# A Mass Formula For Artin–Schreier Curves Over Finite Fields

Anne M. Ho<EMAIL_ADDRESS>University of Tennessee at Knoxville, Department of
Mathematics, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996-1320  and
Rachel Pries<EMAIL_ADDRESS>Colorado State University, Department of
Mathematics, Weber 101, Fort Collins, CO 80523-1874

###### Abstract.

We study a mass formula for Artin–Schreier curves of genus $g$ defined over a
finite field $k$ of characteristic $p$. For an odd prime $p$ and for small
$g$, we determine the number of $k$-isomorphism classes of Artin–Schreier
curves of genus $g$, weighted by the order of the centralizer of the
Artin–Schreier involution in the automorphism group. This extends earlier
results by several authors in characteristic $p=2$.
Keywords: Artin–Schreier curve, finite field, automorphism, Mass formula,
moduli space, arithmetic statistics
MSC10: 11G20, 11T06, 12F10, 14G15, 14H10, 14H37, 62R01.

##  1\. Introduction

Lang and Weil [13] showed that the number of points of a variety over a finite
field gives a lot of information about its geometry. For a prime $p$, let
$k={\mathbb{F}}_{q}$ be a finite field whose cardinality $q$ is a power of
$p$. If $V$ is an irreducible projective algebraic variety of dimension $d$
over $k$, then $q^{d}$ is the dominant term for the number of $k$-points of
$V$. Recently, people developed an interest in the number of $k$-points on
moduli spaces of curves, because this yields information about the cohomology
of these moduli spaces; see, e.g. [21] for a survey of many results in this
area.

Suppose $C$ is a curve over $k$. We assume throughout this paper that $C$ is
smooth, projective, and geometrically irreducible. Let $g$ be the genus of $C$
and suppose that $g\geq 1$.

For small $g$, the number of isomorphism classes $[C]$ of curves $C$ of genus
$g$ over $k$ is known. This number is given by a mass formula, in which the
contribution from $[C]$ is weighted by $|{\rm Aut}_{k}(C)|^{-1}$, where ${\rm
Aut}_{k}(C)$ denotes the automorphism group of $C$ over $k$.

When $g=1$, an elliptic curve $E$ over $k$ defines a $k$-point of the moduli
space $\mathcal{M}_{1,1}$. For every prime power $q$, Howe proved that
$\sum_{[E]\in{\mathcal{M}}_{1,1}(k)/\simeq_{k}}|{\rm Aut}_{k}(E)|^{-1}=q$ [11,
Corollary 2.2].

This was generalized for hyperelliptic curves of arbitrary genus. A
hyperelliptic curve of genus $g$ over $k$ defines a $k$-point of the moduli
space ${\mathcal{H}}_{g}$. As in [1, Definition 2.2], let:

$a({\mathcal{H}}_{g})_{0}\mid_{g}=\sum_{[C]\in{\mathcal{H}}_{g}(k)/\simeq_{k}}|{\rm
Aut}_{k}(C)|^{-1}.$ (1)

Brock and Granville proved that $a({\mathcal{H}}_{g})_{0}\mid_{g}=q^{2g-1}$
when $q$ is odd, [2, Proposition 7.1]. Bergström proved the same formula is
true when $q$ is even, [1, Theorem 10.3].

For $q$ even and $g=2,3$, the fact that
$a({\mathcal{H}}_{g})_{0}\mid_{g}=q^{2g-1}$ was proven earlier: when $g=2$ by
[6, Theorem 18], building on [9, Corollary 5.3]; and when $g=3$ by [17,
Theorem 8].

In characteristic $p=2$, hyperelliptic curves are Artin–Schreier curves by
definition. A curve $C$ is an Artin–Schreier curve if there is a Galois cover
$\pi:C\rightarrow\mathbb{P}^{1}$ of degree $p$. In this paper, we extend the
results about hyperelliptic curves in characteristic $2$ to the case of
Artin–Schreier curves of low genus in arbitrary characteristic $p$. To do
this, we consider the moduli space ${\mathcal{A}S}_{g}$ of Artin–Schreier
curves of genus $g$ as in [16, Proposition 2.7].

We now state the results more precisely. An Artin–Schreier curve $C$ over $k$
has an equation of the form $y^{p}-y=u(x)$ for some rational function $u(x)\in
k(x)$. If $C$ is an Artin–Schreier curve, then the genus of $C$ is a multiple
of $(p-1)/2$ by the wild Riemann-Hurwitz formula. Let $\iota$ denote the order
$p$ automorphism $\iota:=(x,y)\mapsto(x,y+1)$, which generates the Galois
group of $\pi$. Let $\rm{Cent}_{{\rm Aut}_{k}(C)}\langle\iota\rangle$ be the
centralizer of $\langle\iota\rangle$ in ${\rm Aut}_{k}(C)$. The reason for
introducing the centralizer is that $\langle\iota\rangle$ may not be in the
center of ${\rm Aut}_{k}(C)$ when $p$ is odd.

