#  Geometric generalizations of the square sieve,
with an application to cyclic covers

Alina Bucur  - Department of Mathematics, University of California at San
Diego, 9500 Gilman Dr 0112, La Jolla, California 92093, USA
<EMAIL_ADDRESS>,  Alina Carmen Cojocaru  - Department of Mathematics,
Statistics and Computer Science, University of Illinois at Chicago, 851 S
Morgan St, 322 SEO, Chicago, 60607, IL, USA; - Institute of Mathematics
“Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei St, Bucharest,
010702, Sector 1, Romania<EMAIL_ADDRESS>,  Matilde N. Lalín  - Département
de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succ.
Centre-Ville, Montreal, Quebec H3C 3J7, Canada<EMAIL_ADDRESS>and
Lillian B. Pierce
with an appendix by Joseph Rabinoff  - Department of Mathematics, Duke
University, 120 Science Drive, Durham, North Carolina 27708, USA;
<EMAIL_ADDRESS>
###### Abstract.

We formulate a general problem: given projective schemes $\mathbb{Y}$ and
$\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from
$\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in
$\mathbb{X}(K)$ of height at most $B$ have a pre-image under $\eta$ in
$\mathbb{Y}(K)$? This problem is inspired by a well-known conjecture of Serre
on quantitative upper bounds for the number of points of bounded height on an
irreducible projective variety defined over a number field. We give a non-
trivial answer to the general problem when $K=\mathbb{F}_{q}(T)$ and
$\mathbb{Y}$ is a prime degree cyclic cover of
$\mathbb{X}=\mathbb{P}_{K}^{n}$. Our tool is a new geometric sieve, which
generalizes the polynomial sieve to a geometric setting over global function
fields.

##  1\. Introduction

We consider the following general problem: given a morphism between two
projective schemes defined over a global field, how many points in the domain
yield points with bounded height in the image? As we will outline, this
problem is related to a well-known conjecture of Serre formulated over number
fields; in our proposed formulation, it may be viewed as a general version of
a question that arises in a wide array of problems in analytic number theory.

To be precise, let $K$ be a global field of arbitrary characteristic and let
$\mathcal{O}$ be a ring of integers in $K$. We denote by $V_{K}$ the set of
places of $K$. For each $v\in V_{K}$, we denote by $|\cdot|_{v}$ the
associated valuation, normalized such that the product formula holds, namely,
for every $x\in K^{\ast}$,

$\prod_{v\in V_{K}}|x|_{v}=1.$ (1.1)

Using the notation

$[x]:=[x_{0}:x_{1}:\ldots:x_{n}]\in\mathbb{P}^{n}_{K}$

for projective points, we consider the height function

$\operatorname{ht}_{K}:\mathbb{P}^{n}_{K}\rightarrow(0,\infty),$
$\operatorname{ht}_{K}([x]):=\prod_{v\in
V_{K}}\max\\{|x_{0}|_{v},\dots,|x_{n}|_{v}\\},$ (1.2)

and note that it gives rise to the height function

$\operatorname{ht}_{K}:K\rightarrow[0,\infty),$
$\operatorname{ht}_{K}(x):=\left\\{\begin{array}[]{cc}\operatorname{ht}_{K}([x:1])&\text{if}\
x\neq 0,\\\ 0&\text{if}\ x=0.\end{array}\right.$ (1.3)

General Problem

_Given a global field $K$ of arbitrary characteristic, a ring of integers
$\mathcal{O}$ in $K$, projective schemes $\mathbb{X}/K$, $\mathbb{Y}/K$ over
$K$ with fixed models over $\mathcal{O}$, and a $K$-morphism
$\eta:\mathbb{Y}\longrightarrow\mathbb{X}$, defined over $\mathcal{O}$ and of
finite degree, find an upper bound for the cardinality of the set_

$\left\\{[x]\in\mathbb{X}(\mathcal{O}):\operatorname{ht}_{K}([x])<B,\exists[y]\in\mathbb{Y}(\mathcal{O})\;\text{such
that}\;\eta([y])=[x]\right\\}$

_that holds for every $B\geq 1$._

Note that upper bounds for the above cardinality are always given by one of
the two cardinalities below,

$\\#\\{[x]\in\mathbb{X}(K):\operatorname{ht}_{K}([x])<B\\}\leq\\#\\{[x]\in\mathbb{P}_{K}^{n}:\operatorname{ht}_{K}([x])<B\\}.$

For example, when $K$ is a number field of degree $d$ over $\mathbb{Q}$, by
Schanuel’s theorem (e.g. [Ser97, §2.5 p. 17]), there exists an explicit
positive constant $C(K,n)$ such that, as $B\rightarrow\infty$,

$\\#\left\\{[x]\in\mathbb{P}_{K}^{n}:\operatorname{ht}_{K}([x])<B\right\\}\sim
C(K,n)B^{d(n+1)}.$ (1.4)

As a second example, when $K$ is the function field of an absolutely
irreducible projective curve over $\mathbb{F}_{q}$, of genus $g$, as an
immediate consequence of [Ser97, §2.5, Thm. p. 19], there exists an explicit
positive constant $C(K,n)$ such that, as $b\rightarrow\infty$,

$\\#\left\\{[x]\in\mathbb{P}_{K}^{n}:\operatorname{ht}_{K}([x])<q^{b}\right\\}\sim
C(K,n)q^{(b-g)(n+1)}.$ (1.5)

As usual when navigating between the number field and the function field
settings, the parameter $B$ in (1.4) was replaced by $q^{b}$ in (1.5).

In our general problem, for nontrivial choices of $\mathbb{X},\mathbb{Y}$, we
seek a nontrivial upper bound, namely a bound that grows more slowly than the
trivial bound, as $B\rightarrow\infty$ (respectively, as
$q^{b}\rightarrow\infty$ as a function of $b$, or of $q$, or of both $b$ and
$q$).

###  1.1. Serre’s question

Our General Problem has an antecedent in a well-known question of Serre
[Ser97, §13.1 (4) p. 178], which we now recall.

Let $K/\mathbb{Q}$ be a number field of degree $d$, let $n\geq 1$ be an
integer, and let $V$ be an irreducible (non-linear) projective variety in
$\mathbb{P}_{K}^{n}$. Serre seeks an upper bound in $B$ for the cardinality of
the set

$\left\\{[x]\in V(K):\operatorname{ht}_{K}([x])\leq B\right\\}.$ (1.6)

The trivial upper bound for (1.6) is $C(K,n)B^{d(n+1)}$, as mentioned in
(1.4). In [Ser97, §13.1, Thm. 4 p. 178], Serre improves upon the trivial bound
by showing that there exists a constant $0<\gamma<1$ such that, for all $B$,

$\\#\left\\{[x]\in V(K):\operatorname{ht}_{K}([x])\leq
B\right\\}\ll_{n,K,V}(B^{d})^{(n+1)-\frac{1}{2}}(\operatorname{log}B)^{\gamma}.$
(1.7)

Serre deduces (1.7) from a result counting integral points on affine thin
sets, which he proves using the large sieve. A variant due to Cohen [Coh81] of
the result counting integral points on affine thin sets may also be used;
however, Cohen’s result leads to $\gamma=1$. Serre then poses the question of
whether (1.7) can be improved to

$\\#\left\\{[x]\in V(K):\operatorname{ht}_{K}([x])\leq
B\right\\}\ll(B^{d})^{(n+1)-1}(\operatorname{log}B)^{c}$ (1.8)

for some $c\geq 0$, without specifying whether the implied $\ll$-constant
might depend on any of $n,K,V$; see [Ser97, §13.1.3, p. 178]. Additionally,
Serre notes that the logarithmic factor is necessarily present in certain
cases.

Our General Problem is a generalization of Serre’s question and, as a special
case, encompasses a global function field version of Serre’s question.

###  1.2. Main goals

The purpose of the present paper is to investigate the General Problem in a
particular function field setting. Precisely, our goals are two-fold:

  1. (I)

to provide a nontrivial upper bound for the General Problem when
$K=\mathbb{F}_{q}(T)$, $\mathcal{O}=\mathbb{F}_{q}[T]$,
$\mathbb{X}=\mathbb{P}^{n}_{K}$, $\mathbb{Y}$ is a prime degree cyclic cover
of $\mathbb{X}$, and $\eta$ is the natural projection;

  2. (II)

to accomplish (I) by developing a geometric sieve method which generalizes
recent sieve methods (such as the square sieve of Heath-Brown and the
polynomial sieve of Browning) that have been used to improve on Serre’s bound
(1.7).

We will present our main results in the next two sections, according to the
above two goals.

###  1.3. Main results I: counting rational points

We treat the General Problem in the following concrete case:
$K=\mathbb{F}_{q}(T)$, $\mathcal{O}=\mathbb{F}_{q}[T]$,
$\mathbb{X}=\mathbb{P}^{n}_{K}$, $\mathbb{Y}$ a prime degree cyclic cover of
$\mathbb{X}$, and $\eta$ the natural projection.

To be precise, let $q$ be an odd rational prime power, $n\geq 1$ an integer,
$m\geq 2$ an integer, and $\ell\geq 2$ a rational prime such that $\ell\mid
m$. We set

$\mathbb{X}:=\mathbb{P}^{n}_{\mathbb{F}_{q}(T)}$

and take $\mathbb{Y}$ as the projective scheme in the weighted projective
space
$\mathbb{P}_{\mathbb{F}_{q}(T)}^{n+1}\left(1,\ldots,1,\frac{m}{\ell}\right)$
defined by the weighted projective model

$\mathbb{Y}:\qquad X_{n+1}^{\ell}=F(X_{0},\ldots,X_{n})$ (1.9)

for some polynomial $F\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$ of total
degree $m$ in $X_{0},\ldots,X_{n}$. We take

$\eta:\mathbb{Y}\longrightarrow\mathbb{P}^{n}_{\mathbb{F}_{q}(T)}$

as the natural projection defined by

$\eta([x_{0}:x_{1}:\ldots:x_{n}:x_{n+1}]):=[x_{0}:x_{1}:\ldots:x_{n}].$ (1.10)

Our interest is in estimating, from above and as a function of $q^{b}$ ($q$
fixed, $b\rightarrow\infty$), the counting function

$N(\mathbb{Y},\mathbb{F}_{q}(T),\eta;b):=\\#\left\\{[x]\in\mathbb{P}^{n}_{\mathbb{F}_{q}(T)}:\operatorname{ht}_{\mathbb{F}_{q}(T)}[x]<q^{b},\
\exists[y]\in\mathbb{Y}(\mathbb{F}_{q}(T))\;\text{such
that}\;\eta([y])=[x]\right\\},$ (1.11)

or, equivalently, the counting function

$\\#\left\\{(x_{0},\ldots,x_{n})\in\mathbb{F}_{q}[T]^{n+1}:\;\operatorname{deg}_{T}(x_{i})<b\;\forall
0\leq i\leq n,\exists x_{n+1}\in\mathbb{F}_{q}[T]\;\text{such
that}\;x_{n+1}^{\ell}=F(x_{0},\ldots,x_{n})\right\\},$

where $\operatorname{deg}_{T}(x_{i})$ denotes the degree of $x_{i}$ as a
polynomial in $T$.

Note that, in this setting, the trivial bound is

$N(\mathbb{Y},\mathbb{F}_{q}(T),\eta;b)\leq\\#\left\\{[x]\in\mathbb{P}^{n}_{\mathbb{F}_{q}(T)}:\operatorname{ht}_{\mathbb{F}_{q}(T)}([x])<q^{b}\right\\}\leq\left(q^{b}\right)^{(n+1)}$
(1.12)

(see (1.5) and the comment in equation (6.1) of §6).

In contrast, the function field analogue of Serre’s conjecture (1.8) suggests
that it might be possible to prove, under appropriate conditions on $F$, that
there exists some constant $c$ for which

$N(\mathbb{Y},\mathbb{F}_{q}(T),\eta;b)\ll_{\ell,m,n,q,F}\left(q^{b}\right)^{(n+1)-1}b^{c},$
(1.13)

with the implicit constant possibly depending on $\ell,m,n,q,F$.

Our first main theorem improves upon the trivial bound (1.12) and approaches,
in the limit as $n\rightarrow\infty$ (upon omitting an analysis of how the
limit impacts the dependence of the $\ll$-constant on $n$), the upper bound
$\left(q^{b}\right)^{(n+1)-1}$ appearing in (1.13), as long as the defining
polynomial $F$ is Dwork-regular. We now recall the definition of such a
polynomial:

######  Definition 1.1.

Given a field $L$, a fixed algebraic closure $\bar{L}$ of $L$, and an integer
$n\geq 0$, a homogeneous polynomial $H\in L[X_{0},\ldots,X_{n}]$ is called
_Dwork-regular over $L$ with respect to the coordinates $X_{0}$,
$\ldots,X_{n}$_ if there is no solution in $\mathbb{P}^{n}_{\overline{L}}$ to
the system of equations

$H(X_{0},\ldots,X_{n})=0\qquad X_{i}\frac{dH}{dX_{i}}=0,\quad\forall 0\leq
i\leq n.$

The assumption of Dwork-regularity, which implies in particular that the
projective hypersurface defined by $H(X_{0},\ldots,X_{n})=0$ is smooth, arises
in celebrated work of Dwork [Dwo62, §3(a)], is associated with the work of
Kouchnirenko [Kou76, Dfn 1.19], and commonly appears in work relying on Weil-
Deligne bounds, such as that of Katz [Kat08], [Kat09, p. 4]. It is not
particularly restrictive since, when $L$ is infinite, a
$\operatorname{GL}_{n+1}(L)$ change of variables can bring a smooth model into
Dwork-regular form; see Lemma 3.1 of §3.

Our first main theorem is:

######  Theorem 1.2 (Counting Rational Points on a Prime Degree Cyclic Cover
of $\mathbb{P}^{n}_{\mathbb{F}_{q}(T)}$).

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, $\ell\geq 2$ a
rational prime, and $F\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$ a homogeneous
polynomial of degree $m\geq 2$ in $X_{0},\ldots,X_{n}$, with
$\operatorname{char}\mathbb{F}_{q}\nmid m$. Assume that:

  1. (i)

$\ell\mid\operatorname{gcd}(m,q-1)$;

  2. (ii)

$F$ is Dwork-regular over $\mathbb{F}_{q}(T)$ with respect to the coordinates
$X_{0}$, $\ldots,X_{n}$.

Let $\mathbb{Y}$ be the projective scheme in the weighted projective space
$\mathbb{P}_{\mathbb{F}_{q}(T)}^{n+1}\left(1,\ldots,1,\frac{m}{\ell}\right)$
defined by the weighted projective model (1.9). Let $\eta$ be the projection
(1.10). Then, there exists $b(n,q,F)>0$ such that, for any $b\geq b(n,q,F)$,
the quantity $N(\mathbb{X},\mathbb{Y},\mathbb{F}_{q}(T),\eta;b)$ defined in
(1.11) satisfies the bound

$N(\mathbb{Y},\mathbb{F}_{q}(T),\eta;b)\ll_{\ell,m,n,q,F}\;\left(q^{b}\right)^{(n+1)-\frac{n}{n+1}}b^{2},$

where the implicit constant depends on $\ell,m,n,q$, and $F$.

Later on in Theorem 6.1, we will use the function mentioned in the displayed
equation below (1.11) to state a version of Theorem 1.2 in terms of counting
perfect $\ell$-th power values of a homogeneous polynomial
$F\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$.

For precise information on the way in which the implicit constant depends on
$F$, see (10.4). For precise constraints on $b$, see §10.2. For now, let us
note only that $b(n,q,F)$ depends on $n,q$, and a finite set of primes
$\mathcal{P}_{\text{exc}}(F)$ defined in (6.2) that plays the role of the set
of primes of bad reduction for $F$.

To put Theorem 1.2 in context, let us recall the current state of knowledge
toward Serre’s conjecture (1.8) when $K=\mathbb{Q}$. For $n=1,2$, Broberg
[Bro09] proved a weak form of Serre’s conjecture (with $B^{\epsilon}$ in place
of a logarithmic factor) via the determinant method. For $n\geq 3$ and in the
case of cyclic covers of degree $\ell$, the power sieve argument presented by
Munshi in [Mun09] leads to the upper bound

$\ll_{\ell,m,n,F}B^{(n+1)-\frac{n}{n+1}}(\operatorname{log}B)^{\frac{n}{n+1}},$
(1.14)

where $F$ is the defining polynomial of the cover and $m$ is its degree; see
Remark 1.3 for additional notes on this result and [Mun09, Thm. 1.1].
Recently, Bonolis [Bon21] refined the argument given in [Mun09] and obtained
the upper bound

$\ll_{\ell,m,n,F}B^{(n+1)-\frac{n+1}{n+2}}(\operatorname{log}B)^{\frac{n+1}{n+2}}.$
(1.15)

The best known result is due to Heath-Brown and the third author [HBP12], who
proved Serre’s conjecture (1.8) for all $n\geq 9$ in the case of cyclic
covers, by combining a sieve method with the $q$-analogue of Van der Corput’s
method.

Note that, in the limit $n\rightarrow\infty$ and aside from an analysis of how
the limit impacts the dependence of the $\ll$-constant on $n$, the upper bound
(1.14) approaches one of the form conjectured in (1.8). Our result in Theorem
1.2 is thus an analogue over $\mathbb{F}_{q}(T)$ of Munshi’s result over
$\mathbb{Q}$.

It would be interesting to pursue a function field analogue of Bonolis’s
refinement (1.15) of (1.14), or, furthermore, to adapt Heath-Brown and
Pierce’s $q$-analogue of Van der Corput method to the function field setting
of Theorem 1.2 (see Remark 9.2).

######  Remark 1.3.

It is perhaps helpful to clarify a point in the literature on cyclic covers of
$\mathbb{P}^{n}$ over $\mathbb{Q}$. Theorem 1.1 of [Mun09] states a bound of
the strength (1.15), but the argument as written therein proves a result of
the strength (1.14). At the suggestion of Munshi, Bonolis [Bon21] implemented
nontrivial averaging in the relevant sieve inequality in order to prove (1.15)
(as well as a more general result over $\mathbb{Q}$). Our result Theorem 1.2
over function fields is of analogous strength to the result (1.14 ) obtained
by the original argument in [Mun09]. See Remark 9.2 for more details and open
questions.

###  1.4. Main results II: geometric sieve inequalities

Our approach to the General Problem proceeds via a sieve, formulated in the
setting of the General Problem, with $K$ a global field of arbitrary
characteristic, $\mathcal{O}$ a ring of integers in $K$,
$\operatorname{ht}_{K}:K\longrightarrow[0,\infty)$ the height function (1.3)
constructed from valuations that satisfy the product formula (1.1),
$\mathbb{X}/K$ and $\mathbb{Y}/K$ projective schemes over $K$ with fixed
models over ${\mathcal{O}}$, and $\eta:\mathbb{Y}\longrightarrow\mathbb{X}$ a
$K$-morphism, defined over ${\mathcal{O}}$ and of finite degree.

