1 Introduction
For a given smooth projective curve
$C$
defined over a finite field
$\mathbb{F}_{q}$
of genus
$g=g(C)$
, the Hasse–Weil bound says that
$$\begin{eqnarray}|\#C(\mathbb{F}_{q})-(q+1)|\leqslant 2g\sqrt{q}.\end{eqnarray}$$
Here
$C(\mathbb{F}_{q})$
denotes the set of
$\mathbb{F}_{q}$
-rational points on
$C$
and
$q+1=\#\mathbb{P}^{1}(\mathbb{F}_{q})$
. The problem we study in this paper is the distribution of
$\#C(\mathbb{F}_{q})$
in families of covers of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$
of genus
$g$
in the limit
$g\rightarrow \infty$
.
Kurlberg and Rudnick [Reference Kurlberg and Rudnick12] study the ensemble of hyperelliptic curves
$C$
of genus
$g\,\rightarrow \,\infty$
under the assumption that
$q$
is odd. They show that
$\#C(\mathbb{F}_{q})$
distributes as
$\sum _{i=1}^{q+1}Y_{i}$
, with
$Y_{1},\ldots ,Y_{q+1}$
independent and identically distributed (i.i.d.) random variables taking the values
$0,1,2$
with probabilities
$q/2(q+1)$
,
$1/(q+1)$
,
$q/2(q+1)$
, respectively. (In [Reference Kurlberg and Rudnick12], a different normalization is used; the above formulation is similar to the one appearing in [Reference Bucur, David, Feigon and Lalín4, Theorem 1.1] in terms of the trace of Frobenius.)
The result of Kurlberg and Rudnick has been subject to generalizations and extensions in many directions, see for example [Reference Bae and Jung1, Reference Bucur, David, Feigon and Lalín3, Reference Cheong, Wood and Zaman9]. One direction of generalization is to consider abelian covers of
$\mathbb{P}^{1}$
: Artin–Schreier covers [Reference Bucur, David, Feigon and Lalín6, Reference Bucur, David, Feigon, Lalín and Sinha7], biquadratic covers [Reference Lorenzo, Meleleo and Milione13] in odd characteristic, cyclic [Reference Bucur, David, Feigon, Kaplan, Lalín, Ozman and Wood2, Reference Bucur, David, Feigon and Lalín4, Reference Bucur, David, Feigon and Lalín5, Reference Meisner15, Reference Xiong17] and abelian [Reference Meisner14] covers of exponent dividing
$q-1$
. The latter assumption is used in a crucial way in those papers: it implies that
$\mathbb{F}_{q}$
contains a primitive root of unity of order equal to the exponent of the covers, which allows the application of Kummer theory.
To be more precise, for a prime power
$q$
, a non-negative integer
$g$
, and a prime
$\ell$
consider the set of curves
$$\begin{eqnarray}{\mathcal{H}}_{g,\ell }=\{\unicode[STIX]{x1D711}:C\rightarrow \mathbb{P}_{\mathbb{F}_{q}}^{1}:\operatorname{Gal}(\mathbb{F}_{q}(C)/\mathbb{F}_{q}(\mathbb{P}_{\mathbb{F}_{q}}^{1}))\cong \mathbb{Z}/\ell \mathbb{Z},g(C)=g\},\end{eqnarray}$$
of
$\ell$
-cyclic covers of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$
. Here
$\mathbb{F}_{q}(C)$
denotes the function field of
$C$
and
$\mathbb{F}_{q}(\mathbb{P}_{\mathbb{F}_{q}}^{1})=\mathbb{F}_{q}(X)$
is the field of rational functions. The state-of-the-art result [Reference Bucur, David, Feigon, Kaplan, Lalín, Ozman and Wood2, Theorem 1.3] for primes
$\ell$
not dividing
$q$
says that if
$\ell \mid q-1$
and if
$C$
is a random curve chosen uniformly from
${\mathcal{H}}_{g,\ell }$
, then
$$\begin{eqnarray}\operatorname{Prob}(\#C(\mathbb{F}_{q})=N)=\operatorname{Prob}\biggl(\mathop{\sum }_{i=1}^{q+1}Y_{i}=N\biggr)+O\biggl(\frac{1}{g}\biggr),\end{eqnarray}$$
as
$g\rightarrow \infty$
, where
$Y_{1},\ldots ,Y_{q+1}$
are i.i.d. random variables taking the values
$0,1$
and
$\ell$
with probabilities
$(\ell -1)q/\ell (q+\ell -1)$
,
$(\ell -1)/(q+\ell -1)$
, and
$q/\ell (q+\ell -1)$
, respectively.
The primes
$\ell$
to which (1.2) may be applied are bounded by
$q-1$
; in particular, for
$\mathbb{F}_{2}$
the result is empty, for
$\mathbb{F}_{3}$
and
$\mathbb{F}_{5}$
one must have
$\ell =2$
, and the main term of (1.2) coincides with that of [Reference Kurlberg and Rudnick12]. Of course, for larger fields (such as
$\mathbb{F}_{7}$
) (1.2) is more general.
Our main result treats
$q\not \equiv 0,1~\text{mod}~\ell$
. Not every genus may be obtained in this case: if
$n_{q}$
is the multiplicative order of
$q$
modulo
$\ell$
, then the genus
$g$
of an
$\ell$
-cyclic cover must satisfy the congruence
$$\begin{eqnarray}2g+2\ell -2\equiv 0~\text{mod}~(\ell -1)n_{q}\end{eqnarray}$$
(see Proposition 3.2).
Theorem 1.1. Let
$q$
be a prime power and
$\ell$
a prime such that
$q\not \equiv 0,1~\text{mod}~\ell$
. Let
$C$
be a random curve chosen uniformly from
${\mathcal{H}}_{g,\ell }$
(see (1.1)). Then, as
$g\rightarrow \infty$
satisfying the congruence relation (1.3),
$$\begin{eqnarray}\operatorname{Prob}(\#C(\mathbb{F}_{q})=N)=\operatorname{Prob}\biggl(\mathop{\sum }_{i=1}^{q+1}X_{i}=N\biggr)+O(q^{-(1-\unicode[STIX]{x1D716})g/(\ell -1)}).\end{eqnarray}$$
Here the
$X_{1},\ldots ,X_{q+1}$
are i.i.d. random variables taking the values
$0$
and
$\ell$
with probabilities
$(\ell -1)/\ell$
and
$1/\ell$
, respectively.
In (1.2) each of the
$q+1$
random variables
$Y_{i}$
models the number of rational points on
$C$
lying above the
$i$
th rational point
$x_{i}$
of
$\mathbb{P}^{1}(\mathbb{F}_{q})$
(under some arbitrary order). The number of points is
$0$
if
$x_{i}$
is inert in
$C$
(i.e., the fiber is irreducible as an
$\mathbb{F}_{q}$
-scheme),
$1$
if ramified, and
$\ell$
if totally split. The probabilities are derived from the probabilities for the point to have each of these splitting types.
The error term in (1.4) decays exponentially in
$g$
as the error term in [Reference Kurlberg and Rudnick12], while in (1.2) the error term decays linearly in
$g$
. This is another consequence of the fact that the points of
$\mathbb{P}^{1}(\mathbb{F}_{q})$
are always unramified in
$C$
when
$q\not \equiv 0,1~\text{mod}~\ell$
.
We conclude the introduction with a few words on the methods and an outline of the proof. When
$\ell \mid q-1$
, the field
$\mathbb{F}_{q}$
contains a primitive
$\ell$
th root of unity
$\unicode[STIX]{x1D701}_{\ell }$
. Hence the
$\ell$
-cyclic covers of
$\mathbb{P}^{1}$
over
$\mathbb{F}_{q}$
are birationally equivalent to a plane curve of the form
$$\begin{eqnarray}Y^{\ell }=F(X),\quad F\text{ is }\ell \text{th-power free}.\end{eqnarray}$$
This leads to a parameterization of the set or curves
${\mathcal{H}}_{g,\ell }$
. Then one derives an analytic formula for the number of rational points lying above a fixed point of
$\mathbb{P}^{1}(\mathbb{F}_{q})$
in terms of Dirichlet characters. The next step is to apply generating function techniques to derive good asymptotic formulas for each of the terms.
When
$q\not \equiv 0,1~\text{mod}~\ell$
, equation (1.5) generates a non-Galois and in particular non-abelian cover. We use the ray class group and explicit Galois descent to obtain a parametrization of the
$\ell$
-cyclic Galois covers (§2) together with a genus formula (§3). We control the number of rational points lying over the points of
$\mathbb{P}^{1}(\mathbb{F}_{q})$
(Proposition 4.1). From this point, we follow a similar path to that described above: we derive an analytic formula (Proposition 4.2) and then use generating function techniques (§5) to prove Theorem 1.1 (§6).
2 Classifying the curves
In this section we use explicit class field theory to classify the
$\ell$
-cyclic covers
$C\rightarrow \mathbb{P}_{\mathbb{F}_{q}}^{1}$
which in terms of function fields corresponds to
$\mathbb{Z}/\ell \mathbb{Z}$
-extensions of
$K=\mathbb{F}_{q}(X)$
.
2.1 Class field theory preliminaries
We begin by recalling the basic objects and results of class field theory which we use. For more extensive background on this topic we refer the reader to [Reference Cassels and Fröhlich8] or to [Reference Hess and Massierer11], for a more specific treatment in the case of function fields.
For now, we allow
$K$
to be an arbitrary global field and
$L$
a tamely ramified abelian extension of
$K$
. Denote
${\mathcal{D}}(K)$
,
${\mathcal{D}}(L)$
as the groups of divisors of
$K$
and
$L$
, respectively. Define the conorm map on the set of prime divisors in
${\mathcal{D}}(K)$
by
$$\begin{eqnarray}i_{L/K}(P)=\mathop{\sum }_{\mathfrak{P}|P}e(\mathfrak{P}/P)\mathfrak{P},\end{eqnarray}$$
where the sum is over all primes of
$L$
dividing
$P$
. We then extend
$i_{L/K}$
linearly to all of
${\mathcal{D}}(K)$
.
For any effective divisor
$\mathfrak{m}$
of
$K$
, denote
$$\begin{eqnarray}{\mathcal{D}}_{\mathfrak{m}}(L)=\{D\in {\mathcal{D}}(L):\operatorname{supp}(D)\cap \operatorname{supp}(i_{L/K}(\mathfrak{m}))=\emptyset \}.\end{eqnarray}$$
Remark 2.1. If
$L=K$
then
$i_{L/K}$
is the identity map and
${\mathcal{D}}_{\mathfrak{m}}(K)$
is the set of divisors of
${\mathcal{D}}(K)$
that are coprime to
$\mathfrak{m}$
.
Let
$P$
be a prime of
$K$
that is unramified in
$L$
. Then there is a unique automorphism
$\unicode[STIX]{x1D70E}_{P}\in \operatorname{Gal}(L/K)$
, called the Frobenius automorphism, such that
$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{P}(x)\equiv x^{N(P)}~\text{mod}~\mathfrak{P}\end{eqnarray}$$
for all
$x\in {\mathcal{O}}_{\mathfrak{P}}$
and all places
$\mathfrak{P}$
of
$L$
lying above
$P$
, where
$N(P)$
is the norm of
$P$
and
${\mathcal{O}}_{\mathfrak{P}}$
is the local ring at
$\mathfrak{P}$
. Let
$\mathfrak{m}$
be an effective divisor of
$K$
such that
$L/K$
is unramified outside of
$\mathfrak{m}$
and define the Artin map as
where the product is over all primes of
$K$
dividing
$D$
.
Define the ray of
$L$
modulo
$\mathfrak{m}$
as
$$\begin{eqnarray}{\mathcal{P}}_{\mathfrak{m}}(L)=\{(f):f\in L^{\ast },f\equiv 1~\text{mod}~\mathfrak{P}^{\operatorname{ord}_{\mathfrak{P}}(i_{L/K}(\mathfrak{m}))}\text{ for all places }\mathfrak{P}\text{ of }L\},\end{eqnarray}$$
where we use the notation
$(f)$
for the principal divisor of
$f$
. Notice that
${\mathcal{P}}_{\mathfrak{m}}(L)$
is a subgroup of
${\mathcal{D}}_{\mathfrak{m}}(L)$
.
