^{\prime}$, we see that for there to exist a lattice
$\Lambda\in\Sigma(L,s)$ such that $\\#(Y{\mathcal{D}}\cap
s^{-1}(S(\Lambda))>0$, we must have

$Y^{-\Theta_{1}}\ll L_{1}\leq\cdots\leq L_{r}\ll
Y^{\Theta_{2}}\qquad\mbox{and}\qquad L_{1}\cdots L_{r}\ll Y^{r/2}$

for some absolute constants $\Theta_{1},\Theta_{2}>0$. For any such $\Lambda$,
Proposition 5.5 states that

$\\#(Y{\mathcal{D}}\cap s^{-1}S(\Lambda))\ll\frac{Y^{r(r+1)/2}}{(L_{1}\ldots
L_{r})^{r+1}}\prod_{\begin{subarray}{c}1\leq i<j\leq r\\\ i+j\leq
r+1\end{subarray}}\frac{L_{j}}{L_{i}}.$

Thus

$N_{r}(Y,s)\ll\sum_{L}\\#\Sigma(L,s)\frac{Y^{r(r+1)/2}}{(L_{1}\ldots
L_{r})^{r+1}}\prod_{\begin{subarray}{c}1\leq i<j\leq r\\\ i+j\leq
r+1\end{subarray}}\frac{L_{j}}{L_{i}},$

where the sum is over $r$-tuples $L=(L_{1},\ldots,L_{r})$ with $L_{1}\leq
L_{2}\leq\cdots\leq L_{n}$ that partition the region
$\\{(\mu_{1},\ldots,\mu_{r})\in[Y^{-\Theta},Y^{\Theta^{\prime}}]^{r}:\mu_{1}\leq\ldots\leq\mu_{r}\\}$
into dyadic ranges. The sum over $L$ has length $O(\log^{r}Y)$. Using the
upper bound on $\\#\Sigma(L,s)$ in Proposition 5.7, we obtain

$\begin{array}[]{rcl}N_{r}(Y,s)&\ll&\displaystyle\sum_{L}C(r,s)(L_{1}\ldots
L_{r})^{n}\frac{Y^{r(r+1)/2}}{(L_{1}\ldots
L_{r})^{r+1}}\bigg{(}\prod_{\begin{subarray}{c}1\leq i<j\leq r\\\ i+j\leq
r+1\end{subarray}}\frac{L_{j}}{L_{i}}\bigg{)}\bigg{(}\prod_{1\leq i<j\leq
r}\frac{L_{i}}{L_{j}}\bigg{)}\\\\[1.8063pt]
&\ll&\displaystyle\sum_{L}C(r,s)Y^{(n-r-1)r/2}Y^{r(r+1)/2}\\\\[14.45377pt]
&\ll&\displaystyle C(r,s)Y^{nr/2}\log^{r}Y.\end{array}$

This concludes the proof of Theorem 5.1. $\Box$

##  6 A uniformity estimate for even degree polynomials

We fix an even integer $n=2g+2$ with $g\geq 1$. Our goal is to prove Theorem
5(c) by obtaining a bound on the number of integral binary $n$-ic forms having
bounded height having discriminant weakly divisible by the square of a large
squarefree integer.

Throughout this section, we write $V:=V_{n}$ and $W:=W_{n+1}$. Let $m>0$ be an
odd squarefree integer, and let
${\mathcal{W}}_{m}^{\rm{(2)}}:={\mathcal{W}}_{m,n}^{\rm{(2)}}$. We also define
the following auxiliary sets:

$\displaystyle V({\mathbb{Z}})^{\rm red}$ $\displaystyle:=$
$\displaystyle\\{f\in V({\mathbb{Z}}):{\rm Gal}(f(x,1))\neq S_{n}\\},$ (29)
$\displaystyle V({\mathbb{Z}})^{\Delta\text{\,small}}$ $\displaystyle:=$
$\displaystyle\\{f\in V({\mathbb{Z}}):\Delta(f)\leq H(f)^{2n-2-\kappa}\\},$
(30) $\displaystyle{\mathcal{W}}_{m}^{(1\\#)}$ $\displaystyle:=$
$\displaystyle\\{f\in{\mathcal{W}}_{m}^{(2)}:m\mid f(0,1)\\},$ (31)
$\displaystyle{\mathcal{W}}_{m}^{\rm gen}$ $\displaystyle:=$
$\displaystyle\\{f\in{\mathcal{W}}_{m}^{(2)}:\gcd(m,f(0,1))=1\text{ and
}f\notin V({\mathbb{Z}})^{\rm red}\cup
V({\mathbb{Z}})^{\Delta\text{\,small}}\\},$ (32)

where $\kappa>0$ is a small constant (whose exact value will be optimized
later) and ${\rm Gal}$ denotes the Galois group. Then, for any $M>0$, we have
the following containment:

$\bigcup_{\begin{subarray}{c}m>M\\\ {\rm
squarefree}\end{subarray}}{\mathcal{W}}_{m}^{(2)}\;\subset\;V({\mathbb{Z}})^{\rm
red}\cup
V({\mathbb{Z}})^{\Delta\text{\,small}}\cup\bigcup_{\begin{subarray}{c}m>\sqrt{M}\\\
{\rm
squarefree}\end{subarray}}{\mathcal{W}}_{m}^{(1\\#)}\cup\bigcup_{\begin{subarray}{c}m>\sqrt{M}\\\
{\rm squarefree}\end{subarray}}{\mathcal{W}}_{m}^{{\rm gen}}.$ (33)

The number of elements in $V({\mathbb{Z}})^{\rm red}$ having height less than
$X$ was bounded by $O(X^{n}$) in [5]. We next prove a bound on the number of
elements in $V({\mathbb{Z}})^{\Delta\text{\,small}}$ of bounded height.

###### Lemma 6.1

The number of integral binary $n$-ic forms with height less than $X$ and
absolute discriminant less than $X^{2n-2-\kappa}$ is
$O(X^{n+1-\frac{\kappa}{2n-2}})$.