As it will not result in any significant loss in sharpness of the results, we
will, for convenience, count points on $\mathbb{X}$ in the affine sense. To
clarify, using the notation

$\underline{x}:=(x_{0},x_{1},\ldots,x_{n})\in\mathbb{A}^{n+1}_{K}$

for affine points, we will work with the height function in the affine space
$\mathbb{A}_{K}^{n+1}$ given by

$\operatorname{ht}_{K}(\underline{x}):=\max\left\\{\operatorname{ht}_{K}(x_{i}):\underline{x}=(x_{0},\ldots,x_{n})\right\\},$
(1.16)

focus on the set

${\mathcal{A}}=\mathcal{A}(B):=\left\\{\underline{x}\in\mathbb{X}({\mathcal{O}}):\operatorname{ht}_{K}(\underline{x})<B\right\\},$
(1.17)

and seek an upper bound, in terms of $B$, for the cardinality of the set

${\mathcal{S}}({\mathcal{A}})={\mathcal{S}}({\mathcal{A}(B)}):=\left\\{\underline{x}\in{\mathcal{A}}:\;\exists\underline{y}\in\mathbb{Y}({\mathcal{O}})\;\text{such
that}\;\eta(\underline{y})=\underline{x}\right\\}.$ (1.18)

The typical sieve approach is to derive information about
$\mathcal{S}({\mathcal{A}})$ from reductions of $\eta$ modulo primes. Towards
this goal, for each finite place $v\in V_{K}$, we denote by
$({\mathcal{O}}_{v},M_{v})$ the associated discrete valuation ring and by
$k_{v}:={\mathcal{O}}_{v}/M_{v}$ the associated residue field. For all but
finitely many finite places $v$ (in which case we refer to $v$ as a place of
good reduction for $\eta$), we denote by
$\eta_{v}:\mathbb{Y}\rightarrow\mathbb{X}$ the reduction of $\eta$ modulo $v$.
We call $\eta_{v}$ ramified at some
$\underline{x}\in\mathbb{X}({\mathcal{O}}_{v})$ if
$\underline{x}\,(\operatorname{mod}v)\in\mathbb{X}(k_{v})$ is a branch point
for the function $\eta_{v}:\mathbb{Y}\rightarrow\mathbb{X}$. For each
$\underline{x}\in\mathbb{X}({\mathcal{O}})$, we define the set

$V_{K}^{\mathrm{ram}}(\underline{x},\eta):=\left\\{v\in V_{K}:v\ \text{of good
reduction for}\ \eta\ \text{and}\ \eta_{v}\;\text{is ramified at
$\underline{x}\,(\operatorname{mod}v)$}\right\\}.$

Moreover, for any nonempty finite set $\mathcal{P}\subseteq V_{K}$ of finite
places of good reduction for $\eta$, we define the subset

$V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x}):=V_{K}^{\mathrm{ram}}(\underline{x},\eta)\cap{\mathcal{P}}.$
(1.19)

With this notation, we prove the following sieve inequality:

######  Theorem 1.4.

(Geometric Sieve)
Let $K$ be a global field of arbitrary characteristic, $\mathcal{O}$ be the
ring of integers in $K$, and
$\operatorname{ht}_{K}:K\longrightarrow[0,\infty)$ the height function (1.3)
constructed from valuations that satisfy the product formula (1.1). Let
$\mathbb{X}/K$, $\mathbb{Y}/K$ be projective schemes over $K$ with fixed
models over $\mathcal{O}$ and let $\eta:\mathbb{Y}\longrightarrow\mathbb{X}$
be a $K$-morphism, defined over $\mathcal{O}$ and of finite degree. For an
arbitrary $B>0$, define the sets $\mathcal{A}$ and $\mathcal{S}(\mathcal{A})$
as in (1.17), (1.18). Then, for any real number $\alpha\geq 1$ and for any
nonempty finite set ${\mathcal{P}}\subseteq V_{K}$ of finite places of good
reduction for $\eta$,

$\left|{\mathcal{S}}({\mathcal{A}})\right|\leq\frac{2}{|{\mathcal{P}}|}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|+\frac{1}{|{\mathcal{P}}|^{2}}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}I_{\alpha}(\underline{x})^{2},$
(1.20)

where, for any $\underline{x}\in\mathbb{X}({\mathcal{O}})$,

$I_{\alpha}(\underline{x}):=\displaystyle\sum_{v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})}\left(\alpha+\left(\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1\right)\cdot\left(\operatorname{deg}\eta-\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|\right)\right).$

Impetus for inequality (1.20) comes from a sequence of papers on sieve
inequalities, beginning with the square sieve over $\mathbb{Q}$ of [HB84]
(itself inspired by [Hoo78]), which was later developed into the power sieve
over $\mathbb{Q}$ in [Mun09] and [Bra15], and into the square sieve over
$\mathbb{F}_{q}(T)$ in [CD08]. The square and power sieves over $\mathbb{Q}$
were strengthened by being combined with the $q$-analogue of Van der Corput’s
method in [Pie06] and [HBP12]. Most recently, Browning [Bro15] expanded the
square sieve over $\mathbb{Q}$ into a polynomial sieve over $\mathbb{Q}$,
while Bonolis [Bon21] developed a related polynomial sieve over $\mathbb{Q}$
involving expansions via trace functions. It is Browning’s work [Bro15] that
motivates our approach towards generalizing the existing versions of the
square sieve (over $\mathbb{Q}$ and over $\mathbb{F}_{q}(T)$) to a geometric
sieve over global fields. We will make an explicit comparison to the square
sieve, polynomial sieve, and other relatives in §2.4.

Since we will deduce Theorem 1.2 from the above sieve inequality, we now state
the relevant consequence of Theorem 1.4 in the case of cyclic covers of prime
degree that we derive by optimizing the choice of $\alpha$.

######  Theorem 1.5.

(Geometric Sieve for Prime Degree Cyclic Covers of $\mathbb{P}^{n}_{K}$)

Let $K$ be a global field of arbitrary characteristic, let $\mathcal{O}$ be
the ring of integers in $K$, and let
$\operatorname{ht}_{K}:K\longrightarrow[0,\infty)$ be the height function
(1.3) constructed from valuations that satisfy the product formula (1.1). Let
$n,m\geq 1$ be integers, take $\mathbb{X}:=\mathbb{P}^{n}_{K}$, and let
$\mathbb{Y}$ be the projective scheme in the weighted projective space
$\mathbb{P}_{K}^{n+1}\left(1,\dots,1,\frac{m}{\ell}\right)$, defined by a
model

$X_{n+1}^{\ell}=F(X_{0},\ldots,X_{n})$ (1.21)

with $\ell$ prime such that $\ell\mid m$, and with
$F\in{\mathcal{O}}[X_{0},\ldots,X_{n}]$ homogeneous of degree $m$ and having
the property that the hypersurface in $\mathbb{P}^{n}_{\overline{K}}$ defined
by

$F(X_{0},\ldots,X_{n})=0$

is smooth. Let $\eta:\mathbb{Y}\longrightarrow\mathbb{X}$ be the cyclic map of
degree $\ell$ defined over $\mathbb{A}_{\overline{K}}^{n}$ by

$\eta(x_{0},x_{1},\ldots,x_{n},x_{n+1}):=(x_{0},x_{1},\ldots,x_{n}).$ (1.22)

For an arbitrary $B>0$, define ${\mathcal{A}}$ and
${\mathcal{S}}({\mathcal{A}})$ as in (1.17), (1.18). Then, for any nonempty
finite set ${\mathcal{P}}\subseteq V_{K}$ of finite places of good reduction
for $\eta$,

$\displaystyle\left|{\mathcal{S}}({\mathcal{A}})\right|\leq$
$\displaystyle\frac{\left|{\mathcal{A}}\right|}{|{\mathcal{P}}|}(\operatorname{deg}\eta-1)^{2}+\frac{2}{|{\mathcal{P}}|}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|$
$\displaystyle+\max_{\begin{subarray}{c}v_{1},v_{2}\in{\mathcal{P}}\\\
v_{1}\neq
v_{2}\end{subarray}}\left|\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in{\mathcal{A}}\\\
\underline{x}\not\in\mathbb{X}_{\mathcal{O}}^{\mathrm{ram}}(v_{1})\cup\mathbb{X}_{\mathcal{O}}^{\mathrm{ram}}(v_{2})\end{subarray}}\left(\left|\eta_{v_{1}}^{-1}(\underline{x}\,(\operatorname{mod}v_{1}))\right|-1\right)\cdot\left(\left|\eta_{v_{2}}^{-1}(\underline{x}\,(\operatorname{mod}v_{2}))\right|-1\right)\right|,$
(1.23)

where, for each finite place $v\in V_{K}$ of good reduction for $\eta$,

$\mathbb{X}_{{\mathcal{O}}}^{\mathrm{ram}}(v):=\left\\{\underline{x}\in\mathbb{X}({\mathcal{O}}):\eta_{v}\;\text{is
ramified at}\;\underline{x}\,(\operatorname{mod}v)\right\\}.$ (1.24)

On the right-hand side of inequality (1.23), the first term may be regarded as
a main term for which a trivial upper bound suffices. We refer to the second
term as the _ramified sieve term_ and remark that it is similar in size to the
main term. We refer to the third term as the _unramified sieve term_ and
remark that, in applications, the primary difficulty is to bound it
nontrivially, and then to choose the sieving set $\mathcal{P}$ appropriately
to balance the third term’s contribution with that of the first term.

###  1.5. Outline of the paper

In §2, we prove Theorem 1.4 and Theorem 1.5. To prove Theorem 1.5, the
essential point is to compute the optimal choice of $\alpha$ for which to
apply Theorem 1.4. The rest of the paper focuses on proving Theorem 1.2, as
follows. In §3, we gather a number of preliminaries on Dwork-regular
polynomials and on Deligne polynomials. In §4, we recall notation and basic
results related to the function field setting of Theorem 1.2. In §5, we
present results on duals and reductions necessary in our analysis of the
unramified sieve term. In §6, we reformulate Theorem 1.2 as an affine
statement (Theorem 6.1) and bound all but the unramified sieve term in the
inequality (1.23). The remainder of the work is focused on treating the
unramified sieve term. In §7, we introduce background material on Fourier
analysis on function fields and prove that the unramified sieve term can be
stated as a mixed character sum. In §8, we state and verify the Weil-Deligne
bounds we require for the unramified sieve term. In §9, we finally bound the
unramified sieve term. In §10, we optimize the choice of the sieving set, thus
completing the proof of Theorem 1.2. Finally, Appendix §11, written by Joseph
Rabinoff, provides a self-contained account of certain standard facts about
duals and reductions employed in §5.

##  2\. Derivation of the fundamental sieve inequalities

###  2.1. Proof of Theorem 1.4

We begin by proving the most general version of the sieve, as stated in
Theorem 1.4. The proof follows the general framework of [HB84] and [Bro15].

For any fixed real number $\alpha\geq 1$, we consider the sum

$\Sigma:=\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}\left(\displaystyle\sum_{v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})}\left(\alpha+\left(|\eta_{v}^{-1}(\underline{x}(\bmod
v))|-1\right)\cdot\left(\operatorname{deg}\eta-|\eta_{v}^{-1}(\underline{x}(\bmod
v))|\right)\right)\right)^{2}$

and note that each $\underline{x}$ therein is added with a non-negative
weight. Moreover, observe that, if $\underline{x}\in\mathcal{S}(\mathcal{A})$,
then the fiber above $\underline{x}$ has at least one element and at most
$\operatorname{deg}\eta$ elements, that is,

$1\leq|\eta^{-1}(\underline{x})|\leq\operatorname{deg}\eta.$

Therefore, in this case, for each $v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})$, we have

$\alpha+\left(|\eta_{v}^{-1}(\underline{x}(\bmod
v))|-1\right)\cdot\left(\operatorname{deg}\eta-|\eta_{v}^{-1}(\underline{x}(\bmod
v))|\right)\geq 1.$

We deduce that, for every $\underline{x}\in\mathcal{S}(\mathcal{A})$,

$\displaystyle\displaystyle\sum_{v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})}$
$\displaystyle\left(\alpha+\left(|\eta_{v}^{-1}(\underline{x}(\bmod
v))|-1\right)\cdot\left(\operatorname{deg}\eta-|\eta_{v}^{-1}(\underline{x}(\bmod
v))|\right)\right)$ (2.1) $\displaystyle\geq\sum_{v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})}1=|\mathcal{P}|-|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|,$
(2.2)

which implies that

$\displaystyle\Sigma$ $\displaystyle\geq$
$\displaystyle\displaystyle\sum_{\underline{x}\in\mathcal{S}(\mathcal{A})}(|\mathcal{P}|-|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|)^{2}$
(2.3) $\displaystyle=$
$\displaystyle|\mathcal{P}|^{2}\cdot|\mathcal{S}(\mathcal{A})|+\sum_{\underline{x}\in\mathcal{S}(\mathcal{A})}(-2|\mathcal{P}|\cdot|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|+|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|^{2})$
$\displaystyle\geq$
$\displaystyle|\mathcal{P}|^{2}\cdot|\mathcal{S}(\mathcal{A})|-2\sum_{\underline{x}\in\mathcal{S}(\mathcal{A})}|\mathcal{P}|\cdot|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|.$

Rearranging the terms and using the non-negativity of
$|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|$ to enlarge the sum over
$\mathcal{S}(\mathcal{A})$ to $\mathcal{A}$, we deduce that

$|\mathcal{S}(\mathcal{A})|\leq|\mathcal{P}|^{-2}\Sigma+2|\mathcal{P}|^{-1}\sum_{\underline{x}\in\mathcal{A}}|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|,$

which completes the proof of (1.20).

###  2.2. Equivalent formulation of Theorem 1.4

Recalling the notation $\mathbb{X}_{{\mathcal{O}}}^{\mathrm{ram}}(v)$ in
(1.24), we see that the bound for $|\mathcal{S}(\mathcal{A})|$ may be
rewritten by expanding the square inside $\Sigma$. This shows that

$\left|{\mathcal{S}}({\mathcal{A}})\right|\leq\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{v_{1},v_{2}\in{\mathcal{P}}}\left|\displaystyle\sum_{i,j\in\\{0,1,2\\}}c_{i,j}(\alpha)S_{i,j}(v_{1},v_{2})\right|+\frac{2}{\left|{\mathcal{P}}\right|}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|,$
(2.4)

in which, for all $v_{1},v_{2}\in{\mathcal{P}}$ and $i,j\in\\{0,1,2\\}$,

$S_{i,j}(v_{1},v_{2}):=\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in{\mathcal{A}}\\\
\underline{x}\not\in\mathbb{X}_{\mathcal{O}}^{\mathrm{ram}}(v_{1})\cup\mathbb{X}_{\mathcal{O}}^{\mathrm{ram}}(v_{2})\end{subarray}}\left|\eta_{v_{1}}^{-1}(\underline{x}\,(\operatorname{mod}v_{1}))\right|^{i}\cdot\left|\eta_{v_{2}}^{-1}(\underline{x}\,(\operatorname{mod}v_{2}))\right|^{j}$
(2.5)

and

$c_{i,j}(\alpha):=\left\\{\begin{array}[]{cc}(\alpha-\operatorname{deg}\eta)^{2}&\text{if
$(i,j)=(0,0)$,}\\\
(\alpha-\operatorname{deg}\eta)(1+\operatorname{deg}\eta)&\text{if
$(i,j)=(1,0)$ or $(0,1)$,}\\\ (1+\operatorname{deg}\eta)^{2}&\text{if
$(i,j)=(1,1)$,}\\\ -(\alpha-\operatorname{deg}\eta)&\text{if $(i,j)=(2,0)$ or
$(0,2)$,}\\\ -(1+\operatorname{deg}\eta)&\text{if $(i,j)=(2,1)$ or
$(1,2)$,}\\\ 1&\text{if $(i,j)=(2,2)$.}\end{array}\right.$

In this formulation, we see that (2.4) is analogous to the polynomial sieve in
[Bro15, Thm. 1.1], except that our formulation omits the weight that is
typically present in previous sieve inequalities of this type (see §2.4).

###  2.3. Proof of Theorem 1.5: optimizing the choice of $\alpha$

The sieve inequality in Theorem 1.5 follows from inequality (1.20) when
specialized to the case of prime degree cyclic covers of projective space,
once we have computed the optimal choice of $\alpha$ to minimize the first
term on the right-hand side of (1.20).

To compute this choice, recall that, since $\eta$ is a cyclic map of prime
degree $\ell$, for each $v\in V_{K}$ of good reduction for $\eta$ and for each
$\underline{x}\in\mathbb{X}(\mathcal{O})$,

$\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|=\left\\{\begin{array}[]{cc}0&\text{if
$\underline{x}\,(\operatorname{mod}v)\not\in\operatorname{Im}\eta_{v}$,}\\\
1&\text{if $\underline{x}\,(\operatorname{mod}v)\in\operatorname{Im}\eta_{v}$,
$\underline{x}\in\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v)$,}\\\ \ell&\text{if
$\underline{x}\,(\operatorname{mod}v)\in\operatorname{Im}\eta_{v}$,
$\underline{x}\not\in\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v)$.}\end{array}\right.$

Note that in the above we used the primality of $\operatorname{deg}\eta$.

We deduce that, for each $v\in V_{K}$ of good reduction for $\eta$ and for
each $\underline{x}\in\mathbb{X}(\mathcal{O})$,

$\left(\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1\right)^{2}=\left\\{\begin{array}[]{cc}1&\text{if
$\underline{x}\,(\operatorname{mod}v)\not\in\operatorname{Im}\eta_{v}$,}\\\
0&\text{if $\underline{x}\,(\operatorname{mod}v)\in\operatorname{Im}\eta_{v}$,
$\underline{x}\in\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v)$,}\\\
(\ell-1)^{2}&\text{if
$\underline{x}\,(\operatorname{mod}v)\in\operatorname{Im}\eta_{v}$,
$\underline{x}\not\in\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v)$.}\end{array}\right.$
(2.6)

As such, for each $v\in V_{K}$ of good reduction for $\eta$ and for each
$\underline{x}\in\mathbb{X}(\mathcal{O})\backslash\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v),$

$\displaystyle\left(\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1\right)^{2}=(\ell-1)+(\ell-2)\left(\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1\right).$
(2.7)

Noting that $\underline{x}\in{\mathcal{S}}({\mathcal{A}})$ implies
$\underline{x}\,(\operatorname{mod}v)\in\operatorname{Im}\eta_{v}$, we infer
that, for each $v\in V_{K}$ of good reduction for $\eta$ and for each
$\underline{x}\in{\mathcal{S}}({\mathcal{A}})$, we have

$\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|\geq 1.$
(2.8)

Now, recall the notation of the model (1.21) for the prime degree cyclic cover
$\mathbb{Y}$. Motivated by our upcoming expressions in terms of character sums
in §7, for each $v\in V_{K}$ of good reduction for $\eta$ and for each
$\underline{x}\in\mathbb{X}(\mathcal{O})$, we set

$\Psi_{v}(F(\underline{x})):=\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1.$
(2.9)

Then (2.7) implies that, for each $v\in V_{K}$ of good reduction for $\eta$
and for each
$\underline{x}\in\mathbb{X}(\mathcal{O})\backslash\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v)$,

$\displaystyle\Psi_{v}(F(\underline{x}))^{2}=(\ell-1)+(\ell-2)\Psi_{v}(F(\underline{x})),$
(2.10)

while (2.8) implies that, for each $v\in V_{K}$ of good reduction for $\eta$
and for each $\underline{x}\in{\mathcal{S}}({\mathcal{A}})$,

$\Psi_{v}(F(\underline{x}))\geq 0.$

We apply these observations in inequality (1.20) of Theorem 1.4. For this, fix
$\alpha\geq 1$ and $\underline{x}\in{\mathcal{S}}(\mathcal{A})$, and expand