We say that
$\mathfrak{m}$
is a modulus for
$L$
if
${\mathcal{P}}_{\mathfrak{m}}(L)\subset \ker (A_{L/K})$
. In this case, we can view
$A_{L/K}$
as taking elements from the ray class group:
${\mathcal{C}}\ell _{\mathfrak{m}}(L):={\mathcal{D}}_{\mathfrak{m}}(L)/{\mathcal{P}}_{\mathfrak{m}}(L)$
. The following are the class field theory results which we need, borrowed from [Reference Hess and Massierer11].
Proposition 2.2.
(1)
$\mathfrak{m}$
is a modulus of
$L$
if and only if
$L/K$
is unramified outside of
$\operatorname{supp}(\mathfrak{m})$
.(2) If
$L/K$
is unramified outside of
$\operatorname{supp}(\mathfrak{m})$
then
$A_{L/K}:{\mathcal{C}}\ell _{\mathfrak{m}}(K)\rightarrow \operatorname{Gal}(L/K)$
is surjective.(3) Let
$L_{1},L_{2}/K$
be two finite abelian extensions with modulus
$\mathfrak{m}$
. Then
$L_{1}=L_{2}$
if and only if
$\ker (A_{L_{1}/K})=\ker (A_{L_{2}/K})$
.(4) For every effective divisor
$\mathfrak{m}$
and finite index subgroup
$H$
of
${\mathcal{C}}\ell _{\mathfrak{m}}(K)$
there exists a finite abelian extension
$L$
of modulus
$\mathfrak{m}$
such that
$\ker (A_{L/K})=H$
. Moreover,
$n{\mathcal{C}}\ell _{\mathfrak{m}}(K)\subset H$
where
$n$
is the exponent of
$\operatorname{Gal}(L/K)$
.
2.2 Abstract parametrization of geometric
$\ell$
-cyclic extensions
From now on we restrict to the case
$K=\mathbb{F}_{q}(X)$
,
$n=\ell$
a prime not dividing
$q$
, and we want to parametrize the set
$$\begin{eqnarray}{\mathcal{H}}_{\mathfrak{m}}=\{L/K\ell \text{-cyclic Galois extension, unramified outside }\mathfrak{m}\text{ and }L\cap \overline{\mathbb{F}}_{q}=\mathbb{F}_{q}\}.\end{eqnarray}$$
Let
$n_{q}$
be the multiplicative order of
$q$
modulo
$\ell$
. So,
$n_{q}=[K(\unicode[STIX]{x1D707}_{\ell }):K]=[\mathbb{F}_{q}(\unicode[STIX]{x1D707}_{\ell }):\mathbb{F}_{q}]$
where
$\unicode[STIX]{x1D707}_{\ell }$
is the group of
$\ell$
th roots of unity.
First we compute the ray class group in this setting.
Lemma 2.3. Let
$\mathfrak{m}$
be an effective divisor of
$K$
, then the ray class group
${\mathcal{C}}\ell _{\mathfrak{m}}(K)$
is finite and
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{m}}(K)\cong \mathbb{Z}/\ell \mathbb{Z}\times \mathop{\prod }_{\substack{ P\,\in \,\operatorname{supp}(\mathfrak{m}) \\ n_{q}|\deg (P)}}\mathbb{Z}/\ell \mathbb{Z}.\end{eqnarray}$$
Proof. We may write
${\mathcal{D}}_{\mathfrak{m}}={\mathcal{D}}_{\mathfrak{m}}^{0}(K)\times \mathbb{Z}$
with
${\mathcal{D}}_{\mathfrak{m}}^{0}(K)$
the subgroup of divisors of degree
$0$
. Since
${\mathcal{P}}_{m}(K)\subseteq {\mathcal{D}}_{\mathfrak{m}}^{0}(K)$
, we have
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{m}}(K)\cong {\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{ m}}^{0}(K)\times \mathbb{Z}/\ell \mathbb{Z}\end{eqnarray}$$
with
${\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K)={\mathcal{D}}_{\mathfrak{m}}^{0}/{\mathcal{P}}_{\mathfrak{m}}(K)$
. So it suffices to prove that
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{ m}}^{0}(K)\cong \mathop{\prod }_{\substack{ P\,\in \,\operatorname{supp}(\mathfrak{m}) \\ n_{q}|\deg (P)}}\mathbb{Z}/\ell \mathbb{Z}.\end{eqnarray}$$
First, suppose
$\mathfrak{m}=P$
for some prime
$P$
of
$K$
. Since
$\mathbb{F}_{q}[X]$
is a principal ideal domain, we get that
$$\begin{eqnarray}{\mathcal{D}}_{\mathfrak{m}}^{0}(K)=\{(f):f\in K^{\ast },\operatorname{ord}_{P}(f)=0\}=\{(f):f\in K^{\ast },f\not \equiv 0,\infty ~\text{mod}~P\}.\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\displaystyle {\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K) & = & \displaystyle \{(f):f\in K^{\ast },f\not \equiv 0,\infty ~\text{mod}~P\}/\{(f):f\in K^{\ast },f\equiv 1~\text{mod}~P\}\nonumber\\ \displaystyle & \cong & \displaystyle (\mathbb{F}_{q}[X]/P)^{\ast }=\mathbb{F}_{q^{\deg (P)}}^{\ast }.\nonumber\end{eqnarray}$$
Since
$\mathbb{F}_{q^{\deg (P)}}^{\ast }$
is cyclic, and its order is divisible by
$\ell$
if and only if
$n_{q}\mid \deg (P)$
, we have
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{ m}}^{0}(K)\cong \left\{\begin{array}{@{}ll@{}}\mathbb{Z}/\ell \mathbb{Z},\quad & n_{q}|\deg (P),\\ 0,\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$
Likewise, if
$\mathfrak{m}=eP$
for some positive integer
$e$
, then
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}(K)\cong (\mathbb{F}_{q}[X]/P^{e})^{\ast }\cong (\mathbb{F}_{q}[X]/P)^{\ast }\times H\end{eqnarray}$$
for some
$p$
-group
$H$
(where
$p=\operatorname{char}(K)\neq \ell$
). Hence (2.1) holds.
Finally, assume
$\mathfrak{m}=\sum _{P}e_{P}P$
with
$e_{P}\geqslant 0$
and only finitely many non-zero. Then by the Chinese Remainder Theorem,
$$\begin{eqnarray}{\mathcal{D}}_{\mathfrak{m}}^{0}(K)\cong \mathop{\prod }_{P\,\in \,\operatorname{supp}(\mathfrak{m})}{\mathcal{D}}_{e_{P}P}^{0}(K)\quad \text{and}\quad {\mathcal{P}}_{\mathfrak{ m}}(K)\cong \mathop{\prod }_{P\,\in \,\operatorname{supp}(\mathfrak{m})}{\mathcal{P}}_{e_{P}P}(K)\end{eqnarray}$$
and hence, by (2.1)
$$\begin{eqnarray}{\mathcal{C}}\ell _{\mathfrak{m}}^{0}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{ m}}^{0}(K)=\mathop{\prod }_{\substack{ P\,\in \,\operatorname{supp}(\mathfrak{m}) \\ n_{q}|\deg (P)}}\mathbb{Z}/\ell \mathbb{Z},\end{eqnarray}$$
as needed. ◻
Definition 2.4. We say a polynomial
$F$
is
$n$
-divisible if:
(1)
$n\mid \deg (P)$
for all
$P\mid F$
; and(2)
$F$
is monic.
Remark 2.5. A trivial observation, which plays a crucial point in the establishment of the analytic formula below, is that if
$f$
is
$n$
-divisible for
$n>1$
, then
$f(x)\neq 0$
for all
$x\in \mathbb{F}_{q}$
.
Corollary 2.6. For every effective divisor
$\mathfrak{m}$
of
$K$
, there is an
$(\ell -1)$
-to-
$1$
correspondence between pairs
$(F,b)$
with
$F\in \mathbb{F}_{q}[X]$
an
$n_{q}$
-divisible polynomial with prime factors in
$\operatorname{supp}(\mathfrak{m})$
and
$b\in \mathbb{F}_{q^{n_{q}}}^{\ast }$
both
$F,b$
taken up to
$\ell$
th powers in the respective group and between
$\ell$
-cyclic extensions
$L/K$
of modulus
$\mathfrak{m}$
such that
$L\cap \bar{\mathbb{F}}_{q}=\mathbb{F}_{q}$
.
Proof. By Proposition 2.2, the
$\ell$
-cyclic extensions
$L/K$
of modulus
$\mathfrak{m}$
are in bijection with subgroups of index
$\ell$
of
${\mathcal{C}}\ell _{\mathfrak{m}}(K)/\ell {\mathcal{C}}\ell _{\mathfrak{m}}(K)$
. By Lemma 2.3, the latter are in
$1$
-to-
$(\ell -1)$
correspondence with non-zero elements of
$$\begin{eqnarray}\mathbb{Z}/\ell \mathbb{Z}\times \mathop{\prod }_{\substack{ P\,\in \,\operatorname{supp}(\mathfrak{m}) \\ n_{q}|\deg (P)}}\mathbb{Z}/\ell \mathbb{Z}.\end{eqnarray}$$
We identify the first copy of
$\mathbb{Z}/\ell \mathbb{Z}$
with
$\mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
.
Now, a non-zero element
$(b,(a_{P}))$
corresponds to the pair
$(b,\prod _{P}P^{a_{P}})$
bijectively. To conclude the proof, we need to note that there is exactly one
$\ell$
-cyclic extension
$L/K$
with
$L\cap \bar{\mathbb{F}}_{q}\neq \mathbb{F}_{q}$
(namely,
$L=\mathbb{F}_{q^{\ell }}$
) and there are exactly
$\ell -1$
pairs with
$F$
an
$\ell$
th-power, namely
$(1,b)$
,
$b\in \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
.◻
2.3 Explicit correspondence
The correspondence described in Corollary 2.6 is given non-explicitly. In this section, we construct such a correspondence, using explicit Galois descent. We denote the to-be-constructed extension corresponding to
$(F,b)$
by
$L_{F,b}$
. The case
$n_{q}=1$
is treated by Kummer theory.
Proposition 2.7. Assume
$n_{q}=1$
. Then
$L_{F,b}=K(\sqrt[\ell ]{bF})$
.
Proof. This is the main result of Kummer theory, cf. [Reference Dummit and Foote10, Proposition §14.37]. ◻
From now on assume
$n_{q}>1$
. As mentioned above, the extension
$K(\sqrt[\ell ]{F})/K$
is not Galois and in particular not abelian. To deal with this, we extend the base field in order to apply Kummer theory and then descend the extension (using wreath products) to describe the model over the original field.
Let
$K^{\prime }:=\mathbb{F}_{q^{n_{q}}}(X)$
. Then, by the definition of
$n_{q}$
, we have
$\unicode[STIX]{x1D707}_{\ell }\subseteq K^{\prime }$
. Here we can apply Kummer theory, but
$F$
gives rise to many
$\ell$
-cyclic extensions which, in order to descend back to
$K$
, we need to describe explicitly together with the Galois action.
Remark 2.8. One word on the terminology we use; since both
$K$
and
$K^{\prime }$
are rational function fields in
$X$
, we shall use the term “prime polynomial” in
$K$
(respectively
$K^{\prime }$
) usually denoted by
$P$
(respectively
$\mathfrak{P}$
) to indicate both a monic irreducible polynomial in
$X$
with coefficients in
$\mathbb{F}_{q}$
(respectively
$\mathbb{F}_{q^{n_{q}}}$
) and the corresponding prime of the field (which is not
$1/X=\infty$
).