We consider the mass formula:

$Z_{g}(q):=a({\mathcal{A}S}_{g})_{0}\mid_{g}:=\sum_{[C]\in{\mathcal{A}S}_{g}(k)/\simeq_{k}}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1},$ (2)

where $[C]=(C,\iota)$ ranges over the $k$-isomorphism classes of
Artin–Schreier curves with automorphism such that $C$ has genus $g$. Note that
(2) specializes to (1) when $p=2$ since $\iota$ is in the center of ${\rm
Aut}_{k}(C)$ in that case.

The main result of this paper is Theorem 5.3 (see also Theorem 5.2). In it, we
determine the mass formula $Z_{g}(q)$ for an arbitrary odd prime power $q$
when the genus is $g=d(p-1)/2$ for $1\leq d\leq 5$. For $p\geq 7$, the result
states that

$Z_{g}(q)=\begin{cases}1&\rm{if}\ g=1(p-1)/2,\\\ 2q-1&\rm{if}\ g=2(p-1)/2,\\\
2q^{2}-q&\rm{if}\ g=3(p-1)/2,\\\ 4q^{3}-3q^{2}&\rm{if}\ g=4(p-1)/2,\\\
4q^{4}-4q^{3}+q^{2}&\rm{if}\ g=5(p-1)/2.\\\ \end{cases}$

These formulas have geometric significance, as explained in Section 2.6. The
degree of $Z_{g}(q)$ as a polynomial in $q$ is the dimension of
${\mathcal{A}S}_{g}$; if $g=d(p-1)/2$, every irreducible component of
${\mathcal{A}S}_{g}$ has dimension $d-1$ [16, Corollary 3.16]. The leading
coefficient of $Z_{g}(q)$ is the number of irreducible components of
${\mathcal{A}S}_{g}$; this number is determined in [18]. One interesting fact
is that the number of irreducible components is sometimes different for small
primes $p$ and this is reflected in the formulas we find. The lower order
terms of our formulas provide new information about the cohomology of
${\mathcal{A}S}_{g}$.

The proof of Theorem 5.3 builds on Theorem 4.7, in which we re-express
$Z_{g}(q)$ as an average over conjugacy classes in a symmetric group.

To obtain the formulas, we use similar methods as in [6] and [17]. We separate
into cases indexed by discrete information for the cover
$\pi:C\to{\mathbb{P}}^{1}$, namely the ramification data and splitting
behavior. In each case, we study the orbits and stabilizers for the action of
$\rm{PGL}_{2}(k)$ on the Artin–Schreier equations. In Proposition 5.1, we
determine the mass formula for the set of Artin–Schreier curves of arbitrary
genus $g$ over $k$ when the number of branch points of $\pi$ is at most $3$.
This allows us to finish the proof of Theorem 5.3.

We also have some results when $\pi$ has $4$ branch points. This case is more
difficult because it is more complicated to study the group action on covers
branched at $4$-sets of $\mathbb{P}^{1}(\bar{k})$. We count the number of
orbits of $4$-sets in Section 6. This leads to some concluding formulas in
Section 7.1

######  Remark 1.1.

Most of the material in this paper appeared in the thesis of Ho in 2015 but
was not published before. We finished this paper after hearing several people
express interest in moduli spaces of curves with wildly ramified
automorphisms. It is possible that this paper will shed light on the rational
points and cohomology for $B\mu_{p}/{\mathbb{Z}}$, whose fiber over $p$ is not
a Deligne-Mumford stack. See [12] and [7, Section 4.1] for a stacky
perspective on this.

######  Remark 1.2.

Other perspectives on the arithmetic statistics of Artin–Schreier curves can
be found in recent work. For example, the papers [3], [4], and [8] are about
the distribution of the zeros of the $L$-functions. In [5], [19], the authors
study the question of whether a randomly chosen curve is ordinary, or more
generally whether the distribution of the $p$-divisible groups of curves
matches that of the $p$-divisible groups of Dieudonné modules.

We would like to thank Jeff Achter, Jeremy Booher, Bryden Cais, and Beth
Malmskog for many inspiring conversations about Artin–Schreier curves. Pries
was supported by NSF grants DMS-15-02227 and DMS-19-01819.

##  2\. Background on ramification data and splitting behavior

We first clarify notation about automorphisms and isomorphisms in Sections
2.1-2.3. In Sections 2.4 and 2.5, we define the ramification data and
splitting behavior of Artin–Schreier covers. We review geometric results about
the moduli space ${\mathcal{A}S}_{g}$ in Section 2.6. This provides some
context for earlier mass formula results of [6, 17] in characteristic $p=2$,
which we summarize in Section 2.7.

Let $k$ be a finite field of cardinality $q=p^{n}$ and let $\bar{k}$ be an
algebraic closure of $k$. Suppose $\pi:C\to{\mathbb{P}}^{1}$ is an
Artin–Schreier cover over $k$ with affine equation $y^{p}-y=u(x)$ for some
$u(x)\in k(x)$.