$I_{\alpha}(\underline{x})^{2}=\left(\displaystyle\sum_{v\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\text{ram}}(\underline{x})}\left(\alpha+\Psi_{v}(F(\underline{x}))(\ell-1-\Psi_{v}(F(\underline{x})))\right)\right)^{2}$

as

$\displaystyle\sum_{v_{1},v_{2}\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\text{ram}}(\underline{x})}\left(\alpha^{2}+\alpha(\ell-1)\left(\Psi_{v_{1}}(F(\underline{x}))+\Psi_{v_{2}}(F(\underline{x}))\right)\right.\\\
\left.-\alpha\left(\Psi_{v_{1}}(F(\underline{x}))^{2}+\Psi_{v_{2}}(F(\underline{x}))^{2}\right)+(\ell-1)^{2}\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x}))\right.\\\
\left.-(\ell-1)\left(\Psi_{v_{1}}(F(\underline{x}))^{2}\Psi_{v_{2}}(F(\underline{x}))+\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x}))^{2}\right)+\Psi_{v_{1}}(F(\underline{x}))^{2}\Psi_{v_{2}}(F(\underline{x}))^{2}\right).$

Applying (2.10), the above sum simplifies precisely to

$\displaystyle\sum_{v_{1},v_{2}\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\text{ram}}(\underline{x})}\left((\alpha-(\ell-1))^{2}+(\alpha-(\ell-1))\left(\Psi_{v_{1}}(F(\underline{x}))+\Psi_{v_{2}}(F(\underline{x}))\right)+\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x}))\right),$

which reveals that the optimal choice of $\alpha$ to minimize
$I_{\alpha}(\underline{x})^{2}$ is

$\alpha:=\ell-1.$

Using the above choice of $\alpha$ in (1.20) of Theorem 1.4, we obtain that

$\displaystyle\left|{\mathcal{S}}(\mathcal{A})\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}\;I_{\ell-1}(\underline{x})^{2}+\frac{2}{|{\mathcal{P}}|}\;\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\text{ram}}(\underline{x})|$
(2.11) $\displaystyle=$
$\displaystyle\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}\;\displaystyle\sum_{v_{1},v_{2}\in{\mathcal{P}}\backslash
V_{\mathcal{P}}^{\text{ram}}(\underline{x})}\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x}))+\frac{2}{|{\mathcal{P}}|}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\text{ram}}(\underline{x})|.$

We leave the ramified sieve term as is, and treat the first term on the right-
hand side as follows. By interchanging summations, we write this term as

$\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{v_{1},v_{2}\in{\mathcal{P}}}\;\displaystyle\sum_{\underline{x}\in{\mathcal{A}}\backslash\left(\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{1})\cup\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{2})\right)}\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x})).$

Distinguishing between $v_{1}=v_{2}$ and $v_{1}\neq v_{2}$, we obtain that the
above expression equals

$\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{v_{1}\in{\mathcal{P}}}\;\displaystyle\sum_{\underline{x}\in{\mathcal{A}}\backslash\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{1})}\Psi_{v_{1}}(F(\underline{x}))^{2}+\frac{1}{\left|{\mathcal{P}}\right|^{2}}\displaystyle\sum_{\begin{subarray}{c}v_{1},v_{2}\in{\mathcal{P}}\\\
v_{1}\neq
v_{2}\end{subarray}}\;\displaystyle\sum_{\underline{x}\in{\mathcal{A}}\backslash\left(\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{1})\cup\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{2})\right)}\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x})).$
(2.12)

By (2.6), for each summand in the inner sum of the first double sum above we
have

$\Psi_{v_{1}}(F(\underline{x}))^{2}\leq(\ell-1)^{2},$

so (2.12) is bounded from above by

$(\ell-1)^{2}\frac{\left|{\mathcal{A}}\right|}{\left|{\mathcal{P}}\right|}+\max_{\begin{subarray}{c}v_{1},v_{2}\in{\mathcal{P}}\\\
v_{1}\neq
v_{2}\end{subarray}}\left|\displaystyle\sum_{\underline{x}\in{\mathcal{A}}\backslash\left(\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{1})\cup\mathbb{X}_{\mathcal{O}}^{\text{ram}}(v_{2})\right)}\Psi_{v_{1}}(F(\underline{x}))\Psi_{v_{2}}(F(\underline{x}))\right|.$
(2.13)

Inserting this in (2.11) then proves the theorem.

###  2.4. Comparison to square and polynomial sieves

It is informative to compare Theorems 1.4 and 1.5 to their antecedents in a
wide range of settings over $\mathbb{Q}$. These include the square sieve of
Heath-Brown [HB84], which counts perfect square values of a polynomial; the
power sieve in [Mun09], [Bra15], which counts perfect $r$-th power values of a
polynomial; and stronger versions of the square or power sieve, combined with
the $q$-analogue of Van der Corput’s method, in [Pie06], [HBP12]. Moreover,
our work is motivated by the polynomial sieve of Browning [Bro15], also
developed in a different direction involving trace functions, by Bonolis
[Bon21].

All of these works develop a sieve method to tackle a problem roughly of the
following form (over $\mathbb{Q}$): given an appropriate polynomial
$f(z;\underline{x})$ and a set of interest $\mathcal{A}$, how many
$\underline{x}\in\mathcal{A}$ of bounded height have $f(z;\underline{x})$
solvable? This is clearly related to both our aims and our methods, now in the
setting of function fields.

One of the most obvious differences is that we do not include a weight to
count multiplicities of values. To understand the two main roles of the
multiplicity-counting weight in the earlier settings, suppose one fixes a set
$\mathcal{A}$ and aims to bound from above the quantity

$S(\mathcal{A}):=\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{A}\\\
f(z;\underline{x})\;\text{soluble}\end{subarray}}w(\underline{x})$ (2.14)

for a fixed integer-coefficient polynomial
$f(z;\underline{x})=p_{0}(\underline{x})z^{d}+\cdots+p_{d}(\underline{x})$ of
interest, with $\underline{x}=(x_{1},\ldots,x_{n})$. In the most
straightforward comparison to our setting, Browning’s polynomial sieve then
provides an upper bound for $S(\mathcal{A})$ in the form of an inequality like
(2.4), but with $S_{i,j}(v_{1},v_{2})$ containing the coefficient
$w(\underline{x})$ within the sum expressed in (2.5).

In the classical setting of the square sieve, in order to bound from above the
number of perfect square values of a fixed polynomial, say
$F(y_{1},\ldots,y_{k})\in\mathbb{Z}[Y_{1},\ldots,Y_{k}]$ with
$\underline{y}=(y_{1},\ldots,y_{k})$ lying in a certain region $\mathcal{R}$,
we would set $n=1$ and $f(z;x)=z^{2}-x$ and choose
$w(x)=\\#\\{\underline{y}\in\mathcal{R}:F(y_{1},\ldots,y_{k})=x\\}$. Then we
would apply the sieve inequality, including the weight $w$, to bound (2.14)
from above. Similarly for the $r$-power sieve and its variants in [Mun09],
[HBP12], [Bra15], to count perfect $r$-power values of some polynomial
$F(y_{1},\ldots,y_{k})$ of interest, we would set $f(z;x)=z^{r}-x$ and define
$w(x)$ as above. With the more recent polynomial sieve now available in
[Bro15] (and furthermore in our present geometric treatment), this type of
multiplicity-counting weight is no longer as relevant, since one could instead
nominally set $w(x)=1$ and instead choose
$p_{2}(\underline{y})=F(\underline{y})$ for the square sieve or more generally
$p_{r}(\underline{y})=F(\underline{y})$ for the $r$-power sieve. Indeed, in
the application of [Bro15, p. 11], the weight is defined simply to be the
indicator function of a finite set. (Bonolis [Bon21, p. 27] defines a weight
that counts such solutions in a “smoothed” sense.)

This brings us to the second use of the weight: it is used to eliminate what
we have here called the ramified sieve term, that is,

$\frac{2}{|{\mathcal{P}}|}\;\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\text{ram}}(\underline{x})|.$

Previous sieve lemmas in number field settings assume that $w(\underline{x})$
vanishes for $\underline{x}$ sufficiently large, e.g., that
$w(\underline{x})=0$ if
$|\underline{x}|\geq\operatorname{exp}(|\mathcal{P}|)$. In the classical
settings, in the derivation of the sieve inequality, at step (2.1) the
expression $|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|$ counted the
number of primes in the sieving set $\mathcal{P}$ that divided a certain
(nonzero) polynomial expression $H$ in $\underline{x}$, so that for some
degree $D$,

$|V_{\mathcal{P}}^{\mathrm{ram}}(H(\underline{x}))|\leq\omega(H(\underline{x}))=\operatorname{o}(|\underline{x}|^{D})=\operatorname{o}(|\mathcal{P}|),$

under the relevant hypothesis assumed for the support of $w$. (Here, as is
standard, $\omega$ counts the number of distinct prime factors, with
$\omega(m)\ll\operatorname{log}m/\operatorname{log}\operatorname{log}(3m)$ for
any integer $m\geq 1$.) In such a setting, we would conclude in place of (2.3)
that

$\Sigma\geq|\mathcal{P}|^{2}\cdot|\mathcal{S}(\mathcal{A})|+|\mathcal{S}(\mathcal{A})|\cdot\operatorname{o}(|\mathcal{P}|^{2}),$

so that the contribution of the ramified sieve term is dominated by the
unramified sieve term, and hence omitted from the final sieve inequality (see
e.g. [Bro15, Thm. 1.1]). In our present treatment, we prefer not to include
any weight and hence we explicitly record the ramified sieve term, which must
be bounded later.

Finally, in comparison to the polynomial sieve, we have emphasized not the
size of the fiber
$\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|$ but the
quantity $\left|\eta_{v}^{-1}(\underline{x}\,(\operatorname{mod}v))\right|-1$
as in (2.9). As we have seen in the argument of §2.3, at least in the case of
cyclic morphisms, this normalization is more informative of the correct choice
of $\alpha$, rather than the expansion in terms of the fibers, which leads to
the hard-to-interpret coefficients in (2.4).

##  3\. Preliminaries on Dwork-regular polynomials and Deligne polynomials

In this section, we gather general geometric preliminaries needed to prove
Theorem 1.2 via the sieve inequality in Theorem 1.5. To be precise, the
hypothesis of Theorem 1.2 that the relevant polynomial is Dwork-regular will
be applied in the Weil-Deligne bounds that we derive in §8. For this later
purpose, we gather herein general results related to the Dwork-regular
hypothesis.

Let $L$ be any field, with a fixed algebraic closure $\bar{L}$, and let $H\in
L[X_{0},\ldots,X_{n}]$ be a homogeneous polynomial which is Dwork-regular over
$L$ with respect to the coordinates $X_{0},\ldots,X_{n}$. By definition, the
Dwork-regularity of $H$ implies that the system of equations

$H=0,\;\;\frac{dH}{dX_{i}}=0\qquad\forall 0\leq i\leq n$

has no solution in $\mathbb{P}^{n}_{\overline{L}}$. As such, the projective
hypersurface defined by $H(X_{0},\ldots,X_{n})=0$ is smooth over $\bar{L}$.

Following Katz [Kat09, Lemma 3.1] (see also the original proof [Dwo62, pp.
67–68]), we note that, when working over infinite fields, a Dwork-regular
hypothesis is not particularly restrictive:

######  Lemma 3.1.

[Kat09, Lemma 3.1]

Let $L$ be an infinite field, $n\geq 0$ an integer, and $H\in
L[X_{0},\dots,X_{n}]$ a nonzero homogeneous polynomial such that

$W:H(X_{0},\dots,X_{n})=0$

defines a smooth hypersurface in $\mathbb{P}^{n}_{L}$. Then there exists a
change of variables

$X_{i}=\displaystyle\sum_{0\leq j\leq n}a_{i,j}Y_{j}\ \text{for some}\
(a_{i,j})_{i,j}\in\operatorname{GL}_{n+1}(L),$

such that the polynomial $H$ is Dwork-regular over $L$ with respect to
$Y_{0},\dots,Y_{n}$.

###### Proof.

We follow Katz’s proof of [Kat09, Lemma 3.1], noting that while the result in
[Kat09] states that $L$ is algebraically closed, it suffices that $L$ be
infinite.

If $n=0$, then $H$ is Dwork-regular for trivial reasons. Assume now that
$n\geq 1$ and remark that, since $L$ is infinite, for any finite sequence
$(Z_{j})_{j}$ of smooth connected closed subschemes of $\mathbb{P}^{n}_{L}$,
there exists an $L$-rational hyperplane $\mathcal{H}\subset\mathbb{P}^{n}_{L}$
which is transverse to each $Z_{j}$ (that is, $\mathcal{H}\cap Z_{j}$ is
smooth and of codimension $1$ in $Z_{j}$ if $\operatorname{dim}Z_{j}\geq 1$;
$\mathcal{H}\cap Z_{j}=\emptyset$ if $\operatorname{dim}Z_{j}=0$; and there is
no required condition if $Z_{j}=\emptyset$). We apply the remark to the
sequence $(Z_{0,j})_{1\leq j\leq 2}$ consisting of $\mathbb{P}^{n}_{L}$ and
$W$, and obtain a hyperplane $\mathcal{H}_{0}\subset\mathbb{P}^{n}_{L}$ which
is transverse to each $Z_{0,j}$. Next, we apply the remark to the sequence
$(Z_{1,j})_{1\leq j\leq 4}$ consisting of $Z_{0,j}$ and
$Z_{0,j}\cap{\mathcal{H}}_{0}$ for all $j$, that is, consisting of
$\mathbb{P}^{n},W,\mathbb{P}^{n}\cap\mathcal{H}_{0}$, and
$W\cap\mathcal{H}_{0}$, and obtain a hyperplane
$\mathcal{H}_{1}\subset\mathbb{P}^{n}_{L}$ which is transverse to each
$Z_{1,j}$. Continuing in this way, for any integer $2\leq i\leq n$ we apply
the remark to the sequence $(Z_{i,j})_{j}$ consisting of $Z_{i-1,j}$ and
$Z_{i-1,j}\cap{\mathcal{H}}_{i-1}$ for all $j$, and obtain a hyperplane
$\mathcal{H}_{i}\subset\mathbb{P}^{n}_{L}$ which is transverse to each
$Z_{i,j}$. Finally, for each $0\leq i\leq n$, we take $Y_{i}$ to be the linear
form whose vanishing defines $\mathcal{H}_{i}$, and observe that $Y_{0},\ldots
Y_{n}$ give rise to the desired change of variables. ∎

In relation to Dwork-regular polynomials over finite fields and, in
particular, to the Weil-Deligne bounds that will be discussed in §8, we recall
the notion of a Deligne polynomial:

######  Definition 3.2.

Let $k$ be a finite field and let $d,r\geq 1$ be integers. Let $f\in
k[X_{1},\ldots,X_{r}]$ be a polynomial of degree $d$, which we write as

$f=f_{d}+f_{d-1}+\cdots+f_{0}$

for uniquely determined homogeneous polynomials $f_{i}\in
k[X_{1},\ldots,X_{r}]$ with $\operatorname{deg}_{\underline{X}}(f_{i})=i$. We
call $f$ a _Deligne polynomial over $k$_ if:

  1. (i)

$\operatorname{char}k\nmid d$;

  2. (ii)

the equation $f_{d}=0$ defines a smooth, degree $d$ hypersurface in
$\mathbb{P}_{k}^{r-1}$.

######  Remark 3.3.

Let $k$ be a finite field and and let $n\geq 0$ be an integer. Let $H\in
k[X_{0},\ldots,X_{n}]$ be a homogenous polynomial such that
$\operatorname{char}k\nmid\operatorname{deg}_{\underline{X}}(H)$. Given a
nonempty subset $S$ of the set of indices $\\{0,\ldots,n\\}$, let $H_{S}$
denote the homogeneous polynomial obtained by setting $X_{i}=0$ for each
$i\not\in S$. As Katz observes in [Kat09, §3], $H$ is Dwork-regular over $k$
in the coordinates $X_{0},\ldots,X_{n}$ if and only if, for every nonempty
subset $S\subset\\{0,\ldots,n\\}$, $H_{S}$ is a Deligne polynomial over $k$
(in the variables $X_{s}$ for all indices $s\in S$).

We record the following result that we will use in §8.

######  Lemma 3.4.

Let $k$ be a finite field, $n\geq 2$ an integer, and $H\in
k[X_{0},\ldots,X_{n}]$ a Dwork-regular polynomial over $k$ with respect to the
coordinates $X_{0},\ldots,X_{n}$ and such that
$\operatorname{char}k\nmid\operatorname{deg}_{\underline{X}}(H)$. Let
$S\subset\\{0,\ldots,n\\}$ be such that $|S|\geq 2$ and set $f:=H_{S}$, in the
notation of Remark 3.3. For an arbitrary index $j\in S$, consider

$g:=\left.f\right|_{X_{j}=1}$

as a polynomial over $k$ in the variables $(X_{i})_{i\in S\setminus\\{j\\}}$.
Then
$\operatorname{deg}_{\underline{X}}(g)=\operatorname{deg}_{\underline{X}}(H)$
and:

  1. (i)

$g$ is a Deligne polynomial over $k$;

  2. (ii)

the equation $g=0$ defines a smooth hypersurface in $\mathbb{A}_{k}^{|S|-1}$.

###### Proof.

Without loss of generality, we may relabel the variables conveniently as

$H(Y,X_{1},\ldots,X_{r},Z_{1},\ldots,Z_{t})\in
k[Y,X_{1},\ldots,X_{r},Z_{1},\ldots,Z_{t}]$

for some integers $r\geq 1$ and $t\geq 0$ (there may be no $Z_{i}$ variables),
and we may assume that

$f(Y,X_{1},\ldots,X_{r})=H(Y,X_{1},\ldots,X_{r},0,\ldots,0),\qquad
g(X_{1},\ldots,X_{r})=f(1,X_{1},\ldots,X_{r}).$

Correspondingly, $|S|=r+1$.

We denote by $m$ the degree of $H$ in
$Y,X_{1},\ldots,X_{r},Z_{1},\ldots,Z_{t}$, and note that, since $H$ is
homogeneous, the polynomial $f$ is also homogeneous and of degree $m$ in
$Y,X_{1},\ldots,X_{r}$. Initially we see that the polynomial $g$ is of degree
at most $m$ in $X_{1},\ldots,X_{r}$, and not necessarily homogeneous.

We write

$g=g_{m}+\cdots+g_{0},$

where each $g_{i}\in k[X_{1},\ldots,X_{r}]$ is homogeneous of degree $i$ (and
uniquely defined by $g$), and observe that

$f(Y,X_{1},\ldots,X_{r})=g_{m}(X_{1},\ldots,X_{r})+Yg_{m-1}(X_{1},\ldots,X_{r})+\cdots+Y^{m}g_{0}(X_{1},\ldots,X_{r}),$
(3.1)

$g_{m}(X_{1},\ldots,X_{r})=\left.f(Y,X_{1},\ldots,X_{r})\right|_{Y=0}.$ (3.2)

We now observe that $g_{m}\neq 0$, so that $g$ has degree $m$. For otherwise,
$Y$ is a factor of $H$, but this cannot occur (by Lemma 11.1) since $H=0$ is a
smooth hypersurface in $\mathbb{P}_{k}^{n}$, under the Dwork-regularity
hypothesis.