Proof: Set $\eta:=\kappa/(2n-2)$. The number of integral binary $n$-ic forms
$a_{0}x^{n}+\cdots+a_{n}y^{n}$ with height less than $X$ such that
$|a_{0}|\leq X^{1-\eta}$ is $O(X^{n+1-\eta})$. Hence we assume
$|a_{0}|>X^{1-\eta}.$

Now fix integers $a_{0},\ldots,a_{n-1}$ with $|a_{i}|\leq X$ and
$|a_{0}|>X^{1-\eta}$. The discriminant of $a_{0}x^{n}+\cdots+a_{n}y^{n}$ is a
polynomial $F(a_{n})$ in $a_{n}$ of degree $n-1$ with leading coefficient
$C_{n}a_{0}^{n-1}$ for some nonzero constant $C_{n}$. Let
$r_{1},\ldots,r_{n-1}\in{\mathbb{C}}$ be the $n-1$ roots of $F(x)$. Then

$F(a_{n})=C_{n}a_{0}^{n-1}(a_{n}-r_{1})\cdots(a_{n}-r_{n-1}).$

Since $|F(a_{n})|<X^{2n-2-\kappa}$, we have
$(a_{n}-r_{1})\cdots(a_{n}-r_{n-1})\ll X^{n-1-(n-1)\eta}$. Hence
$|a_{n}-r_{i}|\ll X^{1-\eta}$ for some $i=1,\ldots,n-1$. The number of such
integers $a_{n}$ is $O(X^{1-\eta})$. Since there are $O(X^{n})$ choices for
$a_{0},\ldots,a_{n-1}$, we obtain the desired bound. $\Box$

A direct application of a quantitative version of the Ekedahl sieve as in [3,
Theorem 3.3] implies the following bound on the number of elements of bounded
height belonging to ${\mathcal{W}}^{(1\\#)}_{m}$ for large $m$.

###### Lemma 6.2

We have $\displaystyle\\#\\!\\!\bigcup_{\begin{subarray}{c}m>\sqrt{M}\\\
m\;\rm{squarefree}\end{subarray}}\\!\\!\\{f\in{\mathcal{W}}_{m}^{(1\\#)}:H(f)<X\\}=O\Big{(}\frac{X^{n+1}}{\sqrt{M}}+X^{n}\Big{)}.$

To prove Theorem 5(c), it thus remains to obtain an upper bound for

$\\#\bigcup_{\begin{subarray}{c}m>\sqrt{M}\\\
m\;\rm{squarefree}\end{subarray}}\\{f\in{\mathcal{W}}_{m}^{{\rm
gen}}:H(f)<X\\}.$ (34)

In §3.5, we defined a map $\sigma_{m}$ from the set of elements
$f\in{\mathcal{W}}_{m}^{\rm{(2)}}$ with $\gcd(m,f(0,1))=1$ to
$W({\mathbb{Z}})$ such that $f_{\sigma_{m}(f)}=xf$ and $|q|(\sigma_{m}(f))=m.$
For any $M>0$, define the set ${{\mathcal{L}}(M)}$ by

${{\mathcal{L}}(M)}:=\bigcup_{\begin{subarray}{c}m>M\\\ m\;{\rm
squarefree}\end{subarray}}{\rm
SL}_{n+1}({\mathbb{Z}})\cdot\sigma_{m}({\mathcal{W}}_{m}^{{\rm gen}}).$

Then (34) is $\ll$

$\begin{array}[]{rcl}\displaystyle\\#\Bigl{(}{\rm
SL}_{n+1}({\mathbb{Z}})\backslash\\{f\in{{\mathcal{L}}(M)}\colon
H(f)<X\\}\Bigr{)}&\ll&\displaystyle{\mathcal{I}}_{X}({{\mathcal{L}}(M)})\end{array}$
(35)

where

${\mathcal{I}}_{X}({{\mathcal{L}}(M)})=\int_{s\in
T^{\prime}}\\#\bigl{(}s(Y{\mathcal{B}})\cap{{\mathcal{L}}(M)}\bigr{)}\,\delta(s)\,d^{\times}s$

is as defined immediately after (15), where $Y$ is now taken to be
$X^{1/(n+1)}$ throughout this section. Moreover, exactly as in the paragraph
leading up to (16), we break up ${\mathcal{I}}_{X}({{\mathcal{L}}(M)})$ into
three parts—corresponding to the main body, the shallow cusp, and the deep
cusp—and again write

${\mathcal{I}}_{X}({{\mathcal{L}}(M)})={\mathcal{I}}^{\rm
main}_{X}({{\mathcal{L}}(M)})+{\mathcal{I}}^{\rm
scusp}_{X}({{\mathcal{L}}(M)})+{\mathcal{I}}^{\rm
dcusp}_{X}({{\mathcal{L}}(M)}).$

The rest of this section is dedicated to obtaining an upper bound on
${\mathcal{I}}_{X}({{\mathcal{L}}(M)})$. Every element
$(A,B)\in{{\mathcal{L}}(M)}$ satisfies $\det(B)=0$ since $f_{A,B}$ is
divisible by $x$. In §4, we used vanishing conditions on the coefficients
$\\{a_{ij},b_{ij}\\}$ of $W$ to estimate the number of integral pairs $(A,B)$
in skewed domains of $W({\mathbb{R}})$. Now, since we also need to impose the
condition that $B$ has determinant $0$, we use the setup of §5 to count the
number of such $B$’s in skewed bounded domains by fibering over the row space
of $B$.

In §6.1, we thus further break up the three parts of
${\mathcal{I}}_{X}({{\mathcal{L}}(M)})$ into sums over row spaces of the
singular matrix $B$. We also obtain some preliminary bounds on
${\mathcal{I}}_{X}({{\mathcal{L}}(M)})$, and give some conditions that ensure
that a pair $(A,B)$ has discriminant $0$. In §6.2, §6.3, and §6.4, we then
prove the desired upper bounds on ${\mathcal{I}}^{\rm
main}_{X}({{\mathcal{L}}(M)})$, ${\mathcal{I}}^{\rm
scusp}_{X}({{\mathcal{L}}(M)})$, and ${\mathcal{I}}^{\rm
dcusp}_{X}({{\mathcal{L}}(M)})$, respectively. In conjunction with (33), (35),
and Lemmas 6.1–6.2 this will yield Theorem 5(c).

###  6.1 Setup and preliminary bounds

#### Coordinate systems, weight functions, and summing over row spaces

Let $S({\mathbb{Z}})$ denote the set of $(n+1)\times(n+1)$ integral symmetric
matrices. For any primitive lattice $\Lambda$ of ${\mathbb{Z}}^{n+1}$, let
$S(\Lambda)$ denote the sublattice of $S({\mathbb{Z}})$ consisting of elements
$B\in S({\mathbb{Z}})$ with row space contained in $\Lambda$. For
$L=(L_{1},\ldots,L_{n})$ with $L_{i}\in{\mathbb{R}}$ and $L_{1}\leq
L_{2}\leq\cdots\leq L_{n}$ and $s\in T^{\prime}$, let $\Sigma(L,s)$ denote the
set of primitive lattices $\Lambda\subset{\mathbb{Z}}^{n+1}$ of rank $n$ such
that the successive minima $\mu_{1},\ldots,\mu_{n}$ of $s^{-1}\Lambda$ satisfy
$L_{1}\leq\mu_{i}\leq 2L_{i}$ for each $i$. We define
${\mathcal{S}}(L,s)\subset S({\mathbb{Z}})$ by