We denote by
$\unicode[STIX]{x1D719}$
the
$q$
th-power Frobenius. We extend its action from
$\mathbb{F}_{q^{n_{q}}}$
to
$K^{\prime }$
by letting it act trivially on
$X$
. So,
$$\begin{eqnarray}\unicode[STIX]{x1D719}(a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{0})=a_{n}^{q}X^{n}+a_{n-1}^{q}X^{n-1}+\cdots +a_{0}^{q}\end{eqnarray}$$
and
$\operatorname{Gal}(K^{\prime }/K)=\left<\unicode[STIX]{x1D719}\right>$
.
Every
$x\in \mathbb{F}_{q}$
is fixed by
$\unicode[STIX]{x1D719}$
, i.e.,
$x^{q}=x$
. Thus,
$$\begin{eqnarray}\unicode[STIX]{x1D719}(f)(x)=(f(x))^{q},\qquad x\in \mathbb{F}_{q},~f\in \mathbb{F}_{q^{n_{q}}}[X].\end{eqnarray}$$
We begin with a lemma about factorization.
Lemma 2.9. Let
$P$
be a prime polynomial in
$K$
and let
$$\begin{eqnarray}m=\gcd (n_{q},\deg (P)).\end{eqnarray}$$
Then there exist distinct prime polynomials
$\mathfrak{P}_{1},\ldots ,\mathfrak{P}_{m}$
in
$K^{\prime }$
such that
$$\begin{eqnarray}P=\mathfrak{P}_{1}\cdots \mathfrak{P}_{m}\end{eqnarray}$$
and
$\unicode[STIX]{x1D719}(\mathfrak{P}_{i})=\mathfrak{P}_{i+1}$
, where
$\unicode[STIX]{x1D719}$
is as defined in (2.2) and with the convention that
$\mathfrak{P}_{m+1}=\mathfrak{P}_{1}$
.
Proof. Since
$K^{\prime }/K$
is a cyclic unramified extension, each prime can be decomposed as
$$\begin{eqnarray}P=\mathfrak{P}_{1}\cdots \mathfrak{P}_{g},\end{eqnarray}$$
with the
$\mathfrak{P}_{i}$
pairwise distinct. Since the Galois group
$\operatorname{Gal}(K^{\prime }/K)=\left<\unicode[STIX]{x1D719}\right>$
acts transitively on the primes above
$P$
, by relabeling the primes we may assume that
$\mathfrak{P}_{i+1}=\unicode[STIX]{x1D719}(\mathfrak{P}_{i})$
. Finally,
$$\begin{eqnarray}\mathbb{F}_{q^{n_{q}}}[X]/P=(\mathbb{F}_{q}[X]/P)\otimes \mathbb{F}_{q^{n_{q}}}=\mathbb{F}_{q^{\deg (P)}}\otimes \mathbb{F}_{q^{n_{q}}}=\mathop{\prod }_{i=1}^{m}\mathbb{F}_{q^{n_{q}\deg (P)/m}},\end{eqnarray}$$
whence
$g=m$
.◻
We extend the above factorization of primes to
$n_{q}$
-divisible polynomials.
Corollary 2.10. If
$F$
is an
$n_{q}$
-divisible polynomial in
$K$
, then there exist coprime polynomials in
$K^{\prime }$
,
$F_{1},\ldots ,F_{n_{q}}$
such that
$$\begin{eqnarray}\displaystyle F & = & \displaystyle F_{1}\cdots F_{n_{q}},\nonumber\\ \displaystyle F_{i+1} & = & \displaystyle \unicode[STIX]{x1D719}(F_{i}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D719}$
is defined in (2.2) and
$F_{n_{q}+1}:=F_{1}$
. We call (2.4) a
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization and the
$F_{i}$
the corresponding
$\unicode[STIX]{x1D707}_{\ell }$
-stable factors.
Proof. Setting
$F_{i}=\prod _{P|F}\mathfrak{P}_{i}^{\operatorname{ord}_{P}(F)}$
with
$\mathfrak{P}_{i}$
as in Lemmas 2.9 suffices.◻
Remark 2.11. From now on we will fix a prime polynomial
$\mathfrak{P}=\mathfrak{P}_{1}$
in
$K^{\prime }$
dividing each prime
$P$
in
$K$
. This will then fix a
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization and the corresponding
$\unicode[STIX]{x1D707}_{\ell }$
-stable factors, hence we shall routinely refer to “the”
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization of an
$n_{q}$
-divisible polynomial to mean the one induced by our choice. Therefore the explicit correspondence we will now construct will be non-canonical.
From now on we fix our pair
$(F,b)$
and explicitly construct a
$\mathbb{Z}/\ell \mathbb{Z}$
extension
$L^{\prime }=L_{F,b}^{\prime }$
. Since
$F$
is taken up to
$\ell$
th powers, we may assume that
$F$
is an
$\ell$
th-power free polynomial in
$K$
that is
$n_{q}$
-divisible. We also fix notation: let
$F_{1},\ldots ,F_{n_{q}}$
be the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization factors of
$F$
as in (2.4), let
$b_{i}=b^{q^{i-1}}$
, and
$$\begin{eqnarray}u_{i}=\sqrt[\ell ]{b_{i}F_{i}},\end{eqnarray}$$
for
$i=1,\ldots ,n_{q}$
. We put
$$\begin{eqnarray}E^{\prime }=K^{\prime }(u_{1},\ldots ,u_{n_{q}}).\end{eqnarray}$$
Recall that the wreath product
$A\wr B$
of two finite groups
$A$
and
$B$
is defined as the semi-direct product
$A^{B}\rtimes B$
with
$B$
acting on
$A^{B}$
by permuting the indices. To be more precise, if for example
$B=\mathbb{Z}/n_{q}\mathbb{Z}$
and
$x=1\in B$
is a generator, then
$$\begin{eqnarray}(-x)(v_{1},\ldots ,v_{n_{q}})x=(v_{2},\ldots ,v_{n_{q}},v_{1}).\end{eqnarray}$$
Lemma 2.12. Let
$\unicode[STIX]{x1D701}_{\ell }\in \unicode[STIX]{x1D707}_{\ell }$
be a primitive
$\ell$
th root of unity. Then:
(1)
$\operatorname{Gal}(E^{\prime }/K^{\prime })=(\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
with action given by (2.5)$$\begin{eqnarray}(h_{1},\ldots ,h_{n_{q}}).u_{i}=\unicode[STIX]{x1D701}_{\ell }^{q^{i-1}h_{i}}u_{i},\quad i=1,\ldots ,n_{q};\end{eqnarray}$$
(2) the action of
$\unicode[STIX]{x1D719}$
on
$K^{\prime }$
as defined in (2.2) may be lifted to an action on
$E^{\prime }$
by setting (2.6)with our usual convention$$\begin{eqnarray}\unicode[STIX]{x1D719}(u_{i})=u_{i+1},\end{eqnarray}$$
$u_{n_{q}+1}:=u_{1}$
;(3) the extension
$E^{\prime }/K$
is Galois with
$\operatorname{Gal}(E^{\prime }/K)\cong \mathbb{Z}/\ell \mathbb{Z}\wr \mathbb{Z}/n_{q}\mathbb{Z}$
;(4) the fixed field
$E$
of
$\left<\unicode[STIX]{x1D719}\right>$
in
$E^{\prime }$
is $$\begin{eqnarray}E=K(u_{1}+\cdots +u_{n_{q}}).\end{eqnarray}$$
Proof. For (1), since the
$F_{i}$
are pairwise coprime, we have that
$\operatorname{Gal}(E^{\prime }/K^{\prime })=(\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
. We may choose coordinates for
$(\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
to have the stated action.
(2) and (3) are well-known consequences of (1), but we give a full account because we need the precise actions.
For (2), lift
$\unicode[STIX]{x1D719}$
in any way to a map
$\unicode[STIX]{x1D713}$
from
$E^{\prime }$
to its algebraic closure. Since
$$\begin{eqnarray}\unicode[STIX]{x1D713}(u_{i})^{\ell }=\unicode[STIX]{x1D713}(u_{i}^{\ell })=\unicode[STIX]{x1D713}(b_{i}F_{i})=\unicode[STIX]{x1D719}(b_{i}F_{i})=b_{i+1}F_{i+1},\end{eqnarray}$$
we have that
$\unicode[STIX]{x1D713}(u_{i})=\unicode[STIX]{x1D701}_{\ell }^{v_{i+1}}u_{i+1}$
, for some tuple
$(v_{1},\ldots ,v_{n_{q}})\in (\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
(as usual
$v_{n_{q}+1}=v_{1}$
). From this we conclude that
$\unicode[STIX]{x1D713}:E^{\prime }\rightarrow E^{\prime }$
and in particular
$E^{\prime }/K$
is a Galois extension. To get the desired lift, we change
$\unicode[STIX]{x1D713}$
by an element of
$\operatorname{Gal}(E^{\prime }/K^{\prime })$
:
$$\begin{eqnarray}(-v_{1},-v_{2}q^{-1},\ldots ,-v_{n_{q}}q^{1-n_{q}})\unicode[STIX]{x1D713},\end{eqnarray}$$
and this lift maps each
$u_{i}$
to
$u_{i+1}$
. By abuse of notation, we denote the latter lift also by
$\unicode[STIX]{x1D719}$
.
For (3), by (1) and (2), we have that
$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D719}^{-1}(v_{1},\ldots ,v_{n_{q}})\unicode[STIX]{x1D719})(u_{i}) & = & \displaystyle \unicode[STIX]{x1D719}^{-1}(v_{1},\ldots ,v_{n_{q}}).u_{i+1}=\unicode[STIX]{x1D719}^{-1}\unicode[STIX]{x1D701}_{\ell }^{q^{i}v_{i+1}}=\unicode[STIX]{x1D701}_{\ell }^{q^{i-1}v_{i+1}}u_{i}\nonumber\\ \displaystyle & = & \displaystyle (v_{2},\ldots ,v_{n_{q}},v_{1}).u_{i}.\nonumber\end{eqnarray}$$
This shows that the Galois group
$\operatorname{Gal}(E^{\prime }/K)$
is the wreath product.
For (4), put
$u=\sum _{i=1}^{n}u_{i}$
. Since
$K(u)\subseteq E$
, by the fundamental theorem of Galois theory to show equality, it suffices to show that
$u$
is not fixed by any element
$\unicode[STIX]{x1D70E}$
in
$\mathbb{Z}/\ell \mathbb{Z}\wr \mathbb{Z}/n_{q}\mathbb{Z}$
which is not in
$\left<\unicode[STIX]{x1D719}\right>$
. Such
$\unicode[STIX]{x1D70E}$
has the form
$\unicode[STIX]{x1D70E}=((h_{1},\ldots ,h_{n_{q}}),\unicode[STIX]{x1D719}^{k})$
with
$(h_{1},\ldots ,h_{n_{q}})$
a non-zero vector in
$(\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
. Assume by contradiction that
$\unicode[STIX]{x1D70E}(u)=u$
. Now, by (2.5) and since
$u$
is fixed by
$\unicode[STIX]{x1D719}$
,
$$\begin{eqnarray}\mathop{\sum }_{i}u_{i}=u=\unicode[STIX]{x1D70E}(u)=\mathop{\sum }_{i}\unicode[STIX]{x1D701}_{\ell }^{q^{i-1}h_{i}}u_{i}.\end{eqnarray}$$
Since by (1) the
$u_{i}$
are linearly independent over
$K^{\prime }$
, we must have that the coefficients are equal; that is,
$\unicode[STIX]{x1D701}_{\ell }^{q^{i-1}h_{i}}=1$
for all
$i$
. This implies that all
$h_{i}=0$
, in contradiction to the assumption.◻
Now that we have computed the Galois group, to study
$\ell$
-cyclic subextensions we need to find
$\ell$
-cyclic quotients of the wreath product.