In what follows, we check that $g$ is a Deligne polynomial over $k$. By
hypothesis, we have
$\mathrm{char}\,k\nmid\operatorname{deg}_{\underline{X}}(g)$ since
$\operatorname{deg}_{\underline{X}}(g)=m$. To check that $g=0$ defines a
smooth, degree $m$ hypersurface in $\mathbb{P}_{k}^{r-1}$, suppose, for
contradiction, that $[x]=[x_{1}:\ldots:x_{r}]\in\mathbb{P}_{k}^{r-1}$ is a
singular point for $g_{m}(X_{1},\ldots,X_{r})=0$, that is,

$\left.\frac{d}{dX_{i}}g_{m}\right|_{[x]}=0\qquad\forall 1\leq i\leq r.$ (3.3)

and hence also

$g_{m}([x])=0.$ (3.4)

Using (3.2) and observing that $[0:x_{1}:\ldots:x_{r}]$ is well-defined in
$\mathbb{P}_{k}^{r}$, it follows from (3.4) that

$f([0:x_{1}:\ldots:x_{r}])=0.$ (3.5)

Moreover, recalling (3.1), it follows from (3.3) that

$\left.X_{i}\frac{d}{dX_{i}}f\right|_{[0:x_{1}:\ldots:x_{r}]}=0\qquad\forall
1\leq i\leq r.$ (3.6)

Certainly, we also have

$\left.Y\frac{d}{dY}f\right|_{[0:x_{1}:\ldots:x_{r}]}=0.$ (3.7)

Now recall that $f$ is obtained from the polynomial $H$ by setting
$Z_{1}=\ldots=Z_{t}=0$. In particular,
$[u]:=[0:x_{1}:\ldots:x_{r}:0:\ldots:0]$ is well-defined in
$\mathbb{P}_{k}^{r+t}$. It follows from (3.5) and (3.6), as well as trivial
observations analogous to (3.7) in the variables $Z_{1},\ldots,Z_{t}$, that

$\begin{cases}\left.H(Y,X_{1},\ldots,X_{r},Z_{1},\ldots,Z_{t})\right|_{[u]}=0,\\\
\left.Y\frac{d}{dY}H\right|_{[u]}=0,\\\
\left.X_{i}\frac{d}{dX_{i}}H\right|_{[u]}=0\qquad\forall 1\leq i\leq r,\\\
\left.Z_{i}\frac{d}{dZ_{i}}H\right|_{[u]}=0\qquad\forall 1\leq i\leq
t.\end{cases}$ (3.8)

But this contradicts that $H$ is Dwork-regular over $k$. We conclude that $g$
must be Deligne over $k$, confirming part (i) of the lemma.

It remains to check that $g=0$ defines a smooth hypersurface in
$\mathbb{A}_{k}^{r}.$ We proceed again by contradiction. Suppose there exists
a singular point $\underline{x}=(x_{1},\ldots,x_{r})\in\mathbb{A}_{k}^{r}$ for
$g(X_{1},\ldots,X_{r})=0$, that is,

$\displaystyle g(x_{1},\ldots,x_{r})$ $\displaystyle=0,$ (3.9)
$\displaystyle\left.\frac{d}{dX_{i}}g\right|_{\underline{x}}$
$\displaystyle=0\qquad\forall 1\leq i\leq r.$ (3.10)

Since $[1:x_{1}:\ldots:x_{r}]$ is well-defined in $\mathbb{P}_{k}^{r}$, (3.9)
shows that

$f([1:x_{1}:\ldots:x_{r}])=0,$ (3.11)

while (3.1) and (3.10) imply that

$\left.X_{i}\frac{d}{dX_{i}}f\right|_{[1:x_{1}:\ldots:x_{r}]}=0\qquad\forall
1\leq i\leq r.$ (3.12)

By Euler’s identity applied to the homogeneous polynomial $f$ of degree $m$,

$mf(Y,X_{1},\ldots,X_{r})=Y\frac{d}{dY}f(Y,X_{1},\ldots,X_{r})+\displaystyle\sum_{1\leq
i\leq r}X_{i}\frac{d}{dX_{i}}f(Y,X_{1},\ldots,X_{r}).$

Evaluating Euler’s identity at the point $[1:x_{1}:\ldots:x_{r}]$ and then
applying (3.11) and (3.12), we deduce that

$\left.Y\frac{d}{dY}f\right|_{[1:x_{1}:\ldots:x_{r}]}=0.$ (3.13)

Finally, note that $[1:x_{1}:\ldots:x_{r}:0:\ldots:0]$ is well-defined in
$\mathbb{P}_{k}^{r+t}$, and so, by (3.11), (3.12), (3.13), combined with
trivial considerations for $Z_{1},\ldots,Z_{t}$, all the relations (3.8) now
hold with $[u]:=[1:x_{1}:\ldots:x_{r}:0:\ldots:0]\in\mathbb{P}_{k}^{r+t}$,
contradicting that $H$ is Dwork-regular over $k$. Thus we must have that $g=0$
is a smooth hypersurface in $\mathbb{A}_{k}^{r}$, which completes the proof of
part (ii). ∎

##  4\. Preliminaries on function field arithmetic

Since Theorem 1.2 is a result about $\mathbb{F}_{q}(T)$, we gather in this
section standard function field notation and remarks pertinent to our
forthcoming arguments.

We recall the general setting of Section 1: $K$ is a global field of arbitrary
characteristic; ${\mathcal{O}}$ is the ring of integers in $K$; $V_{K}$ is the
set of places of $K$; the valuations $|\cdot|_{v}$ associated to the places
$v\in V_{K}$ are normalized such that the product formula (1.1) holds;
$\operatorname{ht}_{K}$ is the height function on $\mathbb{P}^{n}_{K}$,
defined in (1.2) and its companion height function on $K$, also denoted
$\operatorname{ht}_{K}$, is defined in (1.3).

We now focus on the particular global function field

$K=\mathbb{F}_{q}(T)$

and on the ring

$\mathcal{O}={\mathcal{O}}_{K}=\mathbb{F}_{q}[T],$

where $q$ is the power of an odd rational prime $p$. This setting will be kept
throughout the remainder of the paper, unless explicitly specified otherwise.

In the setting of Theorem 1.2 and Theorem 6.1, we assume that $p\nmid m$,
where $m$ is the degree of the polynomial $F$ therein. Furthermore, we assume
that $\ell\mid(q-1)$, where $\ell$ is a rational prime defined as the degree
of the cyclic cover $\eta$ therein; this divisibility assumption ensures that
$\mathbb{F}_{q}$ contains the $\ell$-th roots of unity. Finally, the
assumption $\ell\mid m$ ensures that the weighted projective space
$\mathbb{P}_{\mathbb{F}_{q}(T)}^{n+1}\left(1,\ldots,1,\frac{m}{\ell}\right)$
is well-defined.

Note that $K=\mathbb{F}_{q}(T)$ is the simplest instance of a global function
field with field of constants $\mathbb{F}_{q}$, and that $\mathcal{O}_{K}$ is
the ring of elements of $K$ which have only $\frac{1}{T}$ as a pole. In
particular, $\mathcal{O}_{K}$ is a Dedekind domain (and, actually, a Euclidean
domain) and plays the role of the ring $\mathbb{Z}$ in the analogy between the
arithmetic of $\mathbb{F}_{q}(T)$ and that of $\mathbb{Q}$.

As usual, we identify a place $v\in V_{K}$ with a generator of its associated
unique maximal ideal. For our particular $K=\mathbb{F}_{q}(T)$, we have either
that $v=\frac{1}{T}$, which we refer to as the place at infinity of $K$, or
that any $v\in V_{K}\backslash\\{\frac{1}{T}\\}$, which we refer to as a
finite place or simply as a prime of $K$, may be thought of as a monic
irreducible polynomial in $\mathbb{F}_{q}[T]$. We recall that, for a nonzero
polynomial $x\in{\mathcal{O}}_{K}=\mathbb{F}_{q}[T]$, we use
$\operatorname{deg}_{T}(x)$ to denote its degree in $T$.

To simplify the exposition, we use the symbol $\infty$ for $\frac{1}{T}$ and
the symbol $\pi$ for an arbitrary prime of $K$ (that is, a monic irreducible
in $\mathcal{O}_{K}$). We denote by $K_{\infty}$ the completion of $K$ with
respect to the topology defined by $|\cdot|_{\infty}$ and recall that

$K_{\infty}=\mathbb{F}_{q}\left(\left(\frac{1}{T}\right)\right)=\left\\{\displaystyle\sum_{j\leq
N}a_{j}T^{j}:N\in\mathbb{Z},a_{j}\in\mathbb{F}_{q}\;\forall j\leq N,a_{N}\neq
0\right\\}.$

We denote by $k_{\pi}$ the residue field $\mathbb{F}_{q}[T]/(\pi)$ of $\pi$
and by $\overline{k}_{\pi}$ a fixed algebraic closure of $k_{\pi}$, and recall
that $k_{\pi}$ is a finite field with $q^{\operatorname{deg}_{T}(\pi)}$
elements. We denote by $\operatorname{ord}_{\pi}(x)$ the power of $\pi$ that
exactly divides $x$.

The absolute values $|\cdot|_{\infty}$, $|\cdot|_{\pi}$ on $K$ are defined by

$|0|_{\infty}:=0,\quad\left|\frac{x}{y}\right|_{\infty}:=q^{\operatorname{deg}_{T}x-\operatorname{deg}_{T}y},$

and

$|0|_{\pi}:=0,\quad\left|\frac{x}{y}\right|_{\pi}:=q^{\operatorname{ord}_{\pi}(y)-\operatorname{ord}_{\pi}(x)},$

where $x,y\in\mathbb{F}_{q}[T]\backslash\\{0\\}$. With these definitions, the
associated valuations satisfy the product formula (1.1), the height function
on $\mathbb{P}_{K}^{n}$ becomes

$\operatorname{ht}_{K}([x_{0}:x_{1}:\ldots:x_{n}])=\max\\{|x_{i}|_{\infty}:0\leq
i\leq n\\}$

for any $[x_{0}:x_{1}:\ldots:x_{n}]\in\mathbb{P}_{K}^{n}$, and the height
function on $K$ becomes

$\operatorname{ht}_{K}(x)=|x|_{\infty}$

for any $x\in K^{\ast}$.

Remark that, since $K=\mathbb{F}_{q}(T)$ has class number 1, for any
$[x]=[x_{0}:x_{1}:\ldots:x_{n}]\in\mathbb{P}_{K}^{n}$ we can find
$x_{0}^{\prime},x_{1}^{\prime},\ldots,x_{n}^{\prime}\in{\mathcal{O}}_{K}$ with
$\operatorname{gcd}(x_{0}^{\prime},x_{1}^{\prime},\ldots,x_{n}^{\prime})=1$,
such that $[x]=[x_{0}^{\prime}:x_{1}^{\prime}:\ldots:x_{n}^{\prime}]$ and
$\operatorname{ht}_{K}([x])=\max\\{|x_{i}^{\prime}|_{\infty}:0\leq i\leq
n\\}$. Reasoning similarly, we can extend the height $\operatorname{ht}_{K}$
to $\mathbb{A}_{K}^{n+1}$ by setting
$\operatorname{ht}_{K}((x_{0},x_{1},\ldots,x_{n})):=\max\\{\operatorname{ht}_{K}(x_{i}):0\leq
i\leq n\\}$ for any
$\underline{x}:=(x_{0},x_{1},\ldots,x_{n})\in\mathbb{A}_{K}^{n+1}$, which is
consistent with (1.16).

In order to employ Fourier analysis on $K$, we extend $|\cdot|_{\infty}$ to
$K_{\infty}^{n+1}$ by

$|(x_{0},x_{1},\ldots,x_{n})|_{\infty}:=\max\left\\{|x_{i}|_{\infty}:0\leq
i\leq n\right\\}$

and consider the additive character

$\psi_{\infty}:K_{\infty}\longrightarrow\mathbb{C}^{\ast},$
$\psi_{\infty}\left(\displaystyle\sum_{j\leq
N}a_{j}T^{j}\right):=\operatorname{exp}\left(\frac{2\pi
i\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}(a_{-1})}{p}\right),$ (4.1)

where $\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}$ is the trace map.

##  5\. Preliminaries on duals and reductions

Our proof of Theorem 1.2 will be an application of the sieve inequality in
Theorem 1.5 for $K=\mathbb{F}_{q}(T)$. To choose a sieving set ${\mathcal{P}}$
and to state the appropriate Weil-Deligne bounds needed to estimate the
resulting unramified sieve term, we require certain standard facts about duals
and reductions. An appendix by Joseph Rabinoff provides a self-contained
reference for all the facts we require; we now summarize the consequences for
our specific application.

######  Proposition 5.1.

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, and
$H\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$ a homogeneous polynomial of degree
$m:=deg_{\underline{X}}(H)$ with $m\geq 2$. Denote by $W$ the projective
hypersurface defined in $\mathbb{P}_{\mathbb{F}_{q}(T)}^{n}$ by

$W:\quad H(X_{0},\ldots,X_{n})=0$

and assume that $W$ is smooth over $\mathbb{F}_{q}(T)$. Then the following
properties hold.

  1. (i)

$W$ and its dual $W^{\ast}$ are geometrically integral, with
$\operatorname{dim}W=\operatorname{dim}W^{\ast}=n-1$.

  2. (ii)

There exists an absolutely irreducible, homogeneous polynomial
$H^{\ast}\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}],$ which depends only on $H$,
such that $W^{\ast}$ is defined in $\mathbb{P}_{\mathbb{F}_{q}(T)}^{n}$ by

$W^{\ast}:\quad H^{\ast}(X_{0},\ldots,X_{n})=0.$

  3. (iii)

There exists a finite set of finite places
$\mathcal{P}\subseteq\mathbb{F}_{q}[T],$ which depends only on $H$ and
$H^{*}$, such that:

    1. (iii.1)

for all finite places $\pi\not\in\mathcal{P}$,
$\operatorname{deg}_{\underline{X}}(H(\operatorname{mod}\pi))=\operatorname{deg}_{\underline{X}}(H)$
and the projective variety $W_{\pi}$ defined by the equation

$W_{\pi}:\quad H(X_{0},\ldots,X_{n})\equiv 0\,(\operatorname{mod}\pi)$

is a smooth and geometrically integral hypersurface;

    2. (iii.2)

for all finite places $\pi\not\in\mathcal{P}$, the dual variety
$(W_{\pi})^{*}$ is geometrically integral and is defined by the equation

$(W_{\pi})^{\ast}:\quad H^{\ast}(X_{0},\ldots,X_{n})\equiv
0\,(\operatorname{mod}\pi);$

in particular, $(W_{\pi})^{*}=(W^{*})_{\pi}$ and the notation $W_{\pi}^{*}$ is
well-defined.

    3. (iii.3)

for all finite places $\pi\not\in\mathcal{P}$,
$\operatorname{dim}W_{\pi}=\operatorname{dim}W^{\ast}_{\pi}=n-1.$

###### Proof.

Assertion (i) is the statement of Proposition 11.2(1). As observed in the
appendix, for any field $k$, a hypersurface in $\mathbb{P}_{k}^{n}$ is the
zero set of a nonzero homogeneous polynomial in $k[X_{0},\ldots,X_{n}]$. Thus
implies that $W^{*}$ is defined by some
$H^{*}\in\mathbb{F}_{q}(T)[X_{0},\ldots,X_{n}]$; after clearing denominators,
we may assume $H^{*}\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$. Absolute
irreducibility of $H^{*}$ is equivalent to geometric irreducibility of
$W^{*}$, so this proves (ii). Let $\mathcal{P}$ be the finite set $S$ in
Proposition 11.5. Then (iii.1) and (iii.2) amount to statements (2) and (3) of
Proposition 11.5, using Lemma 11.1 and Proposition 11.2(1) for geometric
integrality. Assertion (iii.3) is a consequence of (iii.1) and (iii.2). ∎

We also require that for all but finitely many finite places, the reduction of
a Dwork-regular polynomial remains Dwork-regular. This can be proved
analogously to the method given in the appendix for Proposition 11.2(2), but
we also prove it in an elementary way here.

######  Lemma 5.2.

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, and
$H\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$ a homogeneous polynomial of degree
$m:=\operatorname{deg}_{\underline{X}}(H)$ with $m\geq 2$. Assume that $H$ is
Dwork-regular over $\mathbb{F}_{q}(T)$ with respect to the variables
$X_{0},\ldots,X_{n}$. Then there exists a finite set of finite places
$\mathcal{P}\subseteq\mathbb{F}_{q}[T]$, which depends only on $H$, such that
for all finite places $\pi\not\in\mathcal{P}$,
$\operatorname{deg}_{\underline{X}}(H(\operatorname{mod}\pi))=m$ and
$H(\operatorname{mod}\pi)$ is Dwork-regular over the residue field $k_{\pi}$
with respect to the variables $X_{0},\ldots,X_{n}$.

###### Proof of Lemma 5.2.

We use the notation $K=\mathbb{F}_{q}(T)$,
${\mathcal{O}}_{K}=\mathbb{F}_{q}[T]$ introduced in §4. For each
$j=0,\ldots,n$ we consider the polynomial $\left.H\right|_{X_{j}=1}$ and
obtain a set $\mathcal{P}_{j}\subseteq{\mathcal{O}}_{K}$ of finite places we
will exclude. For notational convenience, we argue explicitly in the case
$j=0$; all other cases are analogous.

Consider the ideal $I$ of $K[X_{1},\ldots,X_{n}]$ defined by

$I:=\left(H(1,X_{1}\ldots,X_{n}),\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n}),X_{1}\frac{dH}{dX_{1}}(1,X_{1},\ldots,X_{n}),\ldots,X_{n}\frac{dH}{dX_{n}}(1,X_{1},\ldots,X_{n})\right).$

Since $H$ is Dwork-regular over $K$ with respect to $X_{0},\ldots,X_{n}$, the
generators

$H(1,X_{1},\ldots,X_{n})$, $\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})$,
$X_{1}\frac{dH}{dX_{1}}(1,X_{1},\ldots,X_{n})$,
$\ldots,X_{n}\frac{dH}{dX_{n}}(1,X_{1},\ldots,X_{n})$

have no common zero in $\overline{K}^{n}$. By Weak Nullstellensatz (see
[Lan02, Thm. 1.4, p. 379]), this implies that $1\in I$, hence there exist

$G(X_{1},\ldots,X_{n})$, $G_{0}(X_{1},\ldots,X_{n})$,
$\ldots,G_{n}(X_{1},\ldots,X_{n})\in K[X_{1},\ldots,X_{n}]$

such that

$H(1,X_{1},\ldots,X_{n})G(X_{1},\ldots,X_{n})+\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})G_{0}(X_{1},\ldots,X_{n})\\\
+\displaystyle\sum_{1\leq i\leq
n}X_{i}\frac{dH}{dX_{i}}(1,X_{1},\ldots,X_{n})G_{i}(X_{1},\ldots,X_{n})=1.$
(5.1)

Taking $\overline{G}=\overline{G}(T)\in\mathcal{O}_{K}$ such that

$\displaystyle H(1,X_{1},\ldots,X_{n})G(X_{1},\ldots,X_{n})\overline{G}(T)$
$\displaystyle\in$ $\displaystyle\mathcal{O}_{K}[X_{1},\ldots,X_{n}],$
$\displaystyle\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})G_{0}(X_{1},\ldots,X_{n})\overline{G}(T)$
$\displaystyle\in$ $\displaystyle\mathcal{O}_{K}[X_{1},\ldots,X_{n}],$
$\displaystyle
X_{i}\frac{dH}{dX_{i}}(1,X_{1},\ldots,X_{n})G_{i}(X_{1},\ldots,X_{n})\overline{G}(T)$
$\displaystyle\in$
$\displaystyle\mathcal{O}_{K}[X_{1},\ldots,X_{n}]\;\quad\forall 1\leq i\leq
n,$

equation (5.1) becomes an identity in $\mathbb{F}_{q}[T,X_{1},\ldots,X_{n}]$:

$H(1,X_{1},\ldots,X_{n})G(X_{1},\ldots,X_{n})\overline{G}(T)+\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})G_{0}(X_{1},\ldots,X_{n})\overline{G}(T)\\\
+\displaystyle\sum_{1\leq i\leq
n}X_{i}\frac{dH}{dX_{i}}(1,X_{1},\ldots,X_{n})G_{i}(X_{1},\ldots,X_{n})\overline{G}(T)=\overline{G}(T).$
(5.2)

Now define a set of exceptional finite places

$\mathcal{P}_{0}:=\left\\{\pi\in\mathcal{O}_{K}:\pi\mid\overline{G}\right\\}\displaystyle\cup\left\\{\pi\in\mathcal{O}_{K}:\pi\mid
C\;\text{for each
coefficient}\;C=C(T)\in\mathcal{O}_{K}\;\text{of}\;H(X_{0},\ldots,X_{n})\right\\}.$

Note that $\mathcal{P}_{0}$ is a finite set determined by $H$ and
$\overline{G}$. For each $\pi\not\in\mathcal{P}_{0}$, we have
$\overline{G}\,(\operatorname{mod}\pi)\in(k_{\pi})^{\ast},$ since
$\pi\nmid\overline{G}$. Using the latter in (5.2), we obtain

$\displaystyle H(1,X_{1},\ldots,X_{n})G(X_{1},\ldots,X_{n})$ $\displaystyle+$
$\displaystyle\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})G_{0}(X_{1},\ldots,X_{n})$
$\displaystyle+$ $\displaystyle\displaystyle\sum_{1\leq i\leq
n}X_{i}\frac{dH}{dX_{i}}(1,X_{1},\ldots,X_{n})G_{i}(X_{1},\ldots,X_{n})\equiv
1\,(\operatorname{mod}\pi).$

In particular, the ideal of $k_{\pi}[X_{1},\ldots,X_{n}]$ generated by

$H(1,X_{1},\ldots,X_{n})\,(\operatorname{mod}\pi),\frac{dH}{dX_{0}}(1,X_{1},\ldots,X_{n})\,(\operatorname{mod}\pi),\\\
X_{1}\frac{dH}{dX_{1}}(1,X_{1},\ldots,X_{n})\,(\operatorname{mod}\pi),\ldots,X_{n}\frac{dH}{dX_{n}}(1,X_{1},\ldots,X_{n})\,(\operatorname{mod}\pi)$
(5.3)

equals the whole ring. By Weak Nullstellensatz (again, see [Lan02, Thm. 1.4,
p. 379]), this implies that the generators (5.3) have no common zero in
$\overline{k}_{\pi}^{n}$.