${\mathcal{S}}(L,s):=\bigcup_{\Lambda\in\Sigma(L,s)}S(\Lambda).$

We next introduce coordinate systems and weight functions. Let

${\mathcal{M}}:=\\{\ell_{ij}:1\leq i\leq n,\;1\leq j\leq n+1\\}$

denote the set of coordinates of $n$-tuples of vectors in
${\mathbb{R}}^{n+1}$. We define

$w_{L}(\ell_{ij}):=L_{i}t_{j}^{-1}.$

The significance of $w_{L}$ is the following. Let $\Lambda\in\Sigma(L,s)$ be a
lattice with an integral basis $\\{\ell_{1},\ldots,\ell_{n}\\}$ such that
$\\{s^{-1}\ell_{1},\ldots,s^{-1}\ell_{n}\\}$ is a Minkowski-reduced basis for
$s^{-1}\Lambda$. Then the $j$th coefficient of $\ell_{i}$ is $\ll
L_{i}t_{j}^{-1}=w_{L}(\ell_{ij})$. In particular, for the absolute value of
the $j$th coefficient of $\ell_{i}$ to be nonzero, we must have
$w_{L}(\ell_{ij})\gg 1$. When $L$ is implicit, we will write $w$ in place of
$w_{L}$.

Let ${\mathcal{K}}$ denote the set of coefficients $\\{a_{ij}:1\leq i\leq
j\leq n+1\\}$, and recall the weight function

$w(a_{ij})=t_{i}^{-1}t_{j}^{-1}.$

Define a partial order on ${\mathcal{K}}$ by setting $a_{ij}\lesssim
a_{i^{\prime}j^{\prime}}$ if $i\leq i^{\prime}$ and $j\leq j^{\prime}$, and on
${\mathcal{M}}$ by setting $\ell_{ij}\lesssim\ell_{i^{\prime}j^{\prime}}$ if
$i\leq i^{\prime}$ and $j\leq j^{\prime}$. The significance of this partial
order is that if $\alpha,\beta\in{\mathcal{K}}$ with $\alpha\lesssim\beta$ and
$s\in T^{\prime}$, then $w(\alpha)\ll w(\beta)$ and similarly
$w_{L}(\alpha)\ll w_{L}(\beta)$ if $\alpha,\beta\in{\mathcal{M}}.$

We say that a subset ${\mathcal{Z}}$ of ${\mathcal{K}}\cup{\mathcal{M}}$ is
saturated if for any $\alpha\in{\mathcal{Z}}$, all the
$\alpha^{\prime}\in{\mathcal{K}}\cup{\mathcal{M}}$ with
$\alpha^{\prime}\lesssim\alpha$ are also contained in ${\mathcal{Z}}$.

Let ${\mathcal{D}}\subset S({\mathbb{R}})$ be a bounded domain such that
${\mathcal{B}}_{1}\subset{\mathcal{D}}\times{\mathcal{D}}$. We pick positive
constants $c_{ij}$ for $1\leq i\leq j\leq n+1$ and $c_{i}^{\prime}$ for $1\leq
i\leq n$ such that:

  1. (a)

if $|Yw(a_{ij})|<c_{ij}$, then the $a_{ij}$–coordinate of any integral element
in $s(Y{\mathcal{D}})$ is $0$;

  2. (b)

if $|w_{L}(\ell_{ij})|<c_{j}^{\prime}$, then the $j$th coefficient of
$\ell_{i}$ for any lattice $\Lambda\in\Sigma(L,s)$ is $0$;

  3. (c)

$c_{i}^{\prime}<c^{\prime}$ for all $i=1,\ldots,n$, where $c^{\prime}$ is the
constant in Lemma 5.6;

  4. (d)

$c_{1}c_{g+1,g+1}\leq c_{g+1}^{\prime\,\,2}$, where $c_{1}$ is the constant in
Proposition 5.4;

  5. (e)

for any $i\leq i^{\prime}$ and $j\leq j^{\prime}$, we have
$w(a_{ij})/c_{ij}\leq w(a_{i^{\prime}j^{\prime}})/c_{i^{\prime}j^{\prime}}$
and $w(\ell_{ij})/c_{j}^{\prime}\leq
w(\ell_{i^{\prime}j^{\prime}})/c_{j^{\prime}}^{\prime}.$

More explicitly, we choose $c_{n+1,n+1}$ and $c_{n}^{\prime}$ to be
sufficiently small and take

$\displaystyle c_{ij}$ $\displaystyle=$ $\displaystyle\Bigl{(}\sup_{s\in
T^{\prime}}\frac{w(a_{ij})}{w(a_{n+1,n+1})}\Bigr{)}c_{n+1,n+1}\quad\mbox{for
}i\leq j\leq n+1;$ $\displaystyle c_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigl{(}\sup_{s\in
T^{\prime}}\frac{t_{i}^{-1}}{t_{n}^{-1}}\Bigr{)}c_{n}^{\prime}\quad\mbox{for
}i\leq n.$

For any nondecreasing $n$-tuple $L$ of positive real numbers, and a saturated
subset ${\mathcal{Z}}$ of ${\mathcal{K}}\cup{\mathcal{M}}$, we define the
following subset $T_{\mathcal{Z}}(L,Y)$ of $T^{\prime}$:

$T_{\mathcal{Z}}(L,Y):=\left\\{s\in T^{\prime}\left|\begin{array}[]{l}s_{i}\ll
X^{\Theta}\;\;\forall i\in\\{1,\ldots,n\\}\\\\[7.22743pt] {\rm
for}\;a_{ij}\in{\mathcal{K}},\;|Yw(a_{ij})|<c_{ij}\;{\rm{iff}}\;a_{ij}\in{\mathcal{Z}}\cap{\mathcal{K}}\\\\[7.22743pt]
{\rm
for}\;\ell_{ij}\in{\mathcal{M}},\;|w_{L}(\ell_{ij})|<c_{j}^{\prime}\;{\rm{iff}}\;\ell_{ij}\in{\mathcal{Z}}\cap{\mathcal{M}}\end{array}\right.\right\\},$
(36)

where $\Theta$ is the absolute constant from Lemma 4.6.

For $X$, $Y$, $L$, ${\mathcal{Z}}$ as above and any subset ${\mathcal{L}}$ of
$W({\mathbb{Z}})$, we define the quantity

$N({\mathcal{L}},L,{\mathcal{Z}},X):=\int_{T_{\mathcal{Z}}(L,Y)}\\#\\{(A,B)\in(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{\mathcal{L}}\mid
B\in{\mathcal{S}}(L,s)\\}\,\delta(s)\,d^{\times}s.$ (37)

In the proof of Theorem 5.1, we showed that unless $Y^{-\Theta_{1}}<L_{1}$ and
$Y^{\Theta_{2}}>L_{n}$ for some absolute positive constants $\Theta_{1}$ and
$\Theta_{2}$, we have ${\mathcal{S}}(L,s)=\emptyset$, which implies that
$N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X)=0$. Therefore,

${\mathcal{I}}_{X}({{\mathcal{L}}(M)})\ll\displaystyle\sum_{L}\sum_{{\mathcal{Z}}}N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X),$

where the inner sum is over saturated subsets ${\mathcal{Z}}$ of
${\mathcal{K}}\cup{\mathcal{M}}$, and the outer sum is over $n$-tuples
$L=(L_{1},\ldots,L_{n})$ with $L_{1}\leq L_{2}\leq\cdots\leq L_{n}$ that
partition the region
$\\{(\mu_{1},\ldots,\mu_{n})\in[Y^{-\Theta_{1}},Y^{\Theta_{2}}]^{n}:\mu_{1}\leq\ldots\leq\mu_{n}\\}$
into dyadic ranges.