Lemma 2.13. The wreath product
$G=\mathbb{Z}/\ell \mathbb{Z}\wr \mathbb{Z}/n_{q}\mathbb{Z}$
has a unique
$\ell$
-cyclic quotient. More precisely, for
$H{\vartriangleleft}G$
we have
$G/H\cong \mathbb{Z}/\ell \mathbb{Z}$
if and only if
$H=\tilde{H}\rtimes \mathbb{Z}/n_{q}\mathbb{Z}$
, where
$$\begin{eqnarray}\tilde{H}=\{h=(h_{1},\ldots ,h_{n_{q}}):h_{1}+\cdots +h_{n_{q}}\equiv 0~\text{mod}~\ell \}\leqslant (\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}.\end{eqnarray}$$
Proof. Let
$G^{\prime }$
be the commutator of
$G$
. Direct computation shows that
$$\begin{eqnarray}G^{\prime }\leqslant \{(h,0):h_{1}+\cdots +h_{n_{q}}=0\}\end{eqnarray}$$
and since the quotient by the right-hand side is abelian we have an equality. Here
$h=(h_{1},\ldots ,h_{n_{q}})\in (\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
. Hence we have the isomorphism
$$\begin{eqnarray}\displaystyle G/G^{\prime } & \rightarrow & \displaystyle \mathbb{Z}/\ell \mathbb{Z}\times \mathbb{Z}/n_{q}\mathbb{Z}\cong \mathbb{Z}/\ell n_{q}\mathbb{Z}\nonumber\\ \displaystyle (h,\unicode[STIX]{x1D70E}) & \mapsto & \displaystyle \biggl(\mathop{\sum }_{j}h_{j},\unicode[STIX]{x1D70E}\biggr)\nonumber\end{eqnarray}$$
which implies that
$G/G^{\prime }$
has a unique
$\ell$
-cyclic quotient as a cyclic group, and so does
$G$
. Moreover this quotient is given by
$(h,\unicode[STIX]{x1D70E})\mapsto \sum _{j}h_{j}$
, and so the kernel is
$\tilde{H}\rtimes \mathbb{Z}/n_{q}\mathbb{Z}$
.◻
We enumerate all the
$\ell$
-cyclic subextensions of
$E^{\prime }/K^{\prime }$
by non-zero vectors
$\mathbf{v}=(v_{1},\ldots ,v_{n_{q}})$
of elements in
$\mathbb{Z}/\ell \mathbb{Z}$
. We write
$$\begin{eqnarray}F_{\mathbf{v}}:=\mathop{\prod }_{i=1}^{n_{q}}(b_{i}F_{i})^{v_{i}^{\prime }},\end{eqnarray}$$
where
$0\leqslant v_{i}^{\prime }\leqslant \ell -1$
is a minimal non-negative representative of
$v_{i}$
,
$i=1,\ldots ,n_{q}$
and we let
$$\begin{eqnarray}L_{\mathbf{v}}=K^{\prime }(\sqrt[\ell ]{F_{\boldsymbol{ v}}})\end{eqnarray}$$
be the corresponding Kummer extension.
Lemma 2.14. Let
$\tilde{H}\leqslant H\leqslant \mathbb{Z}/\ell \mathbb{Z}\wr \mathbb{Z}/n_{q}\mathbb{Z}\cong \operatorname{Gal}(E^{\prime }/K)$
be as in Lemma 2.13 and let
$L^{\prime }={E^{\prime }}^{H}$
and
$\tilde{L}={E^{\prime }}^{\tilde{H}}$
. Then,
$$\begin{eqnarray}\displaystyle \tilde{L} & = & \displaystyle L_{\mathbf{v_{0}}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \tilde{L} & = & \displaystyle L^{\prime }\mathbb{F}_{q^{n_{q}}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle L^{\prime } & = & \displaystyle \mathbb{F}_{q}(X)[Y]/\biggl(\mathop{\prod }_{j=0}^{\ell -1}\biggl(Y-\mathop{\sum }_{k=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{\ell }^{jq^{k}}\sqrt[\ell ]{F_{\mathbf{v}_{k}}}\biggr)\biggr),\end{eqnarray}$$
where
$\mathbf{v}_{k}=(q^{k},q^{k-1},\ldots ,q^{k+1-n_{q}})$
.
Proof. By (2.7), (2.5), and since
$q^{j-1}v_{j}\equiv 1~\text{mod}~\ell$
, we have
$$\begin{eqnarray}h(\sqrt[\ell ]{F_{\mathbf{v}_{0}}})=\mathop{\prod }_{j=1}^{n_{q}}\unicode[STIX]{x1D701}_{\ell }^{q^{j-1}h_{j}v_{j}}\sqrt[\ell ]{F_{\mathbf{v}_{0}}}=\unicode[STIX]{x1D701}_{\ell }^{\mathop{\sum }_{j=1}^{n_{q}}h_{j}}\sqrt[\ell ]{F_{\mathbf{v}_{0}}}=\sqrt[\ell ]{F_{\mathbf{v}_{0}}},\end{eqnarray}$$
for all
$h=(h_{1},\ldots ,h_{n_{q}})\in \tilde{H}$
. Therefore,
$L_{\mathbf{v}_{0}}\subseteq \tilde{L}$
. On the other hand,
$[L_{\mathbf{v}_{0}}:K^{\prime }]=\ell =[\tilde{L}:K^{\prime }]$
, hence
$\tilde{L}=L_{\mathbf{v}_{0}}$
.
Since
$\tilde{H}=H\cap (\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}$
and
$(\mathbb{Z}/\ell \mathbb{Z})^{n_{q}}\cong \operatorname{Gal}(E^{\prime }/K^{\prime })$
, the fundamental theorem of Galois theory implies that
$$\begin{eqnarray}\tilde{L}=LK^{\prime }=L\mathbb{F}_{q^{n_{q}}}.\end{eqnarray}$$
Finally, let
$u\in L$
be the trace of
$\sqrt[\ell ]{F_{\mathbf{v}_{0}}}\in \tilde{L}$
in the extension
$\tilde{L}/L$
. Since
$\operatorname{Gal}(\tilde{L}/L)$
is generated by the restriction of
$\unicode[STIX]{x1D719}$
, by (2.6), we have
$$\begin{eqnarray}u=\mathop{\sum }_{k=0}^{n_{q}-1}\sqrt[\ell ]{F_{\boldsymbol{ v}_{k}}}.\end{eqnarray}$$
The conjugates of
$u$
over
$K$
are
$(j,0,\ldots ,0)u$
,
$j\in \mathbb{Z}/\ell \mathbb{Z}$
which by (2.5) satisfy
$$\begin{eqnarray}(j,0,\ldots ,0)u=\mathop{\sum }_{k=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{\ell }^{jq^{k}}\sqrt[\ell ]{F_{\mathbf{v}_{k}}}.\end{eqnarray}$$
Thus
$u$
generates the
$\ell$
-cyclic extension
$L/K$
and has
$$\begin{eqnarray}\mathop{\prod }_{j=0}^{\ell -1}\biggl(Y-\mathop{\sum }_{k=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{\ell }^{jq^{k}}\sqrt[\ell ]{F_{\mathbf{v}_{k}}}\biggr)\end{eqnarray}$$
as its minimal polynomial. ◻
We summarize the construction in the following diagram.
Now we are ready to introduce an explicit description of the correspondence of Corollary 2.6.
Proposition 2.15. The correspondence
$$\begin{eqnarray}(F,b)\mapsto L^{\prime }=L_{F,b}^{\prime }\end{eqnarray}$$
induces an
$(\ell -1)$
-to-
$1$
map between the sets in Corollary 2.6.
Proof. Let
$\mathfrak{m}$
be an effective divisor and let
$F$
and
$G$
be two
$n_{q}$
-divisible polynomials that are supported on
$\mathfrak{m}$
. For each prime polynomial
$P\in \mathbb{F}_{q}[X]$
of degree divisible by
$n_{q}$
, let
$P=\mathfrak{P}_{1}\cdots \mathfrak{P}_{n_{q}}$
be the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization. Let
$F=F_{1}\cdots F_{n_{q}}$
and
$G=G_{1}\cdots G_{n_{q}}$
be the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorizations of
$F$
and
$G$
. So,
$\mathfrak{P}_{1}$
appears only in
$F_{1}$
if
$P\mid F$
and only in
$G_{1}$
if
$P\mid G$
.
Assume that
$L_{F,b}^{\prime }=L_{G,c}^{\prime }$
. By (2.8), (2.9), and (2.10) there exists
$0\leqslant r\leqslant \ell -1$
such that
$F_{\mathbf{v}_{0}}$
and
$G_{\mathbf{v}_{0}}^{r}$
are equal up to
$\ell$
th powers in
$K^{\prime }=\mathbb{F}_{q^{n_{q}}}(X)$
. Comparing leading coefficients gives that
$b^{n_{q}}=c^{n_{q}r}$
and so
$b=c^{r}$
up to
$\ell$
th powers.
Let
$P$
be a prime dividing
$F$
(respectively
$G$
) with multiplicity
$\unicode[STIX]{x1D6FC}$
(respectively
$\unicode[STIX]{x1D6FD}$
). Then
$\mathfrak{P}_{1}$
divides
$F_{1}$
(respectively
$G_{1}$
) with the same multiplicity. So
$\unicode[STIX]{x1D6FC}\equiv r\unicode[STIX]{x1D6FD}~\text{mod}~\ell$
. This implies that
$F=G^{r}$
up to
$\ell$
th powers. So we have that up to respective
$\ell$
th powers,
$(F,b)=(G^{r},c^{r})$
. This implies the correspondence between the pairs and extensions is
$(\ell -1)$
-to-
$1$
and by Corollary 2.6 we also obtain all the extensions.◻
Remark 2.16. This shows that without loss of generality we may assume that
$L_{F,b}$
, the extension corresponding to
$(F,b)$
implied by Corollary 2.6, is the same as
$L_{F,b}^{\prime }$
, the extensions constructed in Lemma 2.14. Thus, we will denote both as simply
$L_{F,b}$
.
Remark 2.17. Let us translate the above construction in terms of covers and equations. Let
$C_{F,b}$
be the smooth projective model of
$L_{F,b}$
. By (2.11),
$C_{F,b}$
is birationally equivalent to the affine plane curve
$$\begin{eqnarray}\bigg\{\mathop{\prod }_{j=0}^{\ell -1}\biggl(Y-\mathop{\sum }_{k=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{\ell }^{jq^{k}}\sqrt[\ell ]{F_{\mathbf{v}_{k}}}\biggr)=0\bigg\}.\end{eqnarray}$$
By (2.9) and (2.10),
$C_{F,b}$
is geometrically birationally equivalent to
$$\begin{eqnarray}\{Y^{\ell }=F_{\boldsymbol{ v}_{0}}\}.\end{eqnarray}$$
Remark 2.18. If
$n_{q}=1$
, then
$F_{\mathbf{v}_{0}}=bF$
and the characteristic polynomial becomes
$$\begin{eqnarray}\mathop{\prod }_{j=0}^{\ell -1}(Y-\unicode[STIX]{x1D701}_{\ell }^{i}\sqrt[\ell ]{bF})=Y^{\ell }-bF\end{eqnarray}$$
which recovers the statement in Kummer theory.
3 Genus formula
From now on we fix an
$n_{q}$
-divisible polynomial
$F$
which is
$\ell$
th-power free and
$b\in \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
. We let
$L=L_{F,b}$
be as in (2.11) and
$C=C_{F,b}$
the smooth projective model of
$L$
. We also assume that
$$\begin{eqnarray}n_{q}\neq 1.\end{eqnarray}$$
Our goal is to give a formula for the genus
$g=g(C)$
of
$C$
.
The genus of projective curves is preserved under separable base change, so the genus
$g$
equals the genus of the smooth projective curve which is birationally equivalent to the affine plane curve (2.12). By the Riemann–Hurwitz formula the genus of the curve in (2.12) may be computed in terms of a factorization of the
$\ell$
th-power free polynomial
$F_{\mathbf{v}_{0}}$
as explained below.