After carrying out the above procedure for each $j=0,\ldots,n$, we obtain
exceptional sets $\mathcal{P}_{0},\ldots,\mathcal{P}_{n}$. Define

$\mathcal{P}:=\bigcup_{0\leq j\leq n}\mathcal{P}_{j}.$

Then, for every $\pi\not\in\mathcal{P}$,
$\operatorname{deg}_{\underline{X}}(H(\operatorname{mod}\,\pi))=m$ and for
every $0\leq j\leq n$,

$\left.H(X_{0},\ldots,X_{n})\right|_{X_{j}=1}\,(\operatorname{mod}\pi),\left.X_{0}\frac{dH}{dX_{0}}(X_{0},\ldots,X_{n})\right|_{X_{j}=1}\,(\operatorname{mod}\pi),\\\
\left.X_{1}\frac{dH}{dX_{1}}(X_{0},\ldots,X_{n})\right|_{X_{j}=1}\,(\operatorname{mod}\pi),\ldots,\left.X_{n}\frac{dH}{dX_{n}}(X_{0},\ldots,X_{n})\right|_{X_{j}=1}\,(\operatorname{mod}\pi)$
(5.4)

have no common zero in $\overline{k}_{\pi}^{n}$. It remains to prove that, for
any finite place $\pi\not\in\mathcal{P}$, $H(\operatorname{mod}\pi)$ is Dwork-
regular over the residue field $k_{\pi}$ with respect to the variables
$X_{0},\ldots,X_{n}$.

Suppose that there exists a finite place $\pi\not\in\mathcal{P}$ such that $H\
(\operatorname{mod}\pi)\in k_{\pi}[X_{0},\ldots,X_{n}]$ is not Dwork-regular
over $k_{\pi}$ with respect to $X_{0},\ldots,X_{n}$. Then there exists
$[y_{0}:\ldots:y_{n}]$ in $\mathbb{P}_{k_{\pi}}^{n}$ that is a common solution
to $H=X_{0}\frac{dH}{dX_{0}}=\cdots=X_{n}\frac{dH}{dX_{n}}=0$. Choosing $j$
such that $y_{j}\neq 0$ and evaluating each function in (5.4) at
$(y_{0}/y_{j},\ldots,y_{n}/y_{j})$ produces a common zero, a contradiction.
This completes the proof. ∎

In the remainder of the paper, we will use

$\mathcal{P}_{1}(H)$ (5.5)

to denote the finite set provided by Proposition 5.1, and

$\mathcal{P}_{2}(H)$ (5.6)

to denote the finite set provided by Lemma 5.2. Finally, we define the finite
set of exceptional primes

$\mathcal{P}_{\mathrm{exc}}(H):=\mathcal{P}_{1}(H)\cup\mathcal{P}_{2}(H).$
(5.7)

##  6\. Initiating the sieve to prove Theorem 1.2

We are now ready to start the proof of Theorem 1.2 on counting points on
cyclic covers of $\mathbb{P}_{K}^{n}$ for $K=\mathbb{F}_{q}(T)$ and an
arbitrary integer $n\geq 2$.

Recall that, in Theorem 1.5, we take $\mathbb{X}:=\mathbb{P}^{n}_{K}$ and
$\mathbb{Y}$ the projective scheme in the weighted projective space
$\mathbb{P}_{K}^{n+1}\left(1,\dots,1,\frac{m}{\ell}\right)$ defined by a model
$X_{n+1}^{\ell}=F(X_{0},\ldots,X_{n})$ with $\ell$ prime such that $\ell\mid
m$, and with $F\in{\mathcal{O}}_{K}[X_{0},\ldots,X_{n}]$ homogeneous of degree
$m\geq 1$ and having the property that the hypersurface in
$\mathbb{P}^{n}_{\overline{K}}$ defined by $F(X_{0},\ldots,X_{n})=0$ is
smooth. Moreover, recall that $\eta:\mathbb{Y}\longrightarrow\mathbb{X}$ is
the cyclic map of degree $\ell$ defined over $\mathbb{A}_{\overline{K}}^{n}$
by $\eta(x_{0},x_{1},\ldots,x_{n},x_{n+1}):=(x_{0},x_{1},\ldots,x_{n})$.

Note that each element $[y]$ of $\mathbb{Y}$ is of the form
$[y]=[x_{0}:\ldots:x_{n}:x_{n+1}]$, where $[x_{0}:\ldots
x_{n}]\in\mathbb{P}^{n}_{K}$ and where $x_{n+1}\in K$ satisfies the equation

$x_{n+1}^{\ell}=F(x_{0},\ldots,x_{n}).$

In order to bound the number of projective solutions, we will think of the
above equation as an affine model. As such, it is equally natural to restate
the result of Theorem 1.2 as:

######  Theorem 6.1.

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, $\ell\geq 2$ a
rational prime, and $F\in\mathbb{F}_{q}[T][X_{0},\ldots,X_{n}]$ a homogeneous
polynomial of degree $m\geq 2$ in $X_{0},\ldots,X_{n}$, with
$\operatorname{char}\mathbb{F}_{q}\nmid m.$ Assume the conditions:

  1. (i)

$\ell\mid\operatorname{gcd}(m,q-1)$;

  2. (ii)

$F$ is Dwork-regular over $\mathbb{F}_{q}(T)$ with respect to the coordinates
$X_{0}$, $\ldots,X_{n}$.

For every $b>0$, let $M_{n}(F;b)$ denote the cardinality of the set

$\left\\{(x_{0},\ldots,x_{n})\in\mathbb{F}_{q}[T]^{n+1}:\;\operatorname{deg}_{T}(x_{i})<b\;\forall
0\leq i\leq n,\exists x_{n+1}\in\mathbb{F}_{q}[T]\;\text{such
that}\;x_{n+1}^{\ell}=F(x_{0},\ldots,x_{n})\right\\}.$

Then, there exists $b(n,q,F)>0$ such that, for any $b\geq b(n,q,F)$,

$M_{n}(F;b)\ll_{\ell,m,n,q,F}b^{2}\left(q^{b}\right)^{(n+1)-\frac{n}{n+1}},$

in which the implicit constant depends on $\ell,m,n,q$ and $F$.

For precise information on the way in which the implicit constant depends on
$F$, see (10.4). When we assume that $b$ is sufficiently large, this is
relative to $n,q$ and the finite set of primes $\mathcal{P}_{\text{exc}}(F)$
defined in (6.2) below; see §10.2 for details.

###  6.1. Choosing the sieving set

We prove Theorem 6.1 (and hence Theorem 1.2) by means of the geometric sieve
derived in Theorem 1.5. To phrase our goal as a sieve problem, from here
onwards, unless otherwise stated, we keep the setting of Theorem 6.1 and the
notation of §4, and work with the sets

$\displaystyle{\mathcal{A}}$ $\displaystyle:=$
$\displaystyle\left\\{\underline{x}=(x_{0},\ldots,x_{n})\in{\mathcal{O}}_{K}^{n+1:}\;\operatorname{deg}_{T}(x_{i})<b\;\forall
0\leq i\leq n\right\\},$ $\displaystyle{\mathcal{S}}_{F}({\mathcal{A}})$
$\displaystyle:=$
$\displaystyle\left\\{\underline{x}\in{\mathcal{A}}:\;\exists
y\in{\mathcal{O}}_{K}\;\text{such that}\;y^{\ell}=F(\underline{x})\right\\}$

for a fixed arbitrary integer $b>0$. Note that the trivial upper bound for
$|\mathcal{S}_{F}(\mathcal{A})|$ is

$|\mathcal{A}|\leq q^{b(n+1)}.$ (6.1)

To improve upon this bound using Theorem 1.5, we seek a suitable sieving set
${\mathcal{P}}$ of primes $\pi$ of $K$, which we describe below.

We consider the smooth projective hypersurface

$W(F):\quad F(X_{0},\ldots,X_{n})=0,$

whose projective dual we denote by $W(F)^{\ast}$. By parts (i), (ii) of
Proposition 5.1, there exists a homogeneous polynomial
$F^{\ast}\in{\mathcal{O}_{K}}[X_{0},\ldots,X_{n}]$ , absolutely irreducible
over $K$, that defines $W(F)^{\ast}$, that is,

$W(F)^{\ast}:\quad F^{\ast}(X_{0},\ldots,X_{n})=0.$

Recall from (5.7), by part (iii) of Proposition 5.1 and by Lemma 5.2, there
exists a finite set of finite primes

${\mathcal{P}}_{\text{exc}}={\mathcal{P}}_{\text{exc}}(F):=\mathcal{P}_{1}(F)\cup\mathcal{P}_{2}(F)$
(6.2)

satisfying that proposition and also the lemma; here we use the notation
decided upon in (5.5), (5.6), and (5.7).

Upon fixing some positive integer $\Delta$, we define the sieving set of
primes as

${\mathcal{P}}:=\\{\pi\in{\mathcal{O}}_{K}:\operatorname{deg}_{T}(\pi)=\Delta,\pi\not\in{\mathcal{P}}_{\text{exc}}\\}.$
(6.3)

Later on in (10.3), we will choose $\Delta=\Delta(n,b)$ optimally in terms of
$n$ and $b$; see also the observation below (6.4).

To understand the growth of $|{\mathcal{P}}|$, let us recall the Prime
Polynomial Theorem [Ros02, Thm. 2.2, p. 14]:

$\left|\\#\\{\pi\in{\mathcal{O}}_{K}:\operatorname{deg}_{T}(\pi)=\Delta\\}-\frac{q^{\Delta}}{\Delta}\right|\leq\frac{q^{\frac{\Delta}{2}}}{\Delta}+q^{\frac{\Delta}{3}}.$
(6.4)

We infer from (6.4) that, upon taking $\Delta:=\lfloor(nb)/(n+1)\rfloor$ (as
we will do in (10.3)), we have that

$|\mathcal{P}|\geq\frac{q^{\Delta}}{\Delta}-\left(\frac{q^{\frac{\Delta}{2}}}{\Delta}+q^{\frac{\Delta}{3}}\right)-|\mathcal{P}_{\text{exc}}|\geq\frac{q^{\Delta}}{2\Delta}$
(6.5)

for all $b$ that are sufficiently large relative to $n$ and
$|\mathcal{P}_{\text{exc}}|$. We denote the least such $b$ by

$b(n,q,F)$

and refer the reader to §10.2 for specifics. While we still allow $\Delta>0$
to be arbitrary, from now on we assume the lower bound (6.5) for
$|\mathcal{P}|$.

Our task for this particular sieve problem is to bound non-trivially the
right-hand side of the resulting sieve inequality of Theorem 1.5. Recall that
there are three terms – the main term, the ramified term, and the unramified
term. In this section, we bound the first two terms quite easily, and in the
remaining sections we focus on bounding the third term.

We make the notational convention that, for
$\underline{x}:=(x_{0},\ldots,x_{n})\in\mathcal{O}_{K}^{n+1}$, we write

$\operatorname{deg}_{T}(\underline{x})<b$

to mean

$\operatorname{deg}_{T}(x_{i})<b\ \forall 0\leq i\leq n.$

###  6.2. Upper bound for the main sieve term

Assuming $\Delta$ is chosen and $b$ is sufficiently large, relative to $n$ and
$|\mathcal{P}_{\text{exc}}(F)|$, so that (6.5) holds (again, see §10.2 for the
explicit requirements on $b$), then by (6.1) and (6.5), the first term on the
right-hand side of (1.23) is bounded above by

$\frac{|\mathcal{A}|}{|\mathcal{P}|}(\ell-1)^{2}\leq 2\Delta
q^{b(n+1)-\Delta}(\ell-1)^{2}\ll_{\ell}\Delta q^{b(n+1)-\Delta}.$ (6.6)

###  6.3. Upper bound for the ramified sieve term

For the second term on the right-hand side of (1.23), note that for each prime
$\pi\in{\mathcal{O}}_{K}$ of good reduction for $\eta$, the condition that
$\eta_{\pi}$ is ramified at $\underline{x}\,(\operatorname{mod}\pi)$ means
that the extension of residue fields defined by
$\underline{x}\,(\operatorname{mod}\pi)$ is inseparable, which cannot happen
in our case since the residue fields are finite, or that $\eta_{\pi}$ has a
ramification index at $(\underline{x},y)\,(\operatorname{mod}\pi)$ larger than
1, which is equivalent to $y\,(\operatorname{mod}\pi)$ being zero. Then,

$\frac{1}{\left|{\mathcal{P}}\right|}\displaystyle\sum_{\underline{x}\in{\mathcal{A}}}|V_{\mathcal{P}}^{\mathrm{ram}}(\underline{x})|=\frac{1}{|{\mathcal{P}}|}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\end{subarray}}\\#\left\\{\pi\in{\mathcal{O}_{K}}:\operatorname{deg}_{T}(\pi)=\Delta,\pi\not\in{\mathcal{P}}_{\text{exc}},\;F(\underline{x})\equiv
0\,(\operatorname{mod}\pi)\right\\}.$ (6.7)

We observe that for any nonconstant polynomial $u\in\mathcal{O}_{K}$ we have

$\\#\left\\{\pi\in\mathcal{O}_{K}:\pi\mid
u\right\\}\leq\operatorname{deg}_{T}(u).$

Thus, after we fix $\underline{x}\in\mathcal{O}_{K}^{n+1}$ with
$\operatorname{deg}_{T}(\underline{x})<b$ and $F(\underline{x})\neq 0$,

$\displaystyle\\#\left\\{\pi\in\mathcal{O}_{K}:\operatorname{deg}_{T}(\pi)=\Delta,\pi\not\in{\mathcal{P}}_{\text{exc}},\;F(\underline{x})\equiv
0\,(\operatorname{mod}\pi)\right\\}$ $\displaystyle\leq$
$\displaystyle\\#\left\\{\pi\in\mathcal{O}_{K}:F(\underline{x})\equiv
0\,(\operatorname{mod}\pi)\right\\}$ $\displaystyle\leq$
$\displaystyle\operatorname{deg}_{T}(F)+mb,$

where, we emphasize, $m=\operatorname{deg}_{\underline{X}}(F)$ is the degree
of
$F\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]=\mathbb{F}_{q}[T,X_{0},\ldots,X_{n}]$
in $X_{0},\ldots,X_{n}$ and $\operatorname{deg}_{T}(F)$ is the degree of $F$
in $T$.

Applying the above observations to each summand with $F(\underline{x})\neq 0$
on the right-hand side of (6.7), trivially counting the number of
$\underline{x}$ in the sum, and applying (6.5) to bound $1/|\mathcal{P}|$ from
above, we deduce that this contribution to the ramified sieve term is

$\ll_{n,\operatorname{deg}_{T}(F),\operatorname{deg}_{\underline{X}}(F)}b\Delta
q^{b(n+1)-\Delta},$

for all $b\geq b(n,q,F)$.

On the other hand, the contribution to (6.7) from those $\underline{x}$ such
that $F(\underline{x})=0$ is

$\frac{|\mathcal{P}|}{|\mathcal{P}|}\\#\\{\underline{x}\in\mathcal{O}_{K}^{n+1}:\operatorname{deg}_{T}(\underline{x})<b,F(\underline{x})=0\\}\leq\operatorname{deg}_{\underline{X}}(F)q^{bn},$

by the trivial bound (see e.g. Lemma 9.3). Thus in total the ramified sieve
term is

$\ll_{n,\operatorname{deg}_{T}(F),\operatorname{deg}_{\underline{X}}(F)}b\Delta
q^{b(n+1)-\Delta}+q^{bn},$

for all $b\geq b(n,q,F)$.

###  6.4. Remaining work

So far we have proved the upper bound

$\left|{\mathcal{S}}_{F}({\mathcal{A}})\right|\ll_{n,\operatorname{deg}_{T}(F),\operatorname{deg}_{\underline{X}}(F)}b\Delta
q^{b(n+1)-\Delta}+q^{bn}\\\
+\max_{\begin{subarray}{c}\pi_{1},\pi_{2}\in{\mathcal{P}}\\\
\pi_{1}\neq\pi_{2}\end{subarray}}\left|\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\not\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}\left(\left|\eta_{\pi_{1}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{1}))\right|-1\right)\left(\left|\eta_{\pi_{2}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{2}))\right|-1\right)\right|.$
(6.8)

Recall that our goal is to prove an upper bound for
$\mathcal{S}_{F}(\mathcal{A})$ that improves on the trivial upper bound
$q^{b(n+1)}$ recorded in (6.1). The first term on the right-hand side of (6.8)
will be nontrivial as long as $\Delta>0$. Since the trivial upper bound for
the unramified sieve term, the last term on the right-hand side of (6.8), is
$\ll q^{b(n+1)}$, which is as large as the aforementioned trivial bound (6.1),
we seek any bound for the unramified sieve term that improves upon this.
Ultimately, we will choose $\Delta$ to balance our upper-estimates for the two
terms on the right-hand side of (6.8).

To tackle the unramified sieve term, we will break our treatment into two main
steps: in §7, we expand the term into a sum of complete character sums; in §9,
we apply Weil-Deligne bounds (proved in §8) to each of these sums, and then
use point-counting results (recalled in §9.2) to sum up the resulting Weil-
Deligne bounds.