We may therefore bound the main-body, the shallow-cusp, and the deep-cusp
parts of ${\mathcal{I}}_{X}({{\mathcal{L}}(M)})$ in terms of sums over
$N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X)$. We have

$\begin{array}[]{rcl}{\mathcal{I}}_{X}^{\rm
main}({{\mathcal{L}}(M)})&\ll&\displaystyle\sum_{L}\sum_{{\mathcal{Z}}:a_{11}\not\in{\mathcal{Z}}}N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X),\\\\[16.26118pt]
{\mathcal{I}}_{X}^{\rm
scusp}({{\mathcal{L}}(M)})&\ll&\displaystyle\sum_{L}\sum_{\begin{subarray}{c}{\mathcal{Z}}:a_{11}\in{\mathcal{Z}}\\\
a_{g+1,g+1}\not\in{\mathcal{Z}}\end{subarray}}N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X),\\\\[23.48752pt]
{\mathcal{I}}_{X}^{\rm
dcusp}({{\mathcal{L}}(M)})&\ll&\displaystyle\sum_{L}\sum_{{\mathcal{Z}}:a_{g+1,g+1}\in{\mathcal{Z}}}N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X).\end{array}$
(38)

#### A preliminary upper bound

We now prove some preliminary results on
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X))$. We start with an upper bound on
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)$, which also bounds
$N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X)$ by directly counting the number of
possible $A$’s and then using the results of §5 to count $B$’s. For a
saturated subset ${\mathcal{Z}}$ of ${\mathcal{K}}\cup{\mathcal{M}}$, define

$w_{L}({\mathcal{Z}}):=\Bigl{(}\prod_{\alpha\in{\mathcal{Z}}\cap{\mathcal{K}}}w(\alpha)\Bigr{)}\Bigl{(}\prod_{\alpha\in{\mathcal{Z}}\cap{\mathcal{M}}}w_{L}(\alpha)\Bigr{)}.$

In what follows, the $n$-tuple $L$ will be clear from the context, and we
simply write $w$ in place of $w_{L}$.

###### Proposition 6.3

Suppose that ${\mathcal{Z}}$ is a saturated subset of
${\mathcal{K}}\cup{\mathcal{M}}$. Then

$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)\ll
X^{n+1}\,\int_{T_{\mathcal{Z}}(L,Y)}Y^{-\\#({\mathcal{Z}}\cap{\mathcal{K}})}w({\mathcal{Z}})^{-1}\Big{(}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\
i+j>n+1\end{subarray}}\frac{L_{i}}{L_{j}}\Big{)}\,\delta(s)d^{\times}s.$ (39)

Proof: By Proposition 4.4, the number of elements $A\in s(Y{\mathcal{D}})\cap
S({\mathbb{Z}})$ is

$\ll
Y^{(n+1)(n+2)/2}\prod_{a_{ij}\in{\mathcal{Z}}\cap{\mathcal{K}}}(Yw(a_{ij}))^{-1}\ll
Y^{(n+1)(n+2)/2-\\#({\mathcal{Z}}\cap{\mathcal{K}})}w({\mathcal{Z}}\cap{\mathcal{K}})^{-1}.$
(40)

By the definition of $T_{\mathcal{Z}}(L,Y)$, it follows from (27) that for
every $s\in T_{\mathcal{Z}}(L,Y)$, we have

$\\#\Sigma(L,s)\ll(L_{1}\cdots
L_{n})^{n+1}w({\mathcal{Z}}\cap{\mathcal{M}})^{-1}\prod_{1\leq i<j\leq
n}\frac{L_{i}}{L_{j}}.$ (41)

For each $\Lambda\in\Sigma(L,s)$, Proposition 5.5 implies that the number of
integral symmetric matrices $B\in s(Y{\mathcal{D}})$ whose row space is
contained in $\Lambda$ is

$\ll\frac{Y^{n(n+1)/2}}{(L_{1}\cdots
L_{n})^{n+1}}\prod_{\begin{subarray}{c}1\leq i<j\leq n\\\ i+j\leq
n+1\end{subarray}}\frac{L_{j}}{L_{i}}.$ (42)

Combining (40), (41), and (42), and recalling that $X=Y^{n+1}$, gives (39).
$\Box$

#### Conditions for vanishing discriminant

Next, we give some conditions on ${\mathcal{Z}}$ that ensure
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)=0$. We start with the following
algebraic result that gives sufficient conditions on a pair $(A,B)\in
W({\mathbb{C}})$ that ensure it has discriminant $0$.

###### Lemma 6.4

Suppose that $(A,B)$ is an element of $W({\mathbb{C}})$ such that one of the
following three conditions are satisfied:

  * (a)

The kernel of $B$ has dimension at least $2$.

  * (b)

There is a nonzero vector $v\in{\mathbb{C}}^{n+1}$ that is in the kernel of
$B$ and isotropic with respect to $A$.

  * (c)

There exists $k\in\\{1,\ldots,g+1\\}$ such that $a_{ij}=b_{ij}=0$ for all
$1\leq i\leq k$ and all $1\leq j\leq n+1-k$.

Then $\Delta(A,B)=0$.

Proof: This is a standard result in the algebraic geometric theory of pencils
of quadrics. We give another proof using the explicit formula for
$f(x,y)=f_{A,B}(x,y).$ The claim regarding Condition (c) is Lemma 4.5. If the
kernel of $B$ has dimension at least $2$, then the quadratic form defined by
$A$ restricted to the kernel of $B$ admits a nonzero isotropic vector in
${\mathbb{C}}^{n+1}$. Thus Condition (a) implies Condition (b). Suppose now
that Condition (b) is satisfied. Then the $y^{n+1}$-coefficient of $f(x,y)$ is
$0$ since $B$ is singular. The $xy^{n}$-coefficient of $f(x,y)$ equals, up to
sign, the alternating sum of the determinants of the matrices obtained by
replacing the $i$-th column of $B$ by the $i$-th column of $A$. By translating
the vector $v$ to $(1,0,0,\ldots,0)$ using an element of ${\rm
SL}_{n+1}({\mathbb{C}})$, we may assume that the first column (and row) of $B$
is $0$ and the $(1,1)$-entry of $A$ is $0$. It is then easy to see that the
determinant of the matrix obtained by replacing the $i$-th column of $B$ by
the $i$-th column of $A$ is $0$ for any $i$. Hence $\Delta(A,B)=\Delta(f)=0$.
$\Box$

We now translate these conditions into the vanishing of
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)$ for certain sets ${\mathcal{Z}}$. To
this end, define the set
${\mathcal{Z}}_{1}\subset{\mathcal{K}}\cup{\mathcal{M}}$ by

${\mathcal{Z}}_{1}:=\\{a_{ij}\mid i\leq j,\,i+j\leq n\\}\cup\\{\ell_{ij}\mid
i+j\leq n+1\\}.$

###### Lemma 6.5

Let ${\mathcal{Z}}$ be a saturated subset of ${\mathcal{K}}\cup{\mathcal{M}}$
satisfying one of the following two conditions:

  * (a)

The set ${\mathcal{Z}}$ is not contained in ${\mathcal{Z}}_{1}$.