Since
$F$
is
$\ell$
th-power free, there exist monic, square-free polynomials
$f_{1},\ldots ,f_{\ell -1}$
which are pairwise coprime such that
$$\begin{eqnarray}F=f_{1}f_{2}^{2}\cdots f_{\ell -1}^{\ell -1}.\end{eqnarray}$$
Since
$F$
is
$n_{q}$
-divisible, all the
$f_{i}$
are
$n_{q}$
-divisible as well. Hence, by Corollary 2.10, for each
$i$
, we have the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization of
$f_{i}$
:
$$\begin{eqnarray}f_{i}=f_{i,1}\cdots f_{i,n_{q}}.\end{eqnarray}$$
We now define
$$\begin{eqnarray}\displaystyle F_{j}:=f_{1,j}f_{2,j}^{2}\cdots f_{\ell -1,j}^{\ell -1}, & & \displaystyle\end{eqnarray}$$
which gives us the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization of
$F$
, namely
$$\begin{eqnarray}F=F_{1}\cdots F_{n_{q}},\end{eqnarray}$$
and
$F_{i}=\unicode[STIX]{x1D719}^{i-1}(F_{1})$
, with
$\unicode[STIX]{x1D719}$
the
$q$
th-power Frobenius. Taking
$v_{i}$
to be the minimal non-negative integer with
$v_{i}\equiv q^{1-i}~\text{mod}~\ell$
, we get by (2.7) and (3.2) that
$$\begin{eqnarray}\displaystyle F_{\mathbf{v}_{0}}=b^{\prime }\mathop{\prod }_{j=1}^{n_{q}}F_{j}^{v_{j}}=b^{\prime }\mathop{\prod }_{i=1}^{\ell }\mathop{\prod }_{j=1}^{n_{q}}f_{i,j}^{iv_{j}}, & & \displaystyle\end{eqnarray}$$
for some
$b^{\prime }\in \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
. We apply the Riemann–Hurwitz formula, which in this setting gives that
$$\begin{eqnarray}2g+2\ell -2=(\ell -1)\mathop{\sum }_{i=1}^{\ell }\mathop{\sum }_{j=1}^{n_{q}}\deg (f_{i,j})+\left\{\begin{array}{@{}ll@{}}0,\quad & \text{if }\deg (F_{\mathbf{v}_{0}})\equiv 0~\text{mod}~\ell ,\\ \ell -1,\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$
(cf. [Reference Meisner15, equation (1.2)] with our
$\ell$
replacing
$r$
and using the fact that
$\ell$
is coprime to all of the
$v_{i}$
).
Since
$\deg (f_{i,j})=\deg (f_{i,1})$
for all
$j$
,
$$\begin{eqnarray}\deg (F_{\mathbf{v}_{0}})\equiv \mathop{\sum }_{i=1}^{\ell -1}\mathop{\sum }_{j=1}^{n_{q}}iq^{1-j}\deg (f_{i,j})\equiv \mathop{\sum }_{i=1}^{\ell -1}i\deg (f_{i,1})\mathop{\sum }_{j=1}^{n_{q}}q^{1-j}~\text{mod}~\ell .\end{eqnarray}$$
Since we assume
$n_{q}\not =1$
and since
$q^{n_{q}}\equiv 1~\text{mod}~\ell$
, we have
$$\begin{eqnarray}\mathop{\sum }_{j=1}^{n_{q}}q^{1-j}\equiv q\cdot \frac{1-q^{-n_{q}}}{1-q^{-1}}\equiv 0~\text{mod}~\ell .\end{eqnarray}$$
Furthermore, by (3.1),
$\sum _{j=1}^{n_{q}}\deg (f_{i,j})=\deg (f_{i})$
. Plugging the above computations into (3.4) yields the genus formula for
$C$
:
$$\begin{eqnarray}2g+2\ell -2=(\ell -1)\mathop{\sum }_{i=1}^{\ell }\deg (f_{i}).\end{eqnarray}$$
Remark 3.1. The Riemann–Hurwitz formula and (3.5) imply that the only primes that ramify in
$\mathbb{F}_{q}(C)$
are those that divide
$F$
. Consequently, no linear primes ramify in
$\mathbb{F}_{q}(C)$
if
$n_{q}\not =1$
. Hence, all the points of
$\mathbb{P}^{1}(\mathbb{F}_{q})$
are unramified in
$C$
.
We use (3.5) to parametrize the set of curves
${\mathcal{H}}_{g,\ell }$
as defined in (1.1). Define the following set
$$\begin{eqnarray}{\mathcal{F}}_{n_{q}}(D)=\left\{(f_{1},\ldots ,f_{\ell -1})\in \mathbb{F}_{q}[X]:\begin{array}{@{}l@{}}f_{1},\ldots ,f_{\ell -1}\text{ are square-free},\\ \text{pairwise coprime},n_{q}\text{-divisible},\\ \text{and }\deg (f_{1}\cdots f_{\ell -1})=D\end{array}\right\}.\end{eqnarray}$$
Each tuple
$(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}(D)$
corresponds to the
$\ell$
th-power free
$n_{q}$
-divisible polynomial
$F=f_{1}f_{2}^{2}\cdots f_{\ell -1}^{\ell -1}$
. By (3.5), the genus of a corresponding curve
$C=C_{F,b}$
satisfies
$$\begin{eqnarray}D=\frac{2g+2\ell -2}{\ell -1}.\end{eqnarray}$$
Thus, by Corollary 2.6, the correspondence
$(F,b)\mapsto C_{F,b}$
defines an
$(\ell -1)$
-to-
$1$
correspondence between the sets
$$\begin{eqnarray}{\mathcal{F}}_{n_{q}}(D)\times \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }\rightarrow {\mathcal{H}}_{g,\ell }.\end{eqnarray}$$
Since
$n_{q}$
-divisible polynomials have degree divisible by
$n_{q}$
, we have that
${\mathcal{F}}_{n_{q}}(D)=\emptyset$
if and only if
$D\not \equiv 0~\text{mod}~n_{q}$
. Thus we immediately get the following.
Proposition 3.2. For
$n_{q}>1$
,
${\mathcal{H}}_{g,\ell }=\emptyset$
if
$2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)n_{q}$
.
When
$D\equiv 0~\text{mod}~n_{q}$
or equivalently
$2g+2\ell -2\equiv 0~\text{mod}~(\ell -1)n_{q}$
, it suffices to count the number of points on curves when we parameterize by
${\mathcal{F}}_{n_{q}}(D)$
instead of by
${\mathcal{H}}_{g,\ell }$
. More rigorously, we have the following result.
Proposition 3.3. Let
$g,D$
be related by (3.7) and assume that
$n_{q}>1$
and that
$2g+2\ell -2\equiv 0~\text{mod}~(\ell -1)n_{q}$
. Choose a random
$(f_{1},\ldots ,f_{\ell -1})$
uniformly in
${\mathcal{F}}_{n_{q}}(D)$
and a random
$b\in \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
and put
$F$
and
$C_{F,b}$
as above. Choose a random
$C$
uniformly in
${\mathcal{H}}_{g,\ell }$
. Then, for every
$N\geqslant 0$
,
$$\begin{eqnarray}\operatorname{Prob}(\#C_{F,b}(\mathbb{F}_{q})=N)=\operatorname{Prob}(\#C(\mathbb{F}_{q})=N).\end{eqnarray}$$
4 Number of points formula
The goal of this section is to give an analytic formula for the number of rational points on cyclic covers. Let
$\unicode[STIX]{x1D70B}:C\rightarrow \mathbb{P}_{\mathbb{F}_{q}}^{1}$
be an
$\ell$
-cyclic cover of smooth projective
$\mathbb{F}_{q}$
-curves and let
$$\begin{eqnarray}C^{\prime }=C\times _{\mathbb{ F}_{q}}\mathbb{F}_{q^{n_{q}}}\end{eqnarray}$$
be the scalar extension of
$C$
to
$\mathbb{F}_{q^{n_{q}}}$
. We start by a simple general observation connecting the number of
$\mathbb{F}_{q}$
-rational points of
$C$
and
$\mathbb{F}_{q^{n_{q}}}$
-rational points of
$C^{\prime }$
lying above
$\mathbb{P}^{1}(\mathbb{F}_{q})$
.
It is convenient to introduce the following piece of notation: for
$x\in \mathbb{P}^{1}(\mathbb{F}_{q})$
, let
$$\begin{eqnarray}N_{x}=\#\{y\in C(\mathbb{F}_{q}):\unicode[STIX]{x1D70B}(y)=x\}\quad \text{and}\quad N_{x}^{\prime }=\#\{y\in C^{\prime }(\mathbb{F}_{q^{n_{q}}}):\unicode[STIX]{x1D70B}(y)=x\}\end{eqnarray}$$
be the number of
$\mathbb{F}_{q}$
-rational and
$\mathbb{F}_{q^{n_{q}}}$
-rational points on
$C$
and
$C^{\prime }$
lying above
$x$
, respectively.
Proposition 4.1. Let
$x\in \mathbb{P}^{1}(\mathbb{F}_{q})$
. Then
$N_{x}=N_{x}^{\prime }$
.
Proof. Let
$x\in \mathbb{P}^{1}(\mathbb{F}_{q})$
, let
$O=O_{\mathbb{P}_{\mathbb{F}_{q}}^{1},x}$
be the local ring at
$x$
(which is the localization of
$\mathbb{F}_{q}[X]$
at
$X-x$
if
$x$
is finite or of
$\mathbb{F}_{q}[X^{-1}]$
at
$X^{-1}$
if
$x=\infty$
) and
$\mathfrak{p}$
the corresponding maximal ideal (which is
$(X-x)$
if
$x$
is finite or
$(X^{-1})$
if
$x=\infty$
). Since
$L=\mathbb{F}_{q}(C)$
is Galois with cyclic group of prime order
$\ell$
, there are
$3$
possible factorizations of
$\mathfrak{p}=\mathfrak{P}_{1}^{e}\cdots \mathfrak{P}_{g}^{e}$
, namely:
(1)
$g=\ell$
,
$e=f=1$
, (where
$f=\deg \mathfrak{P}_{i}$
) in which case
$N_{x}=\ell$
;(2)
$e=\ell$
,
$f=g=1$
, in which case
$N_{x}=1$
;(3)
$f=\ell$
,
$e=g=1$
, in which case
$N_{x}=0$
.
Now we base change to
$\mathbb{F}_{q^{n_{q}}}$
, and denote it as before by adding a tag. Since
$n_{q}=[\mathbb{F}_{q^{n_{q}}}:\mathbb{F}_{q}]$
is coprime to
$\ell$
(as we add
$\ell$
th root of unity), we get that
$f=f^{\prime }$
. Since the extension
$\mathbb{F}_{q^{n_{q}}}/\mathbb{F}_{q}$
is unramified, we get that
$e=e^{\prime }$
. Since
$\ell =[L:K]=[L^{\prime }:K^{\prime }]$
, we conclude that
$g=g^{\prime }$
. As
$N_{x}^{\prime }$
is determined by
$e^{\prime },f^{\prime },g^{\prime }$
in the same manner as
$N_{x}$
is determined by
$e,f,g$
, we conclude that
$N_{x}=N_{x}^{\prime }$
.◻
Next we compute the number of rational points in terms of the parametrization of §3. Recall that
$b\in \mathbb{F}_{q^{n_{q}}}^{\ast }/(\mathbb{F}_{q^{n_{q}}}^{\ast })^{\ell }$
,
$$\begin{eqnarray}F=f_{1}f_{2}^{2}\cdots f_{\ell -1}^{\ell -1}\end{eqnarray}$$
for
$(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}(D)$
and that
$C=C_{F,b}$
in the notation of (2.12). Also recall that
$C^{\prime }$
is birationally equivalent to the affine plane curve
$Y^{\ell }=F_{\mathbf{v}_{0}}$
; see (2.12), where
$F_{\mathbf{v}_{0}}$
is given in (2.7).