##  7\. Expansion of unramified sieve term as a mixed character sum

In this section we recall the basic notions of Fourier analysis in our
function field setting and use them to expand the unramified sieve term as a
mixed character sum. For ease of reference, we first record the main outcome
of this section and then proceed with rigorous definitions and derivations.

For any element $\underline{w}\in\mathcal{O}_{K}^{n+1}$, prime
$\pi\in\mathcal{O}_{K}$, and non-principal multiplicative character
$\chi_{\pi}$ of $\mathcal{O}_{K}$, of modulus $\pi$, we define a mixed
character sum relative to a polynomial
$G\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$ by

$S_{G}(\underline{w},\chi_{\pi}):=\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\chi_{\pi}(G(\underline{a}))\;\psi_{\infty}\left(-\frac{\underline{w}\cdot\underline{a}}{\pi}\right),$
(7.1)

where the additive character $\psi_{\infty}(\cdot/\pi)$ is defined in (4.1).

######  Proposition 7.1.

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, $\ell\geq 2$ a
rational prime, and $F\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$ a homogeneous
polynomial of degree $m\geq 2$ in $X_{0},\ldots,X_{n}$, with
$\mathrm{char}K\nmid m$, where, as before, $K=\mathbb{F}_{q}(T)$. Assume the
conditions:

  1. (i)

$\ell\mid\operatorname{gcd}(m,q-1)$;

  2. (ii)

$F$ is Dwork-regular over $K$ with respect to the coordinates $X_{0}$,
$\ldots,X_{n}$.

Let $b,\Delta>0$ be integers and assume that

$b<2\Delta.$

Defining $\mathcal{P}$ as in (6.3), for all primes
$\pi_{1},\pi_{2}\in\mathcal{P}$ with $\pi_{1}\neq\pi_{2}$, the unramified
sieve term can be expanded as

$\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\not\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}\left(\left|\eta_{\pi_{1}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{1}))\right|-1\right)\left(\left|\eta_{\pi_{2}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{2}))\right|-1\right)\\\
=q^{(n+1)(b-\operatorname{deg}_{T}(\pi_{1}\pi_{2}))}\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\end{subarray}}S_{F}\left(\bar{\pi}_{2}\underline{x},\chi_{\pi_{1}}\right)S_{F}\left(\bar{\pi}_{1}\underline{x},\chi_{\pi_{2}}\right),$
(7.2)

in which, for each $\pi_{i}$, the sum is over all non-principal characters
$\chi_{\pi_{i}}$ of order $\ell$.

We note that for any polynomial $F\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$, the
trivial bound for $S_{F}(\underline{w},\chi_{\pi})$, valid for every prime
$\pi\in\mathcal{O}_{K}$ and every $\underline{w}\in\mathcal{O}_{K}^{n+1}$, is

$\left|S_{F}(\underline{w},\chi_{\pi})\right|\leq|\pi|_{\infty}^{n+1}=q^{(n+1)\operatorname{deg}_{T}(\pi)}.$
(7.3)

(Of course, this is far from sharp.) We then see that for all primes
$\pi_{1},\pi_{2}\in\mathcal{P}$ with $\pi_{1}\neq\pi_{2}$, the trivial bound
for the right-hand side of (7.2) is

$\ll\ell^{2}q^{(n+1)2\Delta},$

which is larger than the trivial bound for the left-hand side whenever $b\leq
2\Delta$, as will occur in our ultimate choice for $\Delta$. The
transformation is nevertheless worthwhile, since we have passed from an
incomplete (multiplicative) character sum on the left-hand side to a sum of
complete (mixed) character sums on the right-hand side, to which we can apply
Weil-Deligne bounds.

We will not try to average nontrivially over the characters $\chi_{\pi_{i}}$
or over $\underline{x}$; for our current scope, it will suffice to prove a
nontrivial bound for the individual sums $S_{F}$, which we will return to in
§8.

###  7.1. Expansion in terms of multiplicative characters

We will now rewrite the unramified sieve term in (6.8) in terms of characters.
Let us fix primes $\pi_{1},\pi_{2}\in\mathcal{O}_{K}$ with
$\pi_{1}\neq\pi_{2}$ (the condition $\pi_{1},\pi_{2}\in{\mathcal{P}}$ need
only be specified later). For any $\underline{x}\in\mathcal{O}_{K}^{n+1}$, we
will rewrite each of the quantities
$(\left|\eta_{\pi_{i}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{i}))\right|-1)$
as a character sum by using the following proposition, whose proof we defer to
Section 7.4.

######  Proposition 7.2.

Let $q$ be an odd rational prime power and, as before, take
$K=\mathbb{F}_{q}(T)$. Fix a rational prime $\ell$ with $\ell\mid(q-1)$. For
any prime $\pi\in\mathcal{O}_{K}$ and any $a\in\mathcal{O}_{K}$, we have

$\\#\left\\{y\,(\operatorname{mod}\pi)\in k_{\pi}:a\equiv
y^{\ell}\,(\operatorname{mod}\pi)\right\\}=\displaystyle\sum_{\chi_{\pi}}\chi_{\pi}(a),$

where $\chi_{\pi}$ runs over all multiplicative characters on
$\mathcal{O}_{K}$ of modulus $\pi$ and order dividing $\ell$, whose definition
is extended from $k_{\pi}^{\ast}$ to $\mathcal{O}_{K}$ via the rule

$\forall x\in\mathcal{O}_{K}\;\text{with}\;\pi\mid
x,\;\;\chi_{\pi}(x):=\left\\{\begin{array}[]{cl}0&\text{if $\chi_{\pi}$ is
non-principal,}\\\ 1&\text{if $\chi_{\pi}$ is principal.}\end{array}\right.$
(7.4)

By Proposition 7.2, for each of $\pi_{1},\pi_{2}$ and any
$\underline{x}\in\mathcal{O}_{K}^{n+1}$,

$\displaystyle\left|\eta_{\pi_{i}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{i}))\right|$
$\displaystyle=$ $\displaystyle\\#\left\\{y\,(\operatorname{mod}\pi_{i})\in
k_{\pi_{i}}:F(x_{0},\ldots,x_{n})\equiv
y^{\ell}\,(\operatorname{mod}\pi_{i})\right\\}$ (7.5) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\chi_{\pi_{i}}}\chi_{\pi_{i}}(F(x_{0},\ldots,x_{n})),$

where $\chi_{\pi_{i}}$ runs over all multiplicative characters on
$\mathcal{O}_{K}$ of modulus $\pi_{i}$ and order dividing $\ell$, extended to
$\mathcal{O}_{K}$ as in (7.4). From this, we can write

$\left|\eta_{\pi_{i}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{i}))\right|-1=\displaystyle\sum_{\chi_{\pi_{i}}\not=\chi_{0}}\chi_{\pi_{i}}(F(x_{0},\ldots,x_{n})),$
(7.6)

in which the sum is now over all (non-principal) characters of order $\ell$
and modulus $\pi_{i}$.

###  7.2. Fourier expansion in terms of additive characters

Next, we perform a Fourier expansion in terms of the additive character
defined in Section 4. We defer the proof of the required proposition to
Section 7.5.

######  Proposition 7.3.

Let $q$ be an odd rational prime power and, as before, take
$K=\mathbb{F}_{q}(T)$. For all primes $\pi,\pi^{\prime}\in\mathcal{O}_{K}$
with $\pi\neq\pi^{\prime}$, for all $\chi_{\pi},\chi_{\pi^{\prime}}$ non-
principal multiplicative characters of $\mathcal{O}_{K}$ of moduli $\pi$,
$\pi^{\prime}$ (respectively), for all
$G\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$, and for all integers $b$ such that

$0<b<\operatorname{deg}_{T}(\pi\pi^{\prime}),$

we have

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\end{subarray}}\chi_{\pi}(G(\underline{x}))\chi_{\pi^{\prime}}(G(\underline{x}))=q^{(n+1)(b-\operatorname{deg}_{T}(\pi\pi^{\prime}))}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi\pi^{\prime})-b\end{subarray}}S_{G}\left(\bar{\pi}^{\prime}\underline{x},\chi_{\pi}\right)\;S_{G}\left(\bar{\pi}\underline{x},\chi_{\pi^{\prime}}\right),$

where $\bar{\pi}^{\prime},\bar{\pi}^{\prime}\in\mathcal{O}_{K}$ are determined
by the congruences $\pi\bar{\pi}\equiv 1\,(\operatorname{mod}\pi^{\prime}),$
$\pi^{\prime}\bar{\pi}^{\prime}\equiv 1\,(\operatorname{mod}\pi),$ and the sum
$S_{G}(\underline{w},\chi_{\pi})$ is defined in (7.1).

###  7.3. Application to the unramified sieve term

By (7.6) and Proposition 7.3, we observe that, for any primes
$\pi_{1},\pi_{2}\in\mathcal{P}$ with $\pi_{1}\neq\pi_{2}$,

$\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\end{subarray}}\left(\left|\eta_{\pi_{1}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{1}))\right|-1\right)\left(\left|\eta_{\pi_{2}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{2}))\right|-1\right)\\\
=q^{(n+1)(b-\operatorname{deg}_{T}(\pi_{1}\pi_{2}))}\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\end{subarray}}S_{F}\left(\bar{\pi}_{2}\underline{x},\chi_{\pi_{1}}\right)S_{F}\left(\bar{\pi}_{1}\underline{x},\chi_{\pi_{2}}\right).$
(7.7)

Here we must assume that

$b<2\Delta$ (7.8)

in order to apply Proposition 7.3. We keep this assumption from here onwards.

Recall that the sum over $\underline{x}$ in (6.8) actually restricts to those
$\underline{x}$ for which $F(\underline{x})\not\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})$. We now show that we may make this
restriction without changing the left-hand side of (7.7). Indeed, by
Proposition 7.2 (both its conclusion and the convention (7.4)),

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}$
$\displaystyle\left(\left|\eta_{\pi_{1}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{1}))\right|-1\right)\left(\left|\eta_{\pi_{2}}^{-1}(\underline{x}\,(\operatorname{mod}\pi_{2}))\right|-1\right)$
$\displaystyle=\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}\left(\displaystyle\sum_{\chi_{\pi_{1}}\neq\chi_{0}}\chi_{\pi_{1}}(F(\underline{x}))\right)\left(\displaystyle\sum_{\chi_{\pi_{2}}\neq\chi_{0}}\chi_{\pi_{2}}(F(\underline{x}))\right)$
$\displaystyle=\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}\chi_{\pi_{1}}(F(\underline{x}))\chi_{\pi_{2}}(F(\underline{x}))$
$\displaystyle=\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\ F(\underline{x})\equiv
0\,(\operatorname{mod}\pi_{1}\pi_{2})\end{subarray}}\chi_{\pi_{1}}(0)\chi_{\pi_{2}}(0)=0.$

Thus, (7.7) is actually equal to the unramified sieve term. As such,
Proposition 7.1 is proved, as long as we verify Proposition 7.2 and
Proposition 7.3.

###  7.4. Proof of Proposition 7.2

As in the setting of the proposition, let $q$ be an odd rational prime power,
take $K=\mathbb{F}_{q}(T)$, let $\ell$ be a rational prime with
$\ell\mid(q-1)$, let $\pi\in\mathcal{O}_{K}$ be a prime, and let
$a\in\mathcal{O}_{K}$.

We start by recalling the definition of the $\ell$-th power residue symbol
$\left(\frac{a}{\pi}\right)_{\ell}$, as in [Ros02, pp. 23-24]. If $\pi\mid a$,
we define

$\left(\frac{a}{\pi}\right)_{\ell}:=0;$

if $\pi\nmid a$, we define

$\left(\frac{a}{\pi}\right)_{\ell}:=\alpha,$

where $\alpha$ is the unique element in $\mathbb{F}_{q}^{\ast}$ such that

$a^{\frac{q^{\operatorname{deg}_{T}(\pi)}-1}{\ell}}\equiv\alpha\,(\operatorname{mod}\pi).$

Note that the $\ell$-th power residue symbol takes values in the subgroup
$\mu_{\ell}(\mathbb{F}_{q})$ of $\ell$-th roots of unity in
$\mathbb{F}_{q}^{\ast}$, which embeds into $k_{\pi}^{\ast}.$

An important property of the $\ell$-th power residue symbol is that

$\left(\frac{a}{\pi}\right)_{\ell}=1\;\Leftrightarrow\;X^{\ell}\equiv
a\,(\operatorname{mod}\pi)\;\text{is solvable};$ (7.9)

see [Ros02, Prop. 3.1, p. 24].

Now recall that there is a non-canonical isomorphism between
$\mu_{\ell}(\mathbb{F}_{q})$ and the subgroup
$\mu_{\ell}(\mathbb{C})\leq\mathbb{C}^{\ast}$ of $\ell$-th roots of unity in
$\mathbb{C}^{\ast}$,

$\theta:\mu_{\ell}(\mathbb{F}_{q})\longrightarrow\mu_{\ell}(\mathbb{C}).$

Using the definition and property (7.9) of the $\ell$-th power residue symbol,
as well as the convention

$0^{0}:=1,$ (7.10)

we obtain that

$\displaystyle\sum_{0\leq
i\leq\ell-1}\theta\left(a^{\frac{q^{\operatorname{deg}_{T}(\pi)}-1}{\ell}}\right)^{i}=\left\\{\begin{array}[]{cl}1&\text{if
$\pi\mid a$,}\\\ \ell&\text{if $\pi\nmid a$ and $X^{\ell}\equiv
a\,(\operatorname{mod}\pi)$ is solvable,}\\\ 0&\text{if $\pi\nmid a$ and
$X^{\ell}\equiv a\,(\operatorname{mod}\pi)$ is not
solvable.}\end{array}\right.$ (7.11)

Next remark that, for each integer $0\leq i\leq\ell-1$, the map

$x\mapsto\theta\left(x^{\frac{q^{\operatorname{deg}_{T}(\pi)}-1}{\ell}}\right)^{i}$

may be viewed as a multiplicative character $\chi_{\pi,i}$ of
$\mathcal{O}_{K}$, of modulus $\pi$ and order dividing $\ell$, whose
definition is extended from $k_{\pi}^{\ast}$ to $\mathcal{O}_{K}$ by

$\forall x\in\mathcal{O}_{K}\;\text{with}\;\pi\mid
x,\;\;\chi_{\pi,i}(x):=\left\\{\begin{array}[]{cl}0&\text{if $1\leq
i\leq\ell-1$,}\\\ 1&\text{if $i=0$.}\end{array}\right.$ (7.12)

Furthermore, remark that
$\theta\left(x^{\frac{q^{\operatorname{deg}_{T}(\pi)}-1}{\ell}}\right)^{i}\in\mu_{\ell}(\mathbb{C})\simeq\mathbb{Z}/\ell\mathbb{Z},$
hence its dependence on $i$ is actually a dependence on the residue class
$i\,(\operatorname{mod}\ell)$; in particular, as we vary over $0\leq
i\leq\ell-1$, the characters $\chi_{\ell,i}$ run over all the multiplicative
characters of $\mathcal{O}_{K}$ of modulus $\pi$ and order dividing $\ell$.
Additionally, the extension (7.12) agrees with the convention (7.10).

We deduce from these remarks that (7.11) may be rewritten as

$\displaystyle\sum_{\chi_{\pi}}\chi_{\pi}(a)=\left\\{\begin{array}[]{cl}1&\text{if
$\pi\mid a$,}\\\ \ell&\text{if $\pi\nmid a$ and $X^{\ell}\equiv
a\,(\operatorname{mod}\pi)$ is solvable,}\\\ 0&\text{if $\pi\nmid a$ and
$X^{\ell}\equiv a\,(\operatorname{mod}\pi)$ is not
solvable,}\end{array}\right.$ (7.13)

where $\chi_{\pi}$ runs over all the multiplicative characters of
$\mathcal{O}_{K}$ of modulus $\pi$, order dividing $\ell$, and satisfying
(7.12). Consequently,

$\displaystyle\sum_{\chi_{\pi}}\chi_{\pi}(a)=\\#\left\\{y\,(\operatorname{mod}\pi)\in
k_{\pi}:a\equiv y^{\ell}\,(\operatorname{mod}\pi)\right\\},$

which completes the proof of Proposition 7.2.

###  7.5. Proof of Proposition 7.3

As before, let $q$ be an odd rational prime power and take
$K=\mathbb{F}_{q}(T)$. We first prove a lemma about detecting congruences with
the additive character $\psi_{\infty}$.

######  Lemma 7.4.

Let $u\in\mathcal{O}_{K}$ and $\underline{a}\in\mathcal{O}_{K}^{n+1}$. Let $b$
be an integer such that

$0<b<\operatorname{deg}_{T}(u).$

Then

$\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\
\underline{x}\equiv\underline{a}\,(\operatorname{mod}u)\end{subarray}}1=q^{(n+1)(b-\operatorname{deg}_{T}(u))}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(u)-b\end{subarray}}\psi_{\infty}\left(-\frac{\underline{x}}{u}\cdot\underline{a}\right).$

###### Proof.

We rewrite the left-hand side by using the following indicator functions:

$w_{\infty,b}:\mathcal{O}_{K}\longrightarrow\\{0,1\\},$
$w_{\infty,b}(x):=\left\\{\begin{array}[]{cl}1&\text{if
$\operatorname{deg}_{T}(x)<b$,}\\\ 0&\text{otherwise},\end{array}\right.$

and

$w_{b}:\mathcal{O}_{K}^{n+1}\longrightarrow\\{0,1\\},$
$w_{b}(\underline{x}):=\displaystyle\prod_{1\leq i\leq n}w_{\infty,b}(x_{i}).$

With these definitions, we write

$\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\
\underline{x}\equiv\underline{a}\,(\operatorname{mod}u)\end{subarray}}1=\displaystyle\sum_{\underline{x}^{\prime}\in\mathcal{O}_{K}^{n+1}}w_{b}(\underline{a}+\underline{x}^{\prime}u).$

To understand the above sum, we use Fourier analysis on $K_{\infty}^{n+1}$, as
detailed in [BV15, Section 2]. By [BV15, Lemma 2.1], we obtain

$\displaystyle\displaystyle\sum_{\underline{x}^{\prime}\in\mathcal{O}_{K}^{n+1}}w_{b}(\underline{a}+\underline{x}^{\prime}u)$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\underline{x}^{\prime\prime}\in\mathcal{O}_{K}^{n+1}}\displaystyle\int_{K_{\infty}^{n+1}}w_{b}(\underline{a}+\underline{s}u)\
\psi_{\infty}(\underline{x}^{\prime\prime}\cdot\underline{s})\ d\underline{s}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\underline{x}^{\prime\prime}\in\mathcal{O}_{K}^{n+1}}\displaystyle\int_{\left\\{\underline{s}\in
K_{\infty}^{n+1}:\
\left|\underline{a}+\underline{s}u\right|_{\infty}<q^{b}\right\\}}\psi_{\infty}(\underline{x}^{\prime\prime}\cdot\underline{s})\
d\underline{s}.$