  * (b)

There exists $k\in\\{1,\ldots,g+1\\}$ such that $a_{kk}\in{\mathcal{Z}}$ and
$\ell_{n+1-k,k}\in{\mathcal{Z}}$.

Then $N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)=0$.

Proof: If ${\mathcal{Z}}$ contains some $\ell_{ij}\notin{\mathcal{Z}}_{1}$,
then for every $s\in T_{\mathcal{Z}}(L,Y)$, the set $\Sigma(L,s)$ (and hence
${\mathcal{S}}(L,s)$) is empty by Lemma 5.6. This implies that
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)=0$. If ${\mathcal{Z}}$ contains some
$a_{ij}\notin{\mathcal{Z}}_{1}$, then every integral $(A,B)\in
s(Y{\mathcal{D}}\times s(Y{\mathcal{D}})$ has discriminant $0$ by Condition
(c) of Lemma 6.4. Once again, this implies that
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)=0$.

Let $k$ be an integer satisfying Condition (b) of the lemma, and let $s\in
T_{\mathcal{Z}}(L,Y)$. Let $(A,B)$ be such that $A\in s(Y{\mathcal{D}})$ and
$B\in{\mathcal{S}}(L,s)$. Since $\ell_{n+1-k,k}\in{\mathcal{Z}}$, it follows
that there exists a nonzero vector $v\in{\mathbb{C}}^{n+1}$ of the form
$(v_{1},\ldots,v_{k},0,\ldots,0)$ that is in the kernel of $B$. Since
$a_{kk}\in{\mathcal{Z}}$, it follows that $v$ is isotropic with respect to
$A$. By Condition (b) of Lemma 6.4, it follows that $\Delta(A,B)=0$, implying
that $N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)=0$, as desired. $\Box$

###  6.2 Bounding the number of distinguished elements in the main body

In this subsection, we bound the number of distinguished elements in the main
body:

###### Theorem 6.6

We have ${\mathcal{I}}_{X}^{{\rm
main}}({{\mathcal{L}}(1)})=O\bigl{(}X^{n+1-1/(10n)}\bigr{)}$.

As ${{\mathcal{L}}(M)}\subset{{\mathcal{L}}(1)}$ for $M\geq 1$, it follows
that ${\mathcal{I}}_{X}^{\rm main}({{\mathcal{L}}(M)})$ satisfies the same
bound.

We will use the Selberg sieve to show that distinsuished elements are
negligible in number in the main body. However, applying the Selberg sieve
requires asymptotics along with a power saving error term. Our methods in §5
do not yield such results.

Hence we will instead fiber over $B\in s(Y{\mathcal{D}})\cap S({\mathbb{Z}})$
having determinant $0$, apply the Selberg sieve to prove that there are
negligibly many $A\in s(Y{\mathcal{D}})\cap S({\mathbb{Z}})$ such that $(A,B)$
is distinguished, and then bound the number of possible $B$’s using the
results of Section 5. To carry out the middle step, we require the following
lower bound on the number of nondistinguished elements modulo primes $p$ that
is independent of $p$ and $B$.

###### Lemma 6.7

Let $B_{0}$ be an element in $S({\mathbb{F}}_{p})$ with
${\mathbb{F}}_{p}$-rank $n$. Let $S_{B_{0}}^{{\rm ndist}}({\mathbb{F}}_{p})$
denote the set of elements $A\in S({\mathbb{F}}_{p})$ such that $(A,B)$ has
nonzero discriminant and $A$ and $B$ do not have a common isotropic
$(g+1)$-dimensional subspace. Then

$\frac{\\#S_{B_{0}}^{{\rm
ndist}}({\mathbb{F}}_{p})}{\\#S({\mathbb{F}}_{p})}\gg_{n}1.$

Proof: For an element $B\in S({\mathbb{F}}_{p})$ with ${\mathbb{F}}_{p}$-rank
$n$ and kernel spanned by $v$, let $d(B)$ denote the discriminant of the
corresponding quadratic form on ${\mathbb{F}}_{p}^{n+1}/({\mathbb{F}}_{p}v)$.
If $B_{1},B_{2}\in S({\mathbb{F}}_{p})$ have ${\mathbb{F}}_{p}$-rank $n$ and
$d(B_{1})/d(B_{2})\in{\mathbb{F}}_{p}^{\times 2}$, then $B_{1}$ and $B_{2}$
are ${\rm SL}_{n+1}({\mathbb{F}}_{p})$-equivalent. Indeed, by using ${\rm
SL}_{n+1}({\mathbb{F}}_{p})$ transformations, we may assume the last row and
columns of $B_{1}$ and $B_{2}$ are all $0$. The nondegenerate forms defined by
the top left $n\times n$ blocks of $B_{1}$ and $B_{2}$ have discriminants
$d(B_{1})$ and $d(B_{2})$, which are in the same quadratic residue class.
Hence they are equivalent via an element $\gamma\in{\rm
GL}_{n}({\mathbb{F}}_{p})$. Expanding $\gamma$ to an element in ${\rm
SL}_{n+1}({\mathbb{F}}_{p})$ by appending an additional row and column whose
entries are all $0$, except for the $(n+1,n+1)$-entry which is
$\det\gamma^{-1}$, gives an element in ${\rm SL}_{n+1}({\mathbb{F}}_{p})$ that
takes $B_{1}$ to $B_{2}$.