Let
$\unicode[STIX]{x1D712}_{\ell }$
be a primitive multiplicative character of
$\mathbb{F}_{q^{n_{q}}}$
of order
$\ell$
(there exists such, as
$n_{q}$
by definition is the minimal positive integer with
$q^{n_{q}}\equiv 1~\text{mod}~\ell$
). For
$x\in \mathbb{F}_{q}$
and for a polynomial
$G$
, we define
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{x}(G)=\unicode[STIX]{x1D712}_{\ell }(G(x)),\end{eqnarray}$$
which is a multiplicative Dirichlet character modulo
$X-x$
of order
$\ell$
. We extend this definition to
$x=\infty \in \mathbb{P}^{1}(\mathbb{F}_{q})$
by setting
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{\infty }(G)=\unicode[STIX]{x1D712}_{\ell }(g_{n}),\end{eqnarray}$$
with
$g_{n}$
being the leading coefficient of
$G$
.
Proposition 4.2. We have
$$\begin{eqnarray}\#C(\mathbb{F}_{q})=\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{\ell }^{w}(b)+\mathop{\sum }_{x\in \mathbb{F}_{q}}\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{x}^{w}(F_{\boldsymbol{ v}_{0}}).\end{eqnarray}$$
Proof. In [Reference Meisner15] it is established that for
$x\in \mathbb{P}^{1}(\mathbb{F}_{q})$
,
$$\begin{eqnarray}N_{x}^{\prime }=\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{x}^{w}(F_{\boldsymbol{ v}_{0}}).\end{eqnarray}$$
(To see it follows from [Reference Meisner15], plug
$\ell$
,
$F_{\mathbf{v}_{0}}$
for
$r$
,
$F$
in the notation of [Reference Meisner15]. For finite
$x$
, we note that
$F_{\mathbf{v}_{0}}(x)\neq 0$
since
$f_{i}(x)\neq 0$
as they are
$n_{q}$
-divisible polynomial. Thus the formula is given in the first paragraph of the proof of [Reference Meisner15, Lemma 2.1]. For
$x=\infty$
, see page 536.)
Then, as
$N_{x}=N_{x}^{\prime }$
(Proposition 4.1) we get that
$$\begin{eqnarray}\#C(\mathbb{F}_{q})=\mathop{\sum }_{x\in \mathbb{P}^{1}(\mathbb{F}_{q})}N_{x}=\mathop{\sum }_{x\in \mathbb{P}^{1}(\mathbb{F}_{q})}N_{x}^{\prime }=\mathop{\sum }_{x\in \mathbb{P}^{1}(\mathbb{F}_{q})}\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{x}^{w}(F_{\boldsymbol{ v}_{0}}).\end{eqnarray}$$
The leading coefficient of
$F_{\mathbf{v_{0}}}$
is
$b^{n_{q}}$
and so
$\sum _{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{\ell }^{w}(b)=\sum _{w=0}^{\ell -1}\unicode[STIX]{x1D712}_{\ell }^{w}(b^{n_{q}})$
.◻
5 Set count
In light of the analytic formula (4.4), the study of the distribution of the number of points on
$\ell$
-cyclic covers parameterized by
${\mathcal{F}}_{n_{q}}(D)\times \mathbb{F}_{q^{n_{q}}}^{\ast }$
may be reduced to the computation of the size of sets of the form
$$\begin{eqnarray}\displaystyle \hspace{-8.0pt}{\mathcal{F}}_{n_{q}}^{k}(D) & = & \displaystyle {\mathcal{F}}_{n_{q},\unicode[STIX]{x1D716}}^{k}(D)\nonumber\\ \displaystyle & = & \displaystyle \{(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}(D):\unicode[STIX]{x1D712}_{\ell }(F_{\mathbf{v}_{0}}(x_{i}))=\unicode[STIX]{x1D716}_{i},i=1,\ldots ,k\},\quad\end{eqnarray}$$
where
$x_{1},\ldots ,x_{k}\in \mathbb{F}_{q}$
are pairwise distinct elements,
$\unicode[STIX]{x1D716}=(\unicode[STIX]{x1D716}_{1},\ldots ,\unicode[STIX]{x1D716}_{k})\in \unicode[STIX]{x1D707}_{\ell }^{k}$
is fixed, and, as usual,
$F$
and
$F_{\mathbf{v}_{0}}$
,
$\mathbf{v}_{0}=(1,q^{-1},\ldots ,q^{1-n_{q}})$
are as defined in (4.2) and (2.7), respectively. The computation is done by analyzing an appropriate generating function
${\mathcal{G}}_{k}(u)$
.
Remark 5.1. While the definition of
${\mathcal{F}}_{n_{q}}^{k}(D)$
depends on the choice of
$(\unicode[STIX]{x1D716}_{1},\ldots ,\unicode[STIX]{x1D716}_{k})$
and
$(x_{1},\ldots ,x_{l})$
, its asymptotic size—which is computed below in (5.13)—is independent of these choices. Hence we omit the
$\unicode[STIX]{x1D716}_{i}$
and
$x_{i}$
from the notation.
5.1 Generating series
Let
$\unicode[STIX]{x1D701}_{n_{q}}\in \mathbb{C}$
be a primitive
$n_{q}$
th root of unity and define the following auxiliary functions:
$$\begin{eqnarray}\displaystyle {\mathcal{I}}_{\infty }(F) & := & \displaystyle \unicode[STIX]{x1D707}^{2}(F)\mathop{\prod }_{P|F}\frac{1}{n_{q}}\biggl(\mathop{\sum }_{i=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{n_{q}}^{i\deg (P)}\biggr),\nonumber\\ \displaystyle {\mathcal{I}}_{x_{i}}(F) & := & \displaystyle \frac{1}{\ell }\biggl(\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D716}_{i}^{-w}\unicode[STIX]{x1D712}_{x_{i}}^{w}(F)\biggr),\nonumber\end{eqnarray}$$
with the convention that
$\unicode[STIX]{x1D712}_{x}^{0}$
is the trivial character modulo
$X-x$
. Using these functions we define a function in
$\ell -1$
variables (which are always assumed to be monic polynomials)
$$\begin{eqnarray}{\mathcal{I}}(f_{1},\ldots ,f_{\ell -1})={\mathcal{I}}_{\infty }(f_{1}\cdots f_{\ell -1})\mathop{\prod }_{i=1}^{k}{\mathcal{I}}_{x_{i}}(F_{\mathbf{v}_{0}}),\end{eqnarray}$$
with
$F$
and
$F_{\mathbf{v}_{0}}$
defined as in (4.2) and (2.7).
Lemma 5.2. For a tuple of monic polynomials
$(f_{1},\ldots ,f_{\ell -1})$
we have
$$\begin{eqnarray}{\mathcal{I}}(f_{1},\ldots ,f_{\ell -1})=\left\{\begin{array}{@{}ll@{}}1,\quad & \text{if }(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}^{k}(\deg (f_{1}\cdots f_{\ell -1})),\\ 0,\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$
In particular,
$$\begin{eqnarray}\displaystyle {\mathcal{G}}_{k}(u):=\mathop{\sum }_{f_{1},\ldots ,f_{\ell -1}}{\mathcal{I}}(f_{1},\ldots ,f_{\ell -1})u^{\deg (f_{1}\cdots f_{\ell -1})}=\mathop{\sum }_{D=0}^{\infty }|{\mathcal{F}}_{n_{q}}^{k}(D)|u^{D}, & & \displaystyle\end{eqnarray}$$
where the first sum is over monic polynomials.
Proof. By the orthogonality relations, for a tuple of monic polynomials
$(f_{1},\ldots ,f_{\ell -1})$
, we have that
${\mathcal{I}}_{\infty }(f_{1}\cdots f_{\ell -1})=1$
if
$(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}(D)$
, and
$=0$
otherwise. Similarly, if
$(f_{1},\ldots ,f_{\ell -1})\in {\mathcal{F}}_{n_{q}}(D)$
, then
${\mathcal{I}}_{x_{i}}(F_{\mathbf{v}_{0}})=1$
if
$\unicode[STIX]{x1D712}_{x_{i}}(F_{\mathbf{v}_{0}})=\unicode[STIX]{x1D716}_{i}$
and
$=0$
otherwise. This completes the proof of the lemma.◻
Expanding the product
$\prod _{i}{\mathcal{I}}_{x_{i}}(F)$
gives
$$\begin{eqnarray}\mathop{\prod }_{i=1}^{k}{\mathcal{I}}_{x_{i}}(F)=\frac{1}{\ell ^{k}}\mathop{\prod }_{i=1}^{k}\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D716}_{i}^{-w}\unicode[STIX]{x1D712}_{x_{i}}^{w}(F)=\frac{1}{\ell ^{k}}\mathop{\sum }_{\mathbf{w}}\biggl(\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D716}_{i}^{-w_{i}}\biggr)\unicode[STIX]{x1D712}_{\mathbf{w}}(F),\end{eqnarray}$$
where the sum is over all vectors
$\mathbf{w}=(w_{1},\ldots ,w_{k})$
such that
$0\leqslant w_{i}\leqslant \ell -1$
and
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{\mathbf{w}}(F)=\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D712}_{x_{i}}^{w_{i}}(F).\end{eqnarray}$$
This is a Dirichlet character of modulus
$\prod (X-x_{i})$
and is non-trivial if
$\mathbf{w}\neq \vec{0}$
. Thus we get a decomposition of
${\mathcal{G}}_{k}(u)$
,
$$\begin{eqnarray}\displaystyle {\mathcal{G}}_{k}(u)=\frac{1}{\ell ^{k}}\mathop{\sum }_{\mathbf{w}}\biggl(\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D716}_{i}^{-w_{i}}\biggr){\mathcal{G}}_{\mathbf{w}}(u), & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}{\mathcal{G}}_{\mathbf{w}}(u)=\mathop{\sum }_{f_{1},\ldots ,f_{\ell -1}}{\mathcal{I}}_{\infty }(f_{1}\cdots f_{\ell -1})\unicode[STIX]{x1D712}_{\mathbf{w}}(F_{\mathbf{v}_{0}})u^{\deg (f_{1}\ldots f_{\ell -1})}.\end{eqnarray}$$
5.2 Euler product
Recall that a non-zero multivariate function
$\unicode[STIX]{x1D713}$
is called multiplicative if
$$\begin{eqnarray}\unicode[STIX]{x1D713}(a_{1}b_{1},\ldots ,a_{r}b_{r})=\unicode[STIX]{x1D713}(a_{1},\ldots ,a_{r})\unicode[STIX]{x1D713}(b_{1},\ldots ,b_{r})\end{eqnarray}$$
whenever
$\gcd (a_{1}\cdots a_{r},b_{1}\cdots b_{r})=1$
. These functions are determined by their values on
$(p^{\unicode[STIX]{x1D6FC}_{1}},p^{\unicode[STIX]{x1D6FC}_{2}},\ldots ,p^{\unicode[STIX]{x1D6FC}_{r}})$
and behave as multiplicative functions in one variable. We assume that the reader is familiar with the standard properties of multiplicative functions, or at least that the reader may complete the details of how those are derived from the one variable case; if this is not the case, one may consult the survey paper [Reference Tóth16].
Proposition 5.3. As usual, let
$K=\mathbb{F}_{q}(X)=\mathbb{F}_{q}(\mathbb{P}_{\mathbb{F}_{q}}^{1})$
and
$K^{\prime }=\mathbb{F}_{q^{n_{q}}}(X)$
. For each prime
$P$
of
$K$
we fix a prime
$\mathfrak{P}$
of
$K^{\prime }$
lying above
$P$
. Then we have the Euler decomposition
$$\begin{eqnarray}\displaystyle {\mathcal{G}}_{\mathbf{w}}(u)=\mathop{\prod }_{\substack{ P \\ n_{q}|\deg (P)}}\biggl(1+\mathop{\sum }_{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})}\biggr), & & \displaystyle\end{eqnarray}$$
with
${\mathcal{G}}_{\mathbf{w}}$
and
$\unicode[STIX]{x1D712}_{\mathbf{w}}$
as defined in (5.5) and (5.3).