Making the change of variables

$\underline{t}:=\underline{a}+\underline{s}\ u,$

for which

$d\underline{t}=q^{(n+1)\operatorname{deg}_{T}(u)}\ d\underline{s},$

by [BV15, Lemma 2.3] we infer that

$\displaystyle\displaystyle\int_{\left\\{\underline{s}\in K_{\infty}^{n+1}:\
\left|\underline{a}+\underline{s}u\right|_{\infty}<q^{b}\right\\}}$
$\displaystyle\psi_{\infty}(\underline{x}^{\prime\prime}\cdot\underline{s})\
d\underline{s}$ $\displaystyle=$ $\displaystyle
q^{-(n+1)\operatorname{deg}_{T}(u)}\displaystyle\int_{\left\\{\underline{t}\in
K_{\infty}^{n+1}:\
|\underline{t}|_{\infty}<q^{b}\right\\}}\psi_{\infty}\left(\underline{x}^{\prime\prime}\cdot\frac{\underline{t}-\underline{a}}{u}\right)\
d\underline{t}$ $\displaystyle=$ $\displaystyle
q^{-(n+1)\operatorname{deg}_{T}(u)}\psi_{\infty}\left(-\frac{\underline{x}^{\prime\prime}\cdot\underline{a}}{u}\right)\displaystyle\int_{\left\\{\underline{t}\in
K_{\infty}^{n+1}:\
|\underline{t}|_{\infty}<q^{b}\right\\}}\psi_{\infty}\left(\frac{\underline{x}^{\prime\prime}}{u}\cdot\underline{t}\right)\
d\underline{t}.$

By [BV15, Lemma 2.2],

$\displaystyle\int_{\left\\{\underline{t}\in K_{\infty}^{n+1}:\
|\underline{t}|_{\infty}<q^{b}\right\\}}\psi_{\infty}\left(\frac{\underline{x}^{\prime\prime}}{u}\cdot\underline{t}\right)\
d\underline{t}=\left\\{\begin{array}[]{cl}q^{(n+1)b}&\text{if
$\left|\frac{\underline{x}^{\prime\prime}}{u}\right|_{\infty}<q^{-b}$,}\\\
0&\text{otherwise}.\end{array}\right.$

Hence

$\displaystyle\displaystyle\sum_{\underline{x}^{\prime}\in\mathcal{O}_{K}^{n+1}}w_{b}(\underline{a}+\underline{x}^{\prime}u)$
$\displaystyle=$ $\displaystyle
q^{(n+1)(b-\operatorname{deg}_{T}(u))}\displaystyle\sum_{\begin{subarray}{c}\underline{x}^{\prime\prime}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x}^{\prime\prime})<\operatorname{deg}_{T}(u)-b\end{subarray}}\psi_{\infty}\left(-\frac{\underline{x}^{\prime\prime}}{u}\cdot\underline{a}\right).$

∎

Now we prove Proposition 7.3. Let $\pi,\pi^{\prime}\in\mathcal{O}_{K}$ be
primes with $\pi\neq\pi^{\prime}$, let $\chi_{\pi},\chi_{\pi^{\prime}}$ be
non-principal multiplicative characters of $\mathcal{O}_{K}$ of moduli $\pi$,
$\pi^{\prime}$ (respectively), and let
$G\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$. Fix an integer $b$ such that

$0<b<\operatorname{deg}_{T}(\pi_{1}\pi_{2}).$

We partition $\mathcal{O}_{K}^{n+1}$ according to the residue classes modulo
$\pi\pi^{\prime}$:

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\end{subarray}}\chi_{\pi}(G(\underline{x}))\chi_{\pi^{\prime}}(G(\underline{x}))$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi\pi^{\prime})\in(\mathcal{O}_{K}/(\pi\pi^{\prime}))^{n+1}}\chi_{\pi}(G(\underline{a}))\chi_{\pi^{\prime}}(G(\underline{a}))\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<b\\\
\underline{x}\equiv\underline{a}\,(\operatorname{mod}\pi\pi^{\prime})\end{subarray}}1.$

By Lemma 7.4, the above double sum equals

$q^{(n+1)(b-\operatorname{deg}_{T}(\pi\pi^{\prime}))}\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi\pi^{\prime})\in(\mathcal{O}_{K}/(\pi\pi^{\prime}))^{n+1}}\chi_{\pi}(G(\underline{a}))\chi_{\pi^{\prime}}(G(\underline{a}))\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi\pi^{\prime})-b\end{subarray}}\psi_{\infty}\left(-\frac{\underline{x}}{\pi\pi^{\prime}}\cdot\underline{a}\right).$
(7.14)

Recall that $\pi\neq\pi^{\prime}$. Then, on one hand, there exist uniquely
determined $\overline{\pi}^{\prime}\,(\operatorname{mod}\pi)\in k_{\pi}$,
$\overline{\pi}\,(\operatorname{mod}\pi^{\prime})\in k_{\pi^{\prime}}$ such
that

$\pi\overline{\pi}\equiv 1\,(\operatorname{mod}\pi^{\prime}),$
$\pi^{\prime}\overline{\pi}^{\prime}\equiv 1\,(\operatorname{mod}\pi);$

on the other hand, by the Chinese Remainder Theorem, there exist uniquely
determined elements $\underline{a}_{1}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}$, $\underline{a}_{2}\,(\operatorname{mod}\pi^{\prime})\in
k_{\pi^{\prime}}^{n+1}$ such that

$\underline{a}\equiv\underline{a}_{1}\pi^{\prime}+\underline{a}_{2}\pi\,(\operatorname{mod}\pi\pi^{\prime}).$

Consequently,

$\displaystyle\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi\pi^{\prime})\in(\mathcal{O}_{K}/(\pi\pi^{\prime}))^{n+1}}\chi_{\pi}(G(\underline{a}))\chi_{\pi^{\prime}}(G(\underline{a}))\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi\pi^{\prime})-b\end{subarray}}\psi_{\infty}\left(-\frac{\underline{x}}{\pi\pi^{\prime}}\cdot\underline{a}\right)$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\begin{subarray}{c}\underline{a}_{1}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ \underline{a}_{2}\,(\operatorname{mod}\pi^{\prime})\in
k_{\pi^{\prime}}^{n+1}\end{subarray}}\chi_{\pi}\left(G(\underline{a}_{1}\pi^{\prime})\right)\chi_{\pi^{\prime}}\left(G(\underline{a}_{2}\pi)\right)\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi\pi^{\prime})-b\end{subarray}}\psi_{\infty}\left(-\frac{\underline{x}\cdot\underline{a}_{1}}{\pi}\right)\
\psi_{\infty}\left(-\frac{\underline{x}\cdot\underline{a}_{2}}{\pi^{\prime}}\right).$

Using that $\pi\neq\pi^{\prime}$, we make the change of variables
$\underline{a_{1}}\mapsto\underline{a_{1}}\overline{\pi}^{\prime}\,(\operatorname{mod}{\pi})$
and
$\underline{a_{2}}\mapsto\underline{a_{2}}\overline{\pi}\,(\operatorname{mod}{\pi^{\prime}})$
and deduce that the above expression equals

$\displaystyle\sum_{\begin{subarray}{c}\underline{a}_{1}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ \underline{a}_{2}\,(\operatorname{mod}\pi^{\prime})\in
k_{\pi^{\prime}}^{n+1}\end{subarray}}\chi_{\pi}\left(G(\underline{a}_{1})\right)\chi_{\pi^{\prime}}\left(G(\underline{a}_{2})\right)\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi\pi^{\prime})-b\end{subarray}}\psi_{\infty}\left(-\frac{(\overline{\pi}^{\prime}\underline{x})\cdot\underline{a}_{1}}{\pi}\right)\
\psi_{\infty}\left(-\frac{(\overline{\pi}\underline{x})\cdot\underline{a}_{2}}{\pi^{\prime}}\right).$

This completes the proof of Proposition 7.3.

##  8\. The Weil-Deligne bounds

We now state and prove the Weil-Deligne estimates that will be used to bound
the unramified sieve term in the form of the expansion in Proposition 7.1.

######  Proposition 8.1.

(Weil-Deligne bounds)

Let $q$ be the power of an odd rational prime and take $K=\mathbb{F}_{q}(T).$
Let $n\geq 1$ be an integer and $H\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$ a
homogeneous polynomial of degree $m\geq 2$, which is Dwork-regular over $K$
with respect to the coordinates $X_{0},\ldots,X_{n}$. Assume that
$\operatorname{char}\mathbb{F}_{q}\nmid m.$ Denote by $W$, ${\mathcal{W}}$ the
smooth projective hypersurface, respectively the affine hypersurface, defined
by

$H(X_{0},\ldots,X_{n})=0.$

Denote by $\mathcal{P}_{\mathrm{exc}}(H)\subseteq\mathcal{O}_{K}$ the finite
set of exceptional primes introduced in Proposition 5.1 and Lemma 5.2 (as per
our notational convention (5.7)). For a prime
$\pi\notin\mathcal{P}_{\mathrm{exc}}(H)$, denote by $W_{\pi}$,
${\mathcal{W}}_{\pi}$ the smooth projective hypersurface, respectively the
affine hypersurface, defined by

$H(X_{0},\ldots,X_{n})\equiv 0\,(\operatorname{mod}\pi),$

and denote by $W_{\pi}^{\ast}$, ${\mathcal{W}}_{\pi}^{\ast}$ their duals. Fix
a rational prime $\ell$ with $\ell\mid(q-1)$. Then, for every prime
$\pi\notin\mathcal{P}_{\mathrm{exc}}(H)$, for every (non-principal)
multiplicative character $\chi_{\pi}$ of $\mathcal{O}_{K}$ of modulus $\pi$
and order $\ell$, and for every $\underline{w}\in\mathcal{O}_{K}^{n+1}$, the
mixed character sum $S_{H}(\underline{w},\chi_{\pi})$ defined in (7.1)
satisfies the bounds:

  1. (i)

provided $\underline{w}\equiv\underline{0}\,(\operatorname{mod}\pi)$,

$\left|S_{H}(\underline{w},\chi_{\pi})\right|\ll_{n,\operatorname{deg}_{\underline{X}}(H)}q^{\frac{(n+2)\operatorname{deg}_{T}(\pi)}{2}};$

  2. (ii)

provided $\underline{w}\not\equiv\underline{0}\,(\operatorname{mod}\pi)$,

$\left|S_{H}(\underline{w},\chi_{\pi})\right|\ll_{n,\operatorname{deg}_{\underline{X}}(H)}q^{\frac{(n+2)\operatorname{deg}_{T}(\pi)}{2}};$

  3. (iii)

provided $\underline{w}\not\equiv\underline{0}\,(\operatorname{mod}\pi)$ and
$\underline{w}\not\in{\mathcal{W}}_{\pi}^{\ast}$,

$\left|S_{H}(\underline{w},\chi_{\pi})\right|\ll_{n,\operatorname{deg}_{\underline{X}}(H)}q^{\frac{(n+1)\operatorname{deg}_{T}(\pi)}{2}}.$

The proof of Proposition 8.1 is based on estimates for character sums of
polynomials in several variables, pioneered by Deligne [Del74] and further
generalized by Katz [Kat02], [Kat07].

Relative to the trivial bound for $|S_{H}(\underline{w},\chi_{\pi})|$ given in
(7.3), we see that the non-trivial bounds given in cases (i) and (ii) provide
square-root cancellation in all but one of the $n+1$ variables of the sum,
while case (iii) provides square-root cancellation in all $n+1$ variables. We
think of cases (i) and (ii) as exceptional “bad” cases, and of case (iii) as
the “good” case. In our application in §9, we use that for a fixed prime
$\pi$, as the parameter $\underline{w}$ varies over $\mathcal{O}_{K}^{n+1}$,
case (iii) is generic, with cases (i) and (ii) being rare.

###  8.1. Proof of part (i) of Proposition 8.1

We consider the case
$\underline{w}\equiv\underline{0}\,(\operatorname{mod}\pi)$ and seek to bound

$\left|S_{H}\left(\underline{0},\chi_{\pi}\right)\right|=\left|\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\chi_{\pi}(H(\underline{a}))\right|.$ (8.1)

Our main tool is the following estimate for non-singular multiplicative
character sums, due to Katz:

######  Theorem 8.2.

([Kat02, Thm. 2.1 p. 335])

Let $k$ be a finite field, $r\geq 1$ an integer, and
$\chi:k^{\ast}\longrightarrow\mathbb{C}^{\ast}$ a non-principal multiplicative
character, extended to $k$ by $\chi(0):=0$. Let $f\in k[X_{1},\ldots,X_{r}]$
be a polynomial of degree $d\geq 2$. Assume that:

  1. (i)

$f$ is a Deligne polynomial;

  2. (ii)

the equation $f=0$ defines a smooth hypersurface in $\mathbb{A}^{r}_{k}$.

Then

$\left|\displaystyle\sum_{\underline{a}\in
k^{r}}\chi(f(\underline{a}))\right|\leq(d-1)|k|^{\frac{r}{2}}.$

In order to apply this theorem, we will slice the sum
$S_{H}\left(\underline{0},\chi_{\pi}\right)$ according to which coordinate
$a_{j}$ is $\not\equiv 0\,(\operatorname{mod}\pi)$. Precisely, we write

$S_{H}\left(\underline{0},\chi_{\pi}\right)=\displaystyle\sum_{\begin{subarray}{c}(a_{1},\ldots,a_{n})\,(\operatorname{mod}\pi)\in
k_{\pi}^{n}\\\ a_{0}\not\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}(H(\underline{a}))+\displaystyle\sum_{\begin{subarray}{c}(a_{1},\ldots,a_{n})\,(\operatorname{mod}\pi)\in
k_{\pi}^{n}\\\ a_{0}\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}(H(\underline{a})).$

Recall that, for any non-empty subset $S\subset\\{0,\ldots,n\\}$, $H_{S}$
denotes the homogeneous polynomial obtained by setting $X_{i}=0$ for all
$i\not\in S$. Using this notation, the second term on the right-hand side of
the above equation can be written as

$\displaystyle\sum_{(a_{1},\ldots,a_{n})\,(\operatorname{mod}\pi)\in
k_{\pi}^{n}}\chi_{\pi}(H_{\\{1,\ldots,n\\}}(a_{1},\ldots,a_{n})).$

We iterate the slicing according to whether $a_{1}\not\equiv
0\,(\operatorname{mod}\pi)$ or $a_{1}\equiv 0\,(\operatorname{mod}\pi)$, pass
to $H_{\\{2,\ldots,n\\}}$, and so on. After $n$ steps, we obtain the identity

$S_{H}\left(\underline{0},\chi_{\pi}\right)=\displaystyle\sum_{0\leq j\leq
n-1}\displaystyle\sum_{\begin{subarray}{c}(a_{j+1},\ldots,a_{n})\,(\operatorname{mod}\pi)\in
k_{\pi}^{n-j}\\\ a_{j}\not\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}(H_{\\{j,\ldots,n\\}}(a_{j},\ldots,a_{n}))+\displaystyle\sum_{a_{n}\,(\operatorname{mod}\pi)\in
k_{\pi}}\chi_{\pi}(H_{\\{n\\}}(a_{n})).$ (8.2)

We bound the last term trivially by $q^{\operatorname{deg}_{T}(\pi)}$, which
suffices for case (i). Next, for each $0\leq j\leq n-1$, we treat the inner-
sum of the double sum in a nontrivial way, as follows.

Fix $0\leq j\leq n-1$ and define

$T_{j}:=\displaystyle\sum_{\begin{subarray}{c}(a_{j+1},\ldots,a_{n})\,(\operatorname{mod}\pi)\in
k_{\pi}^{n-j}\\\ a_{j}\not\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}(H_{\\{j,\ldots,n\\}}(a_{j},\ldots,a_{n})).$

Recall from Remark 3.3 that $H$ is Dwork-regular if and only if, for every
nonempty subset $S\subset\\{0,\ldots,n\\}$, $H_{S}$ is a Deligne polynomial in
the variables $X_{s}$ for all $s\in S$. Thus, in the sum $T_{j}$, the
polynomial $H_{\\{j,\ldots,n\\}}$ is a Deligne polynomial. We now write
$T_{j}$ using the multiplicative inverses $\bar{a}_{j}a_{j}\equiv
1\,(\operatorname{mod}\pi)$ and deduce that

$\displaystyle|T_{j}|$ $\displaystyle=$
$\displaystyle\left|\displaystyle\sum_{\begin{subarray}{c}(a_{j+1},\ldots,a_{n})\,(\operatorname{mod}{\pi})\in
k_{\pi}^{n-j}\\\ a_{j}\not\equiv 0\,(\operatorname{mod}{\pi})\in
k_{\pi}\end{subarray}}\chi_{\pi}\left(a_{j}^{m}H_{\\{j,\ldots,n\\}}\left((1,\bar{a}_{j}a_{j+1},\ldots,\bar{a}_{j}a_{n}\right)\right)\right|$
$\displaystyle\leq$ $\displaystyle\displaystyle\sum_{a_{j}\not\equiv
0\,(\operatorname{mod}{\pi})}\left|\chi_{\pi}(a_{j})^{m}\right|\left|\displaystyle\sum_{\begin{subarray}{c}\underline{b}\,(\operatorname{mod}{\pi})\in
k_{\pi}^{n-j}\\\
\end{subarray}}\chi_{\pi}\left(H_{\\{j,\ldots,n\\}}(1,b_{j+1},\ldots,b_{n})\right)\right|.$

Finally, we check that the polynomial

$G_{j}(X_{j+1},\ldots,X_{n}):=H_{\\{j,\ldots,n\\}}(1,X_{j+1},\ldots,X_{n})$

satisfies all the hypotheses of Theorem 8.2, with $k=k_{\pi},$ $r=n-j$, and
$\operatorname{deg}_{\underline{X}}(G_{j})=\operatorname{deg}_{\underline{X}}(H)\geq
2$. This claim is, in fact, justified by Lemma 3.4. Thus we may apply Theorem
8.2 to conclude that

$|T_{j}|\leq(q^{\operatorname{deg}_{T}(\pi)})\cdot(m-1)q^{\frac{n\operatorname{deg}_{T}(\pi)}{2}}.$

Applying the above bound in (8.2) for each $0\leq j\leq n-1$, we deduce that

$|S_{H}(\underline{0},\chi_{\pi})|\leq
n(m-1)q^{\frac{(n+2)\operatorname{deg}_{T}(\pi)}{2}}+q^{\operatorname{deg}_{T}(\pi)},$

which completes the proof for case (i).

###  8.2. Proof of part (ii) of Proposition 8.1

We consider the case
$\underline{w}\not\equiv\underline{0}\,(\operatorname{mod}\pi)$ and seek to
bound

$\left|S_{H}\left(\underline{w},\chi_{\pi}\right)\right|=\left|\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\chi_{\pi}(H(\underline{a}))\psi_{\infty}\left(-\frac{\underline{w}\cdot\underline{a}}{\pi}\right)\right|.$

Our main tool is the following estimate for non-singular additive character
sums, due to Deligne:

######  Theorem 8.3.

([Del74, Thm. 8.4 p. 302])

Let $k$ be a finite field, $r\geq 1$ an integer, and
$\psi:(k,+)\longrightarrow(\mathbb{C}^{\ast},\cdot)$ a non-trivial additive
character. Let $g\in k[X_{1},\ldots,X_{r}]$ be a polynomial of degree $d\geq
2$. Assume that $g$ is a Deligne polynomial over $k$. Then

$\left|\displaystyle\sum_{\underline{a}\in
k^{r}}\psi(g(\underline{a}))\right|\leq(d-1)^{r}|k|^{\frac{r}{2}}.$

We apply Theorem 8.3 to the finite field $k_{\pi}$, the integer $n+1$, the
character $\psi_{\infty}$, and each of $q^{\operatorname{deg}_{T}(\pi)}-1$
instances of polynomials $g$, derived from $H$, as explained in what follows.