Let $B_{0}\in S({\mathbb{F}}_{p})$ have ${\mathbb{F}}_{p}$-rank $n$. For each
binary $n$-ic form $f(x,y)=a_{0}x^{n}+\cdots+a_{n}y^{n}$ over
${\mathbb{F}}_{p}$ that splits completely over ${\mathbb{F}}_{p}$ such that
$\Delta(xf(x,y))\neq 0$ and $a_{0}\neq 0$, we construct a nondistinguished
element $(A_{0},B_{0})$ with $f_{A_{0},B_{0}}=xf(x,y)$. Let $f$ be such a
form. Then $a_{n}\neq 0$. Let $\alpha=d(B_{0})/a_{n}$. As noted in §3.1, there
exist at least two (in fact $2^{n-1}$) ${\rm SL}_{n}({\mathbb{F}}_{p})$-orbits
of $(A,B)\in W_{n}({\mathbb{F}}_{p})$ such that $f_{A,B}=\alpha f(x,y)$. Pick
two inequivalent representatives $(A_{1},B_{1})$ and $(A_{2},B_{2})$. Let
$A_{1}^{\prime}$ and $A_{2}^{\prime}$ be the $(n+1)$-ary quadratic forms
obtained from $A_{1}$ and $A_{2}$, respectively, by appending an additional
row and column whose entries are all $0$ except for the $(n+1,n+1)$-entry
which is $\alpha^{-1}$. Let $B_{1}^{\prime}$ and $B_{2}^{\prime}$ be the
$(n+1)$-ary quadratic forms obtained from $B_{1}$ and $B_{2}$, respectively,
by appending an additional row and column whose entries are all $0$. Then
$f_{A_{1}^{\prime},B_{1}^{\prime}}=f_{A_{2}^{\prime},B_{2}^{\prime}}=xf(x,y).$
Since $(A_{1},B_{1})$ and $(A_{2},B_{2})$ are ${\rm
SL}_{n}({\mathbb{F}}_{p})$-inequivalent, it follows that
$(A_{1}^{\prime},B_{1}^{\prime})$ and $(A_{2}^{\prime},B_{2}^{\prime})$ are
${\rm SL}_{n+1}({\mathbb{F}}_{p})$-inequivalent. Hence, without loss of
generality, we may assume that $(A_{1}^{\prime},B_{1}^{\prime})$ is
nondistinguished. Now $d(B_{1}^{\prime})=\alpha a_{n}=d(B_{0})$, and so there
exists $\gamma\in{\rm SL}_{n+1}({\mathbb{F}}_{p})$ such that $\gamma
B_{1}^{\prime}\gamma^{t}=B_{0}$. Then $A_{0}=\gamma A_{1}^{\prime}\gamma^{t}$
does the job.

We complete the proof of the lemma via the orbit-stabilizer theorem. By the
above construction, there are $\gg_{n}p^{n+1}$ binary $(n+1)$-ic forms
$xf(x,y)$, with $\Delta(xf(x,y))\neq 0$ and $a_{0}\neq 0$, such that there
exists an element $A\in S({\mathbb{F}}_{p})$ with $f_{A,B_{0}}=xf(x,y)$ and
$(A,B_{0})$ nondistinguished. The group
$G_{B_{0}}({\mathbb{F}}_{p})=\\{\gamma\in{\rm
SL}_{n+1}({\mathbb{F}}_{p})\colon\gamma B_{0}\gamma^{t}=B_{0}\\}$ acts on the
set of such $A$ with stabilizer of size $\\#J_{xf}[2]({\mathbb{F}}_{p})$,
where $J_{xf}$ is the Jacobian of the hyperelliptic curve defined by
$z^{2}=xf(x,y)y$. Any element of $\gamma\in G_{B_{0}}({\mathbb{F}}_{p})$
preserves the kernel ${\mathbb{F}}_{p}v$ of $B_{0}$ and stabilizes the
nondegenerate form $b_{0}$ on ${\mathbb{F}}_{p}^{n+1}/({\mathbb{F}}_{p}v)$
induced by $B_{0}$. The determinant $1$ condition then gives

$\\#G_{B_{0}}({\mathbb{F}}_{p})=\\#\text{O}(b_{0})({\mathbb{F}}_{p})=2p^{\frac{n^{2}+n}{2}}\Bigl{(}1-\frac{O(1)}{p^{2}}\Bigr{)}.$

Finally, since $\\#J_{xf}[2]({\mathbb{F}}_{p})\ll_{n}1$, we have

$\\#S_{B_{0}}^{{\rm
ndist}}({\mathbb{F}}_{p})\gg_{n}p^{n+1}p^{n(n+1)/2}=p^{(n+1)(n+2)/2}=\\#S({\mathbb{F}}_{p}),$

as desired. $\Box$

###### Corollary 6.8

Fix $a\in{\mathbb{F}}_{p}^{\times}$ and $B_{0}\in S({\mathbb{F}}_{p})$ with
rank $n$. Let $S_{B_{0}}^{{\rm ndist}}({\mathbb{F}}_{p})_{a_{11}=a}$ denote
the set of all elements $A\in S_{B}^{{\rm ndist}}({\mathbb{F}}_{p})$ with
$a_{11}=a$. Then

$\frac{\\#S_{B}^{{\rm
ndist}}({\mathbb{F}}_{p})_{a_{11}=a}}{\\#S({\mathbb{F}}_{p})/p}\gg_{n}1.$

Proof: Since the property of $(A,B)$ being nondistinguished is preserved when
$A$ is multiplied by an element of ${\mathbb{F}}_{p}^{\times}$, the claim
follows immediately from Lemma 6.7. $\Box$

We now bound the number of pairs $(A,B)$ in the main body where the first row
and column of $B$ are zero.

###### Proposition 6.9

We have

$\int_{\begin{subarray}{c}s\in T^{\prime}\\\ Yw(a_{11})\gg
1\end{subarray}}\\#\bigl{\\{}(A,B)\in(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{{\mathcal{L}}(1)}:b_{1i}=0\,\forall
i\\}\,\delta(s)d^{\times}s\ll X^{n+1-1/(10n)}.$ (43)

Proof: Let $s\in T^{\prime}$ be an element with $Yw(a_{11})\gg 1$. Then

$\begin{array}[]{rcl}\displaystyle\\#\bigl{\\{}A\in s(Y{\mathcal{D}})\cap
S({\mathbb{Z}})\bigr{\\}}&\ll&\displaystyle Y^{(n+1)(n+2)/2};\\\\[1.8063pt]
\displaystyle\\#\bigl{\\{}B\in s(Y{\mathcal{D}})\cap
S({\mathbb{Z}}):b_{1i}=0\,\forall i\bigr{\\}}&\ll&\displaystyle
Y^{n(n+1)/2}\prod_{i=1}^{n+1}w(b_{1i})^{-1}.\end{array}$

For each $B\in s(Y{\mathcal{D}})\cap S({\mathbb{Z}})$ having rank $n$, we
bound the number of $A\in s(Y{\mathcal{D}})\cap S({\mathbb{Z}})$ such that
$(A,B)$ is distinguished. Indeed, after additionally fibering over the
coefficient $a_{11}$, Corollary 6.8 in conjunction with an application of the
Selberg sieve, used as in [28], saves a $1/5$th power of the smallest
coefficient range among the remaining $a_{ij}$. That is, we obtain a saving of
$(Yw(a_{12}))^{-1/5}$.