Remark 5.4. If
$\mathfrak{P}$
and
$\mathfrak{P}^{\prime }$
are two primes over
$P$
, then
$\unicode[STIX]{x1D719}^{k}(\mathfrak{P})=\mathfrak{P}^{\prime }$
for some
$k$
, and so by (2.3), the inner sum in (5.6) is equal to the same sum with
$\mathfrak{P}^{\prime }$
replacing
$\mathfrak{P}$
.
Proof. The function
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(f_{1},\ldots ,f_{\ell -1})={\mathcal{I}}_{\infty }(f_{1}\cdots f_{\ell -1})\unicode[STIX]{x1D712}_{\mathbf{w}}(F_{\mathbf{v}_{0}})\end{eqnarray}$$
is multiplicative, hence to compute the Euler decomposition of
${\mathcal{G}}_{\mathbf{w}}(u)$
, it suffices to evaluate
$\unicode[STIX]{x1D6F9}$
on
$$\begin{eqnarray}(P^{\unicode[STIX]{x1D708}_{1}},P^{\unicode[STIX]{x1D708}_{2}},\ldots ,P^{\unicode[STIX]{x1D708}_{\ell -1}}),\end{eqnarray}$$
with
$P$
a prime polynomial in
$K$
and
$\unicode[STIX]{x1D708}_{i}$
non-negative integers.
Since
${\mathcal{I}}_{\infty }(P^{\unicode[STIX]{x1D708}_{1}},P^{\unicode[STIX]{x1D708}_{2}},\ldots ,P^{\unicode[STIX]{x1D708}_{\ell -1}})=1$
only when
$(\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{\ell -1})=(0,\ldots ,0,1,0,\ldots ,0)$
and
$n_{q}\mid \deg (P)$
(and
$=0$
otherwise), it suffices to evaluate
$\unicode[STIX]{x1D6F9}$
on
$$\begin{eqnarray}e_{i}(P)=(1,\ldots ,1,P,1,\ldots ,1)\end{eqnarray}$$
with
$P$
appearing in the
$i$
th place and
$P$
a prime polynomial in
$K$
.
By Corollary 2.10,
$P$
has the
$\unicode[STIX]{x1D707}_{\ell }$
-stable factorization
$$\begin{eqnarray}P=\mathfrak{P}_{1}\cdots \mathfrak{P}_{n_{q}},\end{eqnarray}$$
in which we may assume without loss of generality that
$\mathfrak{P}_{1}=\mathfrak{P}$
. We note that as
$P$
is prime in
$K$
, the
$\mathfrak{P}_{i}$
are prime in
$K^{\prime }$
. Then
$F=P^{i}$
,
$F_{j}=\mathfrak{P}_{j}^{i}$
, and
$$\begin{eqnarray}F_{\mathbf{v}_{0}}=\mathop{\prod }_{j=1}^{n_{q}}\mathfrak{P}_{j}^{iv_{j}},\end{eqnarray}$$
with
$v_{j}\equiv q^{1-j}~\text{mod}~\ell$
. By (2.3),
$\mathfrak{P}_{j}^{iv_{j}}(x_{r})$
is equal to
$\mathfrak{P}(x_{i})^{i}$
up to
$\ell$
th powers. As
$\unicode[STIX]{x1D712}_{\mathbf{w}}$
is defined modulo
$\prod (X-x_{i})$
and is trivial on
$\ell$
th powers, we conclude that
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(e_{i}(P))=\unicode[STIX]{x1D712}_{\mathbf{w}}(F_{\mathbf{v}_{0}})=\unicode[STIX]{x1D712}_{\mathbf{w}}^{in_{q}}(\mathfrak{P}).\end{eqnarray}$$
Thus we get the following Euler decomposition:
$$\begin{eqnarray}{\mathcal{G}}_{\mathbf{w}}(u)=\mathop{\prod }_{\substack{ P \\ n_{q}|\deg (P)}}\biggl(1+\mathop{\sum }_{i=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{in_{q}}(\mathfrak{P})u^{\deg (P)}\biggr).\end{eqnarray}$$
By Lemma 2.9,
$\deg (P)=n_{q}\deg (\mathfrak{P})$
. Thus, since
$(n_{q},\ell )=1$
, we have that
$\sum _{i=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{in_{q}}=\sum _{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}$
, and the proof is complete.◻
5.3 Analytic continuation
For a non-trivial character
$\unicode[STIX]{x1D712}$
of
$\mathbb{F}_{q^{n_{q}}}[X]\subseteq K^{\prime }$
we define
$$\begin{eqnarray}L_{K^{\prime }}(u,\unicode[STIX]{x1D712})=\mathop{\prod }_{\mathfrak{P}}(1-\unicode[STIX]{x1D712}(\mathfrak{P})u^{\deg (\mathfrak{P})})^{-1},\end{eqnarray}$$
where the product is over all prime polynomials in
$K^{\prime }$
. Then,
$L_{K^{\prime }}(u,\unicode[STIX]{x1D712})$
is a polynomial (due to the orthogonality relations) and its zeros lie on the circle
$|u|=q^{-n_{q}/2}$
(due to the Riemann Hypothesis for curves). We shall use repeatedly that a product
$$\begin{eqnarray}\mathop{\prod }_{P}(1+O(u^{c\deg P}))\end{eqnarray}$$
over prime polynomials absolutely converges in the disk
$|u|<q^{-1/c}$
.
Proposition 5.5. For any
$\mathbf{w}$
, there exists a non-vanishing function
$H_{\mathbf{w}}(u)$
which is analytic in the open disk
$|u|<q^{-1/2}$
such that:
(1) if
$\mathbf{w}\neq \vec{0}$
, then $$\begin{eqnarray}{\mathcal{G}}_{\mathbf{w}}(u)=\mathop{\prod }_{j\in (\mathbb{Z}/\ell \mathbb{Z})^{\ast }/\langle q\rangle }\biggl(\frac{L_{K^{\prime }}(u^{n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{j})}{L_{K^{\prime }}(u^{2n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{2j})}\biggr)H_{\mathbf{w}}(u);\end{eqnarray}$$
(2) if
$\mathbf{w}=\vec{0}$
, then and$$\begin{eqnarray}{\mathcal{G}}_{\vec{0}}(u)=\mathop{\prod }_{j=0}^{n_{q}-1}(1-\unicode[STIX]{x1D701}_{n_{q}}^{j}qu)^{-(\ell -1)/n_{q}}H_{\vec{0}}(u),\end{eqnarray}$$
$H_{\vec{0}}(\unicode[STIX]{x1D701}_{n_{q}}u)=H_{\vec{0}}(u)$
.
In particular,
(3)
${\mathcal{G}}_{\mathbf{w}}(u)$
has a meromorphic continuation to
$|u|<q^{-1/2}$
which is analytic if
$\mathbf{w}\neq \vec{0}$
and has poles of order
$(\ell -1)/n_{q}$
at
$u=(\unicode[STIX]{x1D701}_{n_{q}}^{j}q)^{-1}$
,
$j=0,\ldots ,n_{q}-1$
if
$\mathbf{w}=\vec{0}$
.
Proof. By the Riemann Hypothesis, the poles and zeros of
$$\begin{eqnarray}\frac{L_{K^{\prime }}(u^{n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{j})}{L_{K^{\prime }}(u^{2n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{2j})}\end{eqnarray}$$
lie on the circles
$|u|=q^{-1/4}$
and
$|u|=q^{-1/2}$
, respectively, hence it has an analytic
$n_{q}$
th root in the open disk
$|u|<q^{-1/2}$
and so (3) follows from (1) and (2).
Now we prove (1). Since each irreducible
$P$
in
$\mathbb{F}_{q}[X]$
with
$n_{q}\mid \deg (P)$
has
$n_{q}$
prime divisors in
$\mathbb{F}_{q^{n_{q}}}[X]$
(Lemma 2.9) we get by (5.6) applied to all the primes dividing
$P$
that
$$\begin{eqnarray}{\mathcal{G}}_{\mathbf{w}}(u)^{n_{q}}=\mathop{\prod }_{\substack{ P \\ n_{q}\mid \deg (P)}}\mathop{\prod }_{\mathfrak{P}\mid P}\biggl(1+\mathop{\sum }_{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})}\biggr).\end{eqnarray}$$
We take a
$n_{q}$
th root and break this product into two parts: denote
$$\begin{eqnarray}F_{1}(u)=\mathop{\prod }_{\substack{ P \\ n_{q}\nmid \deg (P)}}\mathop{\prod }_{\mathfrak{P}|P}\biggl(1+\mathop{\sum }_{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})}\biggr)^{-1/n_{q}}\end{eqnarray}$$
and
$$\begin{eqnarray}F_{2}(u)=\mathop{\prod }_{\mathfrak{P}}\biggl(1+\mathop{\sum }_{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})}\biggr)^{1/n_{q}}\end{eqnarray}$$
so
$$\begin{eqnarray}\displaystyle {\mathcal{G}}_{\mathbf{w}}(u)=F_{1}(u)F_{2}(u). & & \displaystyle\end{eqnarray}$$
We study each of the factors starting with
$F_{1}$
. By Lemma 2.9 and Remark 5.4 for
$|u|<q^{-1/2}$
,
$$\begin{eqnarray}F_{1}(u)=\mathop{\prod }_{\substack{ P \\ n_{q}\nmid \deg (P)}}\biggl(1+\mathop{\sum }_{j=1}^{\ell -1}\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{(n_{q}/(\deg (P),n_{q}))\deg (P)}\biggr)^{-(\deg (P),n_{q})/n_{q}}.\end{eqnarray}$$
Since for
$n_{q}\nmid \deg P$
we have
$$\begin{eqnarray}(1+\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{(n_{q}/(\deg (P),n_{q}))\deg (P)})^{-(\deg (P),n_{q})/n_{q}}=1+O(q^{2\deg P}),\end{eqnarray}$$
by (5.8), we get that
$F_{1}(u)$
is analytic in this open disk
$|u|<q^{-1/2}$
.
Next we study
$F_{2}$
. Writing the sum in the definition of
$F_{2}$
as a product (with suitable error term), we get
$$\begin{eqnarray}F_{2}(u)=\mathop{\prod }_{j=1}^{\ell -1}\mathop{\prod }_{\mathfrak{P}}(1+\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})})^{1/n_{q}}\mathop{\prod }_{\mathfrak{P}}(1+O(u^{2n_{q}\deg (\mathfrak{P})})).\end{eqnarray}$$
By (5.8), the right product absolutely converges in
$|u|<q^{-1/2}$
and by (5.7), and the fact that
$\unicode[STIX]{x1D712}_{\mathbf{w}}^{q}(\mathfrak{P})=\unicode[STIX]{x1D712}_{\mathbf{w}}(\mathfrak{P}^{\unicode[STIX]{x1D719}})$
,
$$\begin{eqnarray}\mathop{\prod }_{\mathfrak{P}}(1+\unicode[STIX]{x1D712}_{\mathbf{w}}^{j}(\mathfrak{P})u^{n_{q}\deg (\mathfrak{P})})=\frac{L_{K^{\prime }}(u^{n_{q}},\unicode[STIX]{x1D712}_{\boldsymbol{ w}}^{j})}{L_{K^{\prime }}(u^{2n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{2j})}=\frac{L_{K^{\prime }}(u^{n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{j^{\prime }})}{L_{K^{\prime }}(u^{2n_{q}},\unicode[STIX]{x1D712}_{\mathbf{w}}^{2j^{\prime }})},\end{eqnarray}$$
where
$j^{\prime }$
is any integer
$1\leqslant j^{\prime }\leqslant \ell -1$
such that
$j$
and
$j^{\prime }$
lie in the same
$\langle q\rangle$
-orbit in
$(\mathbb{Z}/\ell \mathbb{Z})^{\ast }$
. Since each
$\langle q\rangle$
-orbit has size
$n_{q}$
, together with the analyticity of
$F_{1}$
, with (5.10), and with (5.9) this finishes the proof of (1).