Recall that the Gauss sum

$\tau(\chi_{\pi}):=\displaystyle\sum_{\alpha\in
k_{\pi}}\chi_{\pi}(\alpha)\psi_{\infty}\left(\frac{\alpha}{\pi}\right)$

satisfies the Riemann Hypothesis (see, for example, [IK04, Prop 11.5 p. 275]):

$\left|\tau(\chi_{\pi})\right|=q^{\frac{\operatorname{deg}_{T}(\pi)}{2}}.$

Note that for each $\underline{a}\,(\operatorname{mod}\pi)\in k_{\pi}^{n+1}$
such that $H(\underline{a})\not\equiv 0\,(\operatorname{mod}\pi)$, we have a
bijection

$\displaystyle k_{\pi}^{\ast}$ $\displaystyle\longrightarrow$ $\displaystyle
k_{\pi}^{\ast},$ $\displaystyle\alpha$ $\displaystyle\mapsto$
$\displaystyle\alpha H(\underline{a})^{-1}.$

Then, using properties of the Gauss sum (e.g. [IR90, Prop. 8.2.2 p. 92]), we
obtain

$\displaystyle S_{H}(\underline{w},\chi_{\pi})$ $\displaystyle=$
$\displaystyle\frac{\chi_{\pi}(-1)\tau(\chi_{\pi})\tau\left(\overline{\chi_{\pi}}\right)}{q^{\operatorname{deg}_{T}(\pi)}}\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\chi_{\pi}\left(H(\underline{a})\right)\psi_{\infty}\left(-\frac{\underline{w}\cdot\underline{a}}{\pi}\right)$
(8.3) $\displaystyle=$
$\displaystyle\frac{\chi_{\pi}(-1)\tau(\chi_{\pi})\tau\left(\overline{\chi_{\pi}}\right)}{q^{\operatorname{deg}_{T}(\pi)}}\displaystyle\sum_{\begin{subarray}{c}\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ H(\underline{a})\not\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}\left(H(\underline{a})\right)\psi_{\infty}\left(-\frac{\underline{w}\cdot\underline{a}}{\pi}\right)$
$\displaystyle=$
$\displaystyle\frac{\chi_{\pi}(-1)\tau(\chi_{\pi})}{q^{\operatorname{deg}_{T}(\pi)}}\displaystyle\sum_{\alpha\in
k_{\pi}^{\ast}}\displaystyle\sum_{\begin{subarray}{c}\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ H(\underline{a})\not\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\chi_{\pi}\left(\alpha^{-1}H(\underline{a})\right)\psi_{\infty}\left(\frac{\alpha-\underline{w}\cdot\underline{a}}{\pi}\right)$
$\displaystyle=$
$\displaystyle\frac{\chi_{\pi}(-1)\tau(\chi_{\pi})}{q^{\operatorname{deg}_{T}(\pi)}}\displaystyle\sum_{\beta\in
k_{\pi}^{\ast}}\chi_{\pi}(\beta)\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\psi_{\infty}\left(\frac{\beta
H(\underline{a})-\underline{w}\cdot\underline{a}}{\pi}\right).$

In the last identity, in order to sum back in the contribution from
$H(\underline{a})\equiv 0\,(\operatorname{mod}{\pi}),$ we used that

$\displaystyle\sum_{\beta\in
k_{\pi}^{\ast}}\chi_{\pi}(\beta)\displaystyle\sum_{\begin{subarray}{c}\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ H(\underline{a})\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\psi_{\infty}\left(\frac{\beta
H(\underline{a})-\underline{w}\cdot\underline{a}}{\pi}\right)=\left(\displaystyle\sum_{\beta\in
k_{\pi}^{\ast}}\chi_{\pi}(\beta)\right)\left(\displaystyle\sum_{\begin{subarray}{c}\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}\\\ H(\underline{a})\equiv
0\,(\operatorname{mod}\pi)\end{subarray}}\psi_{\infty}\left(\frac{-\underline{w}\cdot\underline{a}}{\pi}\right)\right)=0,$

which follows from the orthogonality of the characters $\chi_{\pi}.$ By taking
absolute values in (8.3), we deduce that

$\displaystyle\left|S_{H}(\underline{w},\chi_{\pi})\right|\leq
q^{-\frac{\operatorname{deg}_{T}(\pi)}{2}}\displaystyle\sum_{\beta\in
k_{\pi}^{\ast}}\left|\displaystyle\sum_{\underline{a}\in
k_{\pi}^{n+1}}\psi_{\infty}\left(\frac{\beta
H(\underline{a})-\underline{w}\cdot\underline{a}}{\pi}\right)\right|.$ (8.4)

We estimate the inner sum above using Theorem 8.3 for each of the polynomials
over $k_{\pi}$ defined by the congruence

$g(X_{0},\ldots,X_{n})\equiv\beta
H(X_{0},\ldots,X_{n})-\underline{w}\cdot(X_{0},\ldots,X_{n})\,(\operatorname{mod}\pi)$

and for the additive character $\psi_{\infty}$. Using that $\beta\neq 0$,
$\operatorname{deg}_{\underline{X}}(H)=m\geq 2$, and
$\pi\not\in{\mathcal{P}}_{\text{exc}}(H)$, we obtain that
$\operatorname{deg}_{\underline{X}}(g)=m$. Let us write
$g=g_{0}+\cdots+g_{m}$, where $g_{0}$, $\ldots,$ $g_{m}$ are the uniquely
determined homogeneous polynomials in $k_{\pi}[X_{0},\ldots,X_{n}]$ such that
$\operatorname{deg}_{\underline{X}}(g_{i})=i$. Then $g_{m}\equiv\beta
H\,(\operatorname{mod}\pi)$. We recall that $H$ is Dwork-regular over $K$, and
since $\pi\notin\mathcal{P}_{\text{exc}}(H)$, then $H(\operatorname{mod}\pi)$
is Dwork-regular over $k_{\pi}$, and hence smooth. Since char
$\mathbb{F}_{q}\nmid\operatorname{deg}(H)$, we conclude that
$H(\operatorname{mod}\pi)$ is a Deligne polynomial over $k_{\pi}$, and hence
$g$ also is. By (8.4) and Theorem 8.3, we deduce that

$\displaystyle\left|S_{H}(\underline{w},\chi_{\pi})\right|\leq
q^{-\frac{\operatorname{deg}_{T}(\pi)}{2}}\displaystyle\sum_{\beta\in
k_{\pi}^{\ast}}(m-1)^{n+1}q^{\frac{(n+1)\operatorname{deg}_{T}(\pi)}{2}}<(m-1)^{n+1}q^{\frac{(n+2)\operatorname{deg}_{T}(\pi)}{2}},$

which completes the proof for case (ii).

###  8.3. Proof of part (iii) of Proposition 8.1

We consider the case
$\underline{w}\not\equiv\underline{0}\,(\operatorname{mod}\pi)$,
$\underline{w}\not\in{\mathcal{W}}_{\pi}^{\ast}$, and seek to bound

$\left|S_{H}\left(\underline{w},\chi_{\pi}\right)\right|=\left|\displaystyle\sum_{\underline{a}\,(\operatorname{mod}\pi)\in
k_{\pi}^{n+1}}\chi_{\pi}(H(\underline{a}))\psi_{\infty}\left(-\frac{\underline{w}\cdot\underline{a}}{\pi}\right)\right|$

in such a way that we improve upon the bound in part (ii). Our main tool is
the following estimate for non-singular mixed character sums, again due to
Katz:

######  Theorem 8.4.

([Kat07, Thm. 1.1 p. 3])

Let $k$ be a finite field, $r\geq 1$ an integer,
$\chi:(k^{\ast},\cdot)\longrightarrow(\mathbb{C}^{\ast},\cdot)$ a non-
principal multiplicative character, extended to $k$ by $\chi(0):=0$, and
$\psi:(k,+)\longrightarrow(\mathbb{C}^{\ast},\cdot)$ a non-trivial additive
character. Let $f,g\in k[X_{1},\ldots,X_{r}]$ be polynomials of degrees
$d,e\geq 1$ with leading homogeneous forms $f_{d}$ of degree $d$, $g_{e}$ of
degree $e$ (respectively). Assume that:

  1. (i)

$f$ is a Deligne polynomial;

  2. (ii)

$g$ is a Deligne polynomial;

  3. (iii)

in case $r\geq 2$, the smooth hypersurfaces in $\mathbb{P}_{k}^{r-1}$ defined
by $f_{d}=0$ and by $g_{e}=0$ are transverse, in the sense that their
intersection is smooth and of codimension $2$ in $\mathbb{P}_{k}^{r-1}$.

Then

$\left|\displaystyle\sum_{\underline{a}\in
k^{r}}\chi(f(\underline{a}))\psi(g(\underline{a}))\right|\ll_{r,d,e}|k|^{\frac{r}{2}}.$

We apply Theorem 8.4 to the finite field $k_{\pi}$, the integer $n+1$, the
characters $\psi_{\infty}$, $\chi_{\pi}$, and the polynomials over $k_{\pi}$
defined by the congruences

$\displaystyle f(X_{0},\ldots,X_{n})$ $\displaystyle\equiv$ $\displaystyle
H(X_{0},\ldots,X_{n})\,(\operatorname{mod}\pi),$ $\displaystyle
g(X_{0},\ldots,X_{n})$ $\displaystyle\equiv$ $\displaystyle-
w_{0}X_{0}-w_{1}X_{1}-\cdots-w_{n}X_{n}\,(\operatorname{mod}\pi).$

Note that $f(\operatorname{mod}\pi)$ is homogeneous of degree
$\operatorname{deg}_{\underline{X}}(H)=m\geq 2$ (because
$\pi\not\in{\mathcal{P}}_{\text{exc}}(H)$) and that $g(\operatorname{mod}\pi)$
is homogeneous of degree $1$ (because $\underline{w}\not\equiv
0\,(\operatorname{mod}\pi)$). Recalling that
$\mathrm{char}\,\mathbb{F}_{q}\nmid\operatorname{deg}_{\underline{X}}(H)$ by
hypothesis, we deduce that $f(\operatorname{mod}\pi)$ and
$g(\operatorname{mod}\pi)$ are indeed Deligne polynomials. Since
$\underline{w}\not\in{\mathcal{W}}_{\pi}^{\ast}$, they are also transverse.
Thus Theorem 8.4 applies, giving

$\left|S_{H}(\underline{w},\chi_{\pi})\right|\ll_{n,m}q^{\frac{(n+1)\operatorname{deg}_{T}(\pi)}{2}}.$

With this, we completed the verification of all Weil-Deligne bounds of
Proposition 8.1.

##  9\. Application of the Weil-Deligne bounds to obtain an upper bound for
the unramified sieve term

In this section, we apply the Weil-Deligne bounds of §8 to the unramified
sieve term (given by the right hand side of (7.2)).

######  Proposition 9.1.

Let $q$ be an odd rational prime power, $n\geq 2$ an integer, $\ell\geq 2$ a
rational prime, and $F\in\mathcal{O}_{K}[X_{0},\ldots,X_{n}]$ a homogeneous
polynomial of degree $m\geq 2$ in $X_{0},\ldots,X_{n}$, with
$\mathrm{char}K\nmid m$, where, as before, $K=\mathbb{F}_{q}(T)$. Assume the
conditions:

  1. (i)

$\ell\mid\operatorname{gcd}(m,q-1)$;

  2. (ii)

$F$ is Dwork-regular over $K$ with respect to the coordinates $X_{0}$,
$\ldots,X_{n}$.

Let $b,\Delta>0$ be integers such that

$\Delta<b<2\Delta.$ (9.1)

Defining $\mathcal{P}$ as in (6.3), we have that, for any
$\pi_{1},\pi_{2}\in\mathcal{P}$ with $\pi_{1}\neq\pi_{2}$,

$\left|\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\end{subarray}}S_{F}\left(\bar{\pi}_{2}\underline{x},\chi_{\pi_{1}}\right)S_{F}\left(\bar{\pi}_{1}\underline{x},\chi_{\pi_{2}}\right)\right|\ll_{n,\ell,\operatorname{deg}_{\underline{X}}(F^{*})}q^{(n+1)(3\Delta-b)}+q^{n(3\Delta-b)+2\Delta}.$

Consequently,

$q^{(n+1)(b-2\Delta)}\max_{\begin{subarray}{c}\pi_{1},\pi_{2}\in{\mathcal{P}}\\\
\pi_{1}\neq\pi_{2}\end{subarray}}\left|\displaystyle\sum_{\begin{subarray}{c}\chi_{\pi_{1}}\neq\chi_{0}\\\
\chi_{\pi_{2}}\neq\chi_{0}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\underline{x}\in\mathcal{O}_{K}^{n+1}\\\
\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\end{subarray}}S_{F}\left(\bar{\pi}_{2}\underline{x},\chi_{\pi_{1}}\right)S_{F}\left(\bar{\pi}_{1}\underline{x},\chi_{\pi_{2}}\right)\right|\ll_{n,\ell,\operatorname{deg}_{\underline{X}}(F^{*})}q^{n\Delta+b}.$
(9.2)

To prove Proposition 9.1, we will dominate the absolute value of the double
sum by bringing the absolute value inside both sums – the sum over
$\chi_{\pi_{1}},\chi_{\pi_{2}}$ and the sum over $\underline{x}$; then, we
will estimate the absolute value of the innermost term according to different
cases of $\underline{x}$ suggested by Proposition 8.1. A key ingredient in our
proof is a result on point-counting, which we present in a separate subsection
below, before embarking on the proof of Proposition 9.1.

######  Remark 9.2.

Proposition 9.1 is analogous to the strength of the sieve inequality obtained
by the argument in [Mun09] over $\mathbb{Q}$. In that setting and in the
notation of [Mun09, p. 340], Munshi’s argument yields a term of size
$S(p,q)\ll P^{n}B$ for terms that are “bad” for both $p$ and $q$; ultimately,
this estimate leads to the upper bound (1.14). In order to improve Munshi’s
bound (1.14) to (1.15), one approach is to average nontrivially over pairs of
primes in the sieving set. In our setting over $\mathbb{F}_{q}(T)$ and in our
notation, such an approach would involve using a variant of our geometric
sieve based on (2.12) rather than on (2.13) (e.g., see [CM06, Thm. 2.3.1 p.
21] and [CD08, Thm. 13 p. 844], for such a formulation of the square sieve
over $\mathbb{Q}$ and over $\mathbb{F}_{q}(T)$, respectively), interchanging
the summation over $\chi_{\pi_{1}},\chi_{\pi_{2}}$ with that over
$\underline{x}$, bringing the absolute value only inside the summation over
$\underline{x}$ and keeping it outside the summation over
$\chi_{\pi_{1}},\chi_{\pi_{2}}$, and then aiming to show that each
$\underline{x}$ can be “bad” only for very few primes $\pi_{1},\pi_{2}$. In
the setting over $\mathbb{Q}$, Bonolis carried out such an approach by relying
on Salberger’s work based on the determinant method [Sal07, Thm. 0.1]; see
[Bon21, Eqn. (24)]. It would be interesting to carry out similar nontrivial
averaging in the setting over $\mathbb{F}_{q}(T)$. Moreover, it would be
interesting to adapt to $\mathbb{F}_{q}(T)$ the approach of Heath-Brown and
the third author [HBP12].

###  9.1. Proof of Proposition 9.1: dissecting the inner sum into cases

Let $\pi_{1},\pi_{2}\in\mathcal{P}$ with $\pi_{1}\neq\pi_{2}$ and let
$\chi_{\pi_{1}},\chi_{\pi_{2}}\neq\chi_{0}$ be fixed. We dissect the set
$\\{\underline{x}\in\mathcal{O}_{K}^{n+1}:\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\\}$
into subsets suggested by Proposition 8.1, as follows.

For each $i\in\\{1,2\\}$, we consider the smooth projective hypersurface

$W(F)_{\pi_{i}}:\quad F(X_{0},\ldots,X_{n})\equiv
0\,(\operatorname{mod}\pi_{i}).$

By part (iii) of Proposition 5.1, the reduction modulo $\pi_{i}$ of $F^{\ast}$
remains absolutely irreducible and defines the projective dual
$W(F)_{\pi_{i}}^{\ast}$, that is,

$W(F)_{\pi}^{\ast}:\quad F^{\ast}(X_{0},\ldots,X_{n})\equiv
0\,(\operatorname{mod}\pi_{i}).$

We denote by

${\mathcal{W}}(F)_{\pi_{i}},\;{\mathcal{W}}(F)_{\pi_{i}}^{\ast}$

the _affine_ varieties defined by the homogenous polynomials
$F\,(\operatorname{mod}\pi_{i})$, $F^{\ast}\,(\operatorname{mod}\pi_{i})$
(respectively). We partition
$\\{\underline{x}\in\mathcal{O}_{K}^{n+1}:\operatorname{deg}_{T}(\underline{x})<\operatorname{deg}_{T}(\pi_{1}\pi_{2})-b\\}$
according to the following cases:

  1. (C1)

$\bar{\pi}_{1}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$,
$\bar{\pi}_{2}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$;

  2. (C2)

$\bar{\pi}_{1}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$,
$\bar{\pi}_{2}\underline{x}\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$;

  3. (C2’)

$\bar{\pi}_{1}\underline{x}\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$,
$\bar{\pi}_{2}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$;

  4. (C3)

$\bar{\pi}_{1}\underline{x}\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$,
$\bar{\pi}_{2}\underline{x}\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$.

In the above partition, the case
$\bar{\pi}_{1}\underline{x}\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$ includes the
subcase in which
$\bar{\pi}_{1}\underline{x}\equiv\underline{0}\,(\operatorname{mod}{\pi_{2}})$;
similarly, the case
$\bar{\pi}_{2}\underline{x}\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$ includes the
subcase in which
$\bar{\pi}_{2}\underline{x}\equiv\underline{0}\,(\operatorname{mod}{\pi_{1}})$.
Thus the condition
$\bar{\pi}_{1}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{2}}^{\ast}$ implies
that
$\bar{\pi}_{1}\underline{x}\not\equiv\underline{0}\,(\operatorname{mod}{\pi_{2}})$;
similarly,
$\bar{\pi}_{2}\underline{x}\not\in{\mathcal{W}}(F)_{\pi_{1}}^{\ast}$ implies
that
$\bar{\pi}_{2}\underline{x}\not\equiv\underline{0}\,(\operatorname{mod}{\pi_{1}})$.

We will treat each case separately. (The case (C2’) is analogous to the case
(C2), and thus we only explicitly describe the treatment of (C2).)
Intuitively, recalling the bounds presented in Proposition 8.1, we would think
of case (C1) as the “good-good” case, of (C2) and (C2’) as the “good-bad”
cases, and of (C3) as the “bad-bad” case, mediated by the number of times each
case can occur.

###  9.2. Counting points

We record below a simple counting lemma pertinent to our application of the
Weil-Deligne bound in the “bad” cases (C2), (C2’), and (C3).

######  Lemma 9.3.

Let $A$ be a domain, $n\geq 1$ an integer, and $G\in A[X_{0},\ldots,X_{n}]$ a
homogeneous polynomial of degree $e\geq 1$ in $X_{0},\ldots,X_{n}$. Then, for
any finite subset $S\subseteq A$, we have

$\\#\left\\{(\gamma_{0},\ldots,\gamma_{n})\in
S^{n+1}:G(\gamma_{0},\ldots,\gamma_{n})=0\right\\}\leq e|S|^{n}.$ (9.3)

This can be considered a “trivial bound”, also called the Schwartz-Zippel
bound. We recall the standard proof, which proceeds by induction on $n$ (e.g.,
see [HB02, Thm. 1] for a version of this result when $A=\mathbb{Z}$).

###### Proof.

In what follows, $L$ is the field of fractions of $A$ and $\overline{L}$ is a
fixed algebraic closure of $L$. In our argument below, it suffices that
$G(X_{0},\ldots,X_{n})\in\overline{L}[X_{0},\ldots,X_{n}]$, that is, we do not