Therefore, the left hand side of (43) is

$\begin{array}[]{rcl}&\ll&\displaystyle
Y^{(n+1)^{2}-1/5}\int_{\begin{subarray}{c}s\in T^{\prime}\\\ Yw(a_{11})\gg
1\end{subarray}}w(a_{12})^{-1/5}s_{1}^{n(n+1)}s_{2}^{(n-1)(n+1)}\cdots
s_{n}^{n+1}\delta(s)d^{\times}s\\\\[12.64746pt] &\ll&\displaystyle
Y^{(n+1)^{2}-1/5}\int_{\begin{subarray}{c}s\in T^{\prime}\\\ Yw(a_{11})\gg
1\end{subarray}}s_{1}^{(n-1)/5}\prod_{j=2}^{n}s_{j}^{(2n+2-2j)/5-(n+1)(n+1-j)(j-1)}d^{\times}s.\end{array}$
(44)

In particular, the power of $s_{i}$ above is negative for all
$j\in\\{2,\ldots,n\\}$, and hence the integral over $s_{2},\ldots,s_{n}$ is
absolutely bounded. The condition that $Yw(a_{11})\gg 1$ on the integrand
implies that we have $s_{1}\ll Y^{1/2n}$. Therefore the terms in (44) are

$\ll Y^{(n+1)^{2}-1/5}\int_{1\ll s_{1}\ll
Y^{1/(2n)}}s_{1}^{(n-1)/5}d^{\times}s_{1}\ll
Y^{(n+1)^{2}-1/5+(n-1)/(10n)}=Y^{(n+1)^{2}-(n+1)/(10n)}.$

Since $Y=X^{1/(n+1)}$, we obtain the result. $\Box$

###### Remark 6.10

Our use of the Selberg sieve saves a power of the smallest range of any
coordinate. In the above proof, we fiber over $a_{11}$ because in the region
of the main body close to the cusp, just before we enter the shallow cusp, the
range of $a_{11}$ has size $\ll 1$. In this case, the Selberg sieve gives no
saving at all. Once we fiber over $a_{11}$, the next smallest range is that of
$a_{12}$. Implicit in our proof is an argument that either the range of
$a_{12}$ is large, in which case the Selberg sieve gives the desired saving,
or the number of pairs $(A,B)$ is automatically small.

Proof of Theorem 6.6: Recall from (38) that we have

${\mathcal{I}}_{X}^{\rm
main}({{\mathcal{L}}(1)})\ll\sum_{L}\sum_{{\mathcal{Z}}:a_{11}\notin{\mathcal{Z}}}N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X),$

where the second sum is over all saturated ${\mathcal{Z}}$. Since
${\mathcal{Z}}$ is saturated and $a_{11}\not\in{\mathcal{Z}}$, we have
${\mathcal{Z}}\subset{\mathcal{M}}$. If $\ell_{k,1}=0$ for every
$k=1,\ldots,n$, then $(1,0,\ldots,0)$ is in the kernel of $B$ implying that
the top row of $B$ is zero. The number of such pairs $(A,B)$ has already been
bounded in Proposition 6.9, and hence we may assume that
$\ell_{n,1}\not\in{\mathcal{Z}}$. Fix a nondecreasing $n$-tuple $L$ of
positive real numbers, and a saturated ${\mathcal{Z}}\subset{\mathcal{M}}$
with $\ell_{n,1}\not\in{\mathcal{Z}}$ such that
$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)\neq 0$. We partition the integrand
$T_{\mathcal{Z}}(L,Y)$ into two parts: let $T_{1}$ denote the subset of
$T_{\mathcal{Z}}(L,Y)$ consisting of elements $s$ for $s=(s_{i})_{i}$ with
$s_{n}\geq Y^{\delta}$, and let $T_{2}$ denote the subset of elements $s$ with
$1\ll s_{n}<Y^{\delta}$, where $\delta$ is a positive constant to be optimized
later.

We first bound the contribution to $N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)$
from $T_{1}$. Since $Yw(a_{11})\gg 1$, we have

$\\#\bigl{(}(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{{\mathcal{L}}(1)}\bigr{)}\leq\\#\bigl{(}(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap W_{n+1}({\mathbb{Z}})\bigr{)}\ll Y^{(n+1)(n+2)}$

for $s\in T_{1}$. Integrating over $T_{1}$ gives the bound

$\begin{array}[]{rcl}\displaystyle\int_{s\in
T_{1}}\\#\bigl{(}(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{{\mathcal{L}}(1)}\bigr{)}\,\delta(s)d^{\times}s&\ll&\displaystyle
Y^{(n+1)(n+2)}\int_{s_{1},\ldots,s_{n-1}\gg 1}\int_{s_{n}\geq
Y^{\delta}}\delta(s)d^{\times}s\\\\[7.22743pt] &\ll&\displaystyle
Y^{(n+1)(n+2)}\int_{s_{n}\geq
Y^{\delta}}s_{n}^{-n(n+1)}d^{\times}s_{n}\\\\[7.22743pt] &\ll&\displaystyle
Y^{n+1-n(n+1)\delta}X^{n+1}.\end{array}$ (45)

Next, we consider the contribution from $T_{2}$. Define the map
$\pi:{\mathcal{Z}}_{1}\cap{\mathcal{M}}\to{\mathcal{M}}$ by

$\pi(\ell_{ij})=\left\\{\begin{array}[]{rl}\ell_{n1}&\;{\rm if}\;j=1\mbox{ and
}i\geq 2,\\\ \ell_{i,n+2-i}&\;{\rm otherwise}.\end{array}\right.$

Since we have assumed that $N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)\neq 0$,
Lemma 6.5 implies that ${\mathcal{Z}}\subset{\mathcal{Z}}_{1}$ and so the
image of $\pi$ lies in ${\mathcal{M}}\backslash{\mathcal{Z}}$. Then for any
$\alpha\in{\mathcal{Z}}_{1}\cap{\mathcal{M}}$ and any $s\in
T_{\mathcal{Z}}(L,Y)$, we have $w_{L}(\pi(\alpha))\gg w_{L}(\alpha)$ and
$w_{L}(\pi(\alpha))\gg 1$. These inequalities along with (41) and (42) imply
that for any $s\in T_{\mathcal{Z}}(L,Y)$, the number
$\\#(s(Y{\mathcal{D}})\cap{\mathcal{S}}(L,s))$ of possible $B$’s is

$\ll
Y^{n(n+1)/2}w({\mathcal{Z}}\cap{\mathcal{M}})^{-1}\Bigl{(}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i+j>n+1\end{subarray}}\frac{L_{i}}{L_{j}}\Bigr{)}\\\ \ll
Y^{n(n+1)/2}\Bigl{(}\prod_{\begin{subarray}{c}\ell\in{\mathcal{Z}}_{1}\cap{\mathcal{M}}\\\
\ell\neq\ell_{n1}\end{subarray}}\frac{w(\pi(\ell))}{w(\ell)}\Bigr{)}\Bigl{(}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i+j>n+1\end{subarray}}\frac{L_{i}}{L_{j}}\Bigr{)}.$