###  2.1. Automorphisms of the projective line

We describe $\gamma\in\textup{Aut}_{k}(\mathbb{P}^{1})\simeq{\rm PGL}_{2}(k)$
using fractional linear transformations: for $a,b,c,d\in k$ with $ad-
bc\not=0$, write

$\displaystyle\gamma(x)=\frac{ax+b}{cx+d}.$ (3)

The action of $\gamma$ on $u(x)\in k(x)$ is given by $\gamma\cdot
u(x)=u(\gamma(x))$.

We identify the symmetric group $S_{3}$ with the set of $6$ fractional linear
transformations which stabilize $\\{0,1,\infty\\}$. The orbit of
$x\in\overline{k}-\\{0,1\\}$ under the action of $\sigma\in S_{3}$ is given by

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr\sigma&{\rm
id}&(1,2)&(2,3)&(1,3)&(1,2,3)&(1,3,2)\\\
\hline\cr\sigma(x)&x&1/x&1-x&x/(x-1)&1/(1-x)&(x-1)/x\\\ \hline\cr\end{array}.$

The following result is well-known.

######  Lemma 2.1.

The orbits of $x\in\overline{k}-\\{0,1\\}$ under the $S_{3}$-action each have
size $6$ except:

  1. (1)

if $p=2$, there is one orbit of size $2$, namely
$\\{\zeta_{3},\zeta_{3}^{2}\\}$;

  2. (2)

if $p=3$, there is one orbit of size $1$, namely $\\{-1\\}$;

  3. (3)

if $p\geq 5$, there is one orbit of size $3$, namely $\\{-1,2,1/2\\}$, and one
orbit of size $2$, namely $\\{\zeta_{6},\zeta_{6}^{5}\\}$.

For $t\in k-\\{0,1\\}$, write $B_{t}=\\{0,1,\infty,t\\}$. Let $\Gamma_{t}$ be
the stabilizer of $B_{t}$ under the ${\rm PGL}_{2}(k)$-action. Note
$\Gamma_{t}\hookrightarrow{\rm Sym}(B_{t})\simeq S_{4}$.

Define $\gamma_{1,t},\gamma_{2,t}\in{\rm PGL}_{2}(k)$ by $\gamma_{1,t}(x)=t/x$
and $\gamma_{2,t}(x)=(x-t)/(x-1)$. Then $\gamma_{1,t}$ and $\gamma_{2,t}$
commute, have order 2, and stabilize $B_{t}$. As permutations of $B_{t}$, we
see that $\gamma_{1,t}=(0,\infty)(1,t)$ and $\gamma_{2,t}=(0,t)(1,\infty)$.
Let $K=\langle\gamma_{1,t},\gamma_{2,t}\rangle\simeq C_{2}\times C_{2}$.

Let

$\binom{-3}{q}=\begin{cases}1&\text{ if }q\equiv 1\bmod 3\\\ -1&\text{ if
}q\equiv-1\bmod 3\\\ 0&\text{ if }3\mid q.\end{cases}$

The next lemma can essentially be found in [22, Table 1].

######  Lemma 2.2.

If $t\in k-\\{0,1\\}$, then $\Gamma_{t}=K$ except in the following cases.

  1. (1)

If $p=2$ and $t\in\\{\zeta_{3},\zeta_{3}^{2}\\}$, then $\Gamma_{t}\simeq
A_{4}$.

The number of orbits with $\Gamma_{t}=K$ is $(q-3+\binom{-3}{q})/6$.

  2. (2)

If $p=3$ and $t=-1$, then $\Gamma_{t}\simeq S_{4}$. The number of orbits with
$\Gamma_{t}=K$ is $(q-3)/6$.

  3. (3)

If $p\geq 5$ and $t\in\\{-1,2,1/2\\}$, then $\Gamma_{t}\simeq D_{4}$. If
$p\geq 5$ and $t\in\\{\zeta_{6},\zeta_{6}^{5}\\}$, then $\Gamma_{t}\simeq
A_{4}$.

The number of orbits with $\Gamma_{t}=K$ is $(q-6-\binom{-3}{q})/6$.

###### Proof.

It is clear that $K\subset\Gamma_{t}$. If $\Gamma_{t}\not=K$, then
$\Gamma_{t}$ is isomorphic to $D_{4}$, $A_{4}$, or $S_{4}$. This implies that
some $\gamma\in\Gamma_{t}$ fixes an element of $B_{t}$. Without loss of
generality, we can suppose that $\gamma$ fixes $t$ and stabilizes
$\\{0,1,\infty\\}$. This is only possible if $t$ is one of the special values
from Lemma 2.1 and that result also shows the following.

If $p=2$ and $t\in\\{\zeta_{3},\zeta_{3}^{2}\\}$, then $\Gamma_{t}$ contains a
$3$-cycle and has index $2$ in ${\rm Sym}(B_{t})$, and so $\Gamma_{t}\simeq
A_{4}$. The same is true if $p\geq 5$ and $