For (2), we use (5.6), to get that
$$\begin{eqnarray}{\mathcal{G}}_{\vec{0}}(u)=\mathop{\prod }_{\substack{ P \\ n_{q}|\deg (P)}}(1+(\ell -1)u^{\deg (P)}).\end{eqnarray}$$
By the orthogonality relations
$1_{\{n\equiv 0~\text{mod}~n_{q}\}}=(1/n_{q})\sum _{j=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{n_{q}}^{jn}$
we have
$$\begin{eqnarray}{\mathcal{G}}_{\vec{0}}(u)=\mathop{\prod }_{P}\biggl(1+\frac{\ell -1}{n_{q}}\mathop{\sum }_{j=0}^{n_{q}-1}\unicode[STIX]{x1D701}_{n_{q}}^{j\deg (P)}u^{\deg (P)}\biggr).\end{eqnarray}$$
Then we define
$H_{\vec{0}}(u)$
by the equation
$$\begin{eqnarray}\displaystyle {\mathcal{G}}_{\vec{0}}(u) & = & \displaystyle \mathop{\prod }_{j=0}^{n_{q}-1}\biggl(\mathop{\prod }_{P}(1-(\unicode[STIX]{x1D701}_{n_{q}}^{j}u)^{\deg (P)})\biggr)^{-(\ell -1)/n_{q}}H_{\vec{0}}(u)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{j=0}^{n_{q}-1}\biggl(\frac{1}{1-q\unicode[STIX]{x1D701}_{n_{q}}^{j}u}\biggr)^{(\ell -1)/n_{q}}H_{\vec{0}}(u).\nonumber\end{eqnarray}$$
As
$H_{\vec{0}}(u)=\prod _{P}(1+O(u^{2\deg (P)}))$
holds true, (5.8) implies that
$H_{\vec{0}}(u)$
is analytic in
$|u|<q^{-1/2}$
. Further,
$H_{\vec{0}}(\unicode[STIX]{x1D701}_{n_{q}}u)=H_{\vec{0}}(u)$
since both
${\mathcal{G}}_{\mathbf{w}}$
and
$\prod _{j=0}^{n_{q}-1}(1/(1-q\unicode[STIX]{x1D701}_{n_{q}}^{j}u))^{(\ell -1)/n_{q}}$
satisfy this relation.◻
5.4 Computing
$|{\mathcal{F}}_{n_{q}}^{k}(D)|$
We determine the size of the set
${\mathcal{F}}_{n_{q}}^{k}(D)$
using the residue theorem applied to the contour
$C_{\unicode[STIX]{x1D716}}=\{u:|u|=q^{-1/2-\unicode[STIX]{x1D716}}\}$
and to the function
${\mathcal{G}}_{k}(u)/u^{D+1}$
. By Proposition 5.5, by (5.2), and by (5.4) we have
$$\begin{eqnarray}\displaystyle \frac{1}{2\unicode[STIX]{x1D70B}i}\oint _{C_{\unicode[STIX]{x1D716}}}\frac{{\mathcal{G}}_{k}(u)}{u^{D+1}}\,du & = & \displaystyle \operatorname{Res}_{u=0}\biggl(\frac{{\mathcal{G}}_{k}(u)}{u^{D+1}}\biggr)+\mathop{\sum }_{j=0}^{n_{q}-1}\operatorname{Res}_{u=(\unicode[STIX]{x1D701}_{n_{q}}^{j}q)^{-1}}\biggl(\frac{{\mathcal{G}}_{k}(u)}{u^{D+1}}\biggr)\nonumber\\ \displaystyle & = & \displaystyle |{\mathcal{F}}_{n_{q}}^{k}(D)|+\frac{1}{\ell ^{k}}\mathop{\sum }_{j=0}^{n_{q}-1}\operatorname{Res}_{u=(\unicode[STIX]{x1D701}_{n_{q}}^{j}q)^{-1}}\biggl(\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}}\biggr).\end{eqnarray}$$
We compute each of the terms separately. Put
$N=(\ell -1)/n_{q}-1$
and
$u_{j}=(\unicode[STIX]{x1D701}_{n_{q}}^{j}q)^{-1}$
. Since
${\mathcal{G}}_{\vec{0}}$
has a pole at
$u=u_{j}$
of order
$N+1$
and using Proposition 5.5, we have
$$\begin{eqnarray}\displaystyle N!\cdot \operatorname{Res}_{u=u_{j}}\biggl(\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}}\biggr) & = & \displaystyle \frac{d^{N}}{du^{N}}\biggl[(u-u_{j})^{N+1}\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}}\biggr]\biggr|_{u=u_{j}}\nonumber\\ \displaystyle & = & \displaystyle \frac{d^{N}}{du^{N}}\mathop{\biggl[\frac{u_{j}^{N+1}}{u^{D+1}}\mathop{\prod }_{\substack{ i=0 \\ i\not =j}}^{n_{q}-1}u_{i}^{N+1}(u_{i}-u)^{-(N+1)}H_{\vec{0}}(u)\biggr]\biggr|}\nolimits_{u=u_{j}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{m=0}^{N}\binom{N}{m}\frac{d^{m}}{du^{m}}\frac{u_{j}^{N+1}}{u^{D+1}}\biggr|_{u=u_{j}}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{d^{N-m}}{du^{N-m}}\mathop{\biggl[\mathop{\prod }_{\substack{ i=0 \\ i\not =j}}^{n_{q}-1}\biggl(\frac{1}{1-u/u_{i}}\biggr)^{N+1}H_{\vec{0}}(u)\biggr]\biggr|}\nolimits_{u=u_{j}}\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{n_{q}}P(D)u_{j}^{-D}=-\frac{1}{n_{q}}P(D)q^{D}\unicode[STIX]{x1D701}_{n_{q}}^{jD},\end{eqnarray}$$
where
$P(D)$
is a polynomial of degree
$N$
which is independent of
$j$
.
Since
${\mathcal{G}}_{k}(u)$
is continuous on
$C_{\unicode[STIX]{x1D716}}$
, it is
$O(1)$
there, and we have
$$\begin{eqnarray}\biggl|\frac{1}{2\unicode[STIX]{x1D70B}i}\oint _{C_{\unicode[STIX]{x1D716}}}\frac{{\mathcal{G}}_{k}(u)}{u^{D+1}}\,du\biggr|=O(q^{(1/2+\unicode[STIX]{x1D716})D}).\end{eqnarray}$$
Plug this bound and (5.12) into (5.11) to get
$$\begin{eqnarray}|{\mathcal{F}}_{n_{q}}^{k}(D)|=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{\ell ^{k}}P(D)q^{D}+O(q^{(1/2+\unicode[STIX]{x1D716})D}),\quad & D\equiv 0~\text{mod}~n_{q},\\ 0,\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
We emphasize that the main term is independent of the actual choice of the values
$\unicode[STIX]{x1D716}_{1},\ldots ,\unicode[STIX]{x1D716}_{k}$
of the characters. The error term may depend on this choice.
6 Proof of Theorem 1.1
Fix
$D\equiv 0~\text{mod}~n_{q}$
. Let
$N\geqslant 0$
and
$\unicode[STIX]{x1D716}_{0},\unicode[STIX]{x1D716}_{1},\ldots ,\unicode[STIX]{x1D716}_{q}\in \unicode[STIX]{x1D707}_{\ell }$
such that
$$\begin{eqnarray}\mathop{\sum }_{i=0}^{q}\mathop{\sum }_{w=0}^{\ell -1}\unicode[STIX]{x1D716}_{i}^{w}=N.\end{eqnarray}$$
Conditioning on
$b$
, (5.13) gives that the probability
$\unicode[STIX]{x1D70B}$
that
$\unicode[STIX]{x1D712}_{x_{i}}(F_{\mathbf{v}_{0}})=\unicode[STIX]{x1D716}_{i}$
for
$i=1,\ldots ,q$
and
$\unicode[STIX]{x1D712}_{\ell }(b)=\unicode[STIX]{x1D716}_{0}$
is given by
$$\begin{eqnarray}\unicode[STIX]{x1D70B}=\frac{1}{\ell ^{q+1}}+O(q^{D(-1/2+\unicode[STIX]{x1D716})}).\end{eqnarray}$$
Thus, (4.4) implies that
$$\begin{eqnarray}\operatorname{Prob}(\#C_{F,b}(\mathbb{F}_{q})=N)=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D716}_{0},\ldots ,\unicode[STIX]{x1D716}_{q}\in \unicode[STIX]{x1D707}_{\ell } \\ \mathop{\sum }_{i}\mathop{\sum }_{w}\unicode[STIX]{x1D716}_{i}^{w}=N}}\frac{1}{\ell ^{q+1}}+O(q^{D(-1/2+\unicode[STIX]{x1D716})}).\end{eqnarray}$$
Putting
$\unicode[STIX]{x1D716}_{i}^{\prime }=\sum _{w=0}^{\ell -1}\unicode[STIX]{x1D716}_{i}^{w}$
, we have that
$\unicode[STIX]{x1D716}_{i}^{\prime }\in \{0,\ell \}$
. Moreover,
$\unicode[STIX]{x1D716}_{i}^{\prime }=0$
if and only if
$\unicode[STIX]{x1D716}_{i}\neq 1$
. This means that if
$\unicode[STIX]{x1D716}_{i}^{\prime }=0$
there are
$\ell -1$
different choices for
$\unicode[STIX]{x1D716}_{i}$
, whereas if
$\unicode[STIX]{x1D716}_{i}^{\prime }=\ell$
, then there is one choice:
$\unicode[STIX]{x1D716}_{i}=1$
. We plug this into the previous equation and get
$$\begin{eqnarray}\displaystyle \operatorname{Prob}(\#C_{F,b}(\mathbb{F}_{q})=N) & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D716}_{0}^{\prime },\ldots ,\unicode[STIX]{x1D716}_{q}^{\prime }\in \{0,\ell \} \\ \mathop{\sum }_{i}\unicode[STIX]{x1D716}_{i}^{\prime }=N}}\frac{(\ell -1)^{\#\{i:\unicode[STIX]{x1D716}_{i}^{\prime }=0\}}}{\ell ^{q}}+O(q^{D(-1/2+\unicode[STIX]{x1D716})})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ \unicode[STIX]{x1D716}_{0}^{\prime },\ldots ,\unicode[STIX]{x1D716}_{q}^{\prime }\in \{0,\ell \} \\ \mathop{\sum }_{i}\unicode[STIX]{x1D716}_{i}^{\prime }=N}}\biggl(\frac{\ell -1}{\ell }\biggr)^{\#\{i:\unicode[STIX]{x1D716}_{i}^{\prime }=0\}}\biggl(\frac{1}{\ell }\biggr)^{\#\{i:\unicode[STIX]{x1D716}_{i}^{\prime }=\ell \}}\nonumber\\ \displaystyle & & \displaystyle +\,O(q^{D(-1/2+\unicode[STIX]{x1D716})})\nonumber\\ \displaystyle & = & \displaystyle \operatorname{Prob}\biggl(\mathop{\sum }_{i=0}^{q}X_{i}=N\biggr),\nonumber\end{eqnarray}$$
with
$X_{i}$
as in the formulation of the theorem. Since by Proposition 3.3
$$\begin{eqnarray}\operatorname{Prob}(\#C_{F,b}(\mathbb{F}_{q})=N)=\operatorname{Prob}(\#C(\mathbb{F}_{q})=N)\end{eqnarray}$$
with
$C$
a random curve in
${\mathcal{H}}_{g,\ell }$
, this finishes the proof.◻
Acknowledgements
The authors thank Dan Haran for a useful conversation on wreath product actions on fields, Chantal David for crucial comments, and Will Sawin for finding a critical mistake in a previous version.
L.B.S. was partially supported by a grant of the Israel Science Foundation and by the Simons CRM Scholars program. P. M. has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement number 320755.