For each possible $B$, applying the Selberg sieve using Lemma 6.7 gives us a
bound of

$\ll Y^{(n+1)(n+2)/2}Y^{-1/5}w(a_{11})^{-1/5}$

for the number of possible choices for $A$. Therefore,

$\\#\bigl{(}(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{{\mathcal{L}}(1)}\bigr{)}\ll
Y^{-1/5}X^{n+1}w(a_{11})^{-1/5}\Bigl{(}\prod_{\begin{subarray}{c}\ell\in{\mathcal{Z}}_{1}\cap{\mathcal{M}}\\\
\ell\neq\ell_{n1}\end{subarray}}\frac{w(\pi(\ell))}{w(\ell)}\Bigr{)}\Bigl{(}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i+j>n+1\end{subarray}}\frac{L_{i}}{L_{j}}\Bigr{)},$

for $s\in T_{2}$. We compute the ratio of these weights: For any $i\geq 2$ and
$j=1$, we have

$\frac{w(\pi(\ell_{i1}))}{w(\ell_{i1})}=\frac{w(\ell_{n1})}{w(\ell_{i1})}=\frac{L_{n}}{L_{i}}.$

For any other $i,j$, we have

$\frac{w(\pi(\ell_{ij}))}{w(\ell_{ij})}=\frac{w(\ell_{i,n+2-i})}{w(\ell_{ij})}=\frac{t_{j}}{t_{n+2-i}}.$

As the $L_{i}$ are nondecreasing and positive, we multiply by the Haar measure
character $\delta(s)$ to obtain

$\begin{array}[]{rcl}&&\displaystyle
w(a_{11})^{-1/5}\Bigl{(}\prod_{\begin{subarray}{c}\ell\in{\mathcal{Z}}_{1}\cap{\mathcal{M}}\\\
\ell\neq\ell_{n1}\end{subarray}}\frac{w(\pi(\ell))}{w(\ell)}\Bigr{)}\Bigl{(}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\
i+j>n+1\end{subarray}}\frac{L_{i}}{L_{j}}\Bigr{)}\,\delta(s)\\\\[14.45377pt]
&\leq&\displaystyle w(a_{11})^{-1/5}\Bigl{(}\prod_{\begin{subarray}{c}i,j\geq
1\\\ i+j\leq
n+1\end{subarray}}\frac{t_{j}}{t_{n+2-i}}\Bigr{)}\Bigl{(}\prod_{i=2}^{n}\frac{t_{n+2-i}}{t_{1}}\Bigr{)}\Bigl{(}\prod_{1\leq
j<i\leq n+1}\frac{t_{i}}{t_{j}}\Bigr{)}\\\\[14.45377pt] &=&\displaystyle
w(a_{11})^{-1/5}\prod_{i=2}^{n}\frac{t_{n+2-i}}{t_{1}}\\\\[14.45377pt]
&=&\displaystyle w(a_{11})^{-1/5}s_{1}^{-(n+1)(n-1)}s_{2}^{-(n+1)(n-2)}\cdots
s_{n-1}^{-(n+1)}.\end{array}$

The powers of $s_{i}$ in the above expression are negative for $1\leq i\leq
n-1$, while the power of $s_{n}$ is $2/5$. Integrating over $T_{2}$ now gives
the bound

$\begin{array}[]{rcl}\displaystyle\int_{s\in
T_{1}}\\#\bigl{(}(s(Y{\mathcal{D}})\times
s(Y{\mathcal{D}}))\cap{{\mathcal{L}}(1)}\bigr{)}\,\delta(s)d^{\times}s&\ll&\displaystyle
Y^{-1/5}X^{n+1}\int_{1\ll s_{n}\ll
Y^{\delta}}s_{n}^{2/5}d^{\times}s_{n}\\\\[12.64746pt] &\ll&\displaystyle
Y^{-1/5+(2\delta)/5}X^{n+1}.\end{array}$ (46)

Combining (45) and (46) and choosing $\delta=\frac{5n+6}{5n^{2}+5n+2}$ yields

$N({{\mathcal{L}}(1)},L,{\mathcal{Z}},X)\ll X^{n+1-\frac{n-2}{5n^{2}+5n+2}}.$

The summation of this bound over the $O(1)$ different possible
${\mathcal{Z}}$’s and the $O(Y^{\epsilon})$ different possible $L$’s, in
conjunction with the bound in Proposition 6.9, implies Theorem 6.6. $\Box$

###  6.3 Bounding the number of distinguished elements in the shallow cusp

In this subsection, we bound the number of distinguished elements having large
$q$-invariant that lie in the shallow cusp of the fundamental domain.

###### Theorem 6.11

Let $\eta>0$ be any real number. Assume that $M>X^{\eta}$. Then

${{\mathcal{I}}_{X}^{{\rm
scusp}}({{\mathcal{L}}(M)})=O\bigl{(}X^{n+1-\min(\eta,1)/(22n^{6})}\bigr{)}}.$

We will take $\eta=1/4$ when we prove Theorem 5 in §6.5.

####  6.3.1 A preliminary bound of $O_{\epsilon}(X^{n+1+\epsilon})$

We again use (38) to write

${\mathcal{I}}_{X}^{\rm
scusp}({{\mathcal{L}}(M)})\ll\sum_{L,{\mathcal{Z}}}N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X),$

where the sum is over nondecreasing $n$-tuples $L=(L_{1},\ldots,L_{n})$ of
positive real numbers that partition the region
$\\{(\mu_{1},\ldots,\mu_{n})\in[Y^{-\Theta_{1}},Y^{\Theta_{2}}]^{n}:\mu_{1}\leq\mu_{2}\leq\ldots\leq\mu_{n}\\}$
into dyadic ranges, and over saturated
${\mathcal{Z}}\subset{\mathcal{K}}\cup{\mathcal{M}}$ such that
$a_{11}\in{\mathcal{Z}}$ and $a_{g+1,g+1}\not\in{\mathcal{Z}}$. By Lemma 6.5,
we have $N({{\mathcal{L}}(M)},L,{\mathcal{Z}},X)>0$ only when
${\mathcal{Z}}\subset{\mathcal{Z}}_{1}$, which we henceforth assume.

For $k\in\\{0,\ldots,g\\}$, define the map
$\pi_{k}:{\mathcal{Z}}_{1}\to{\mathcal{K}}\cup{\mathcal{M}}$ by

$\pi_{k}(a_{ij})=a_{n+1-j,j},\qquad\pi_{k}(\ell_{ij})=\left\\{\begin{array}[]{rl}\ell_{n+1-j,j}&\;{\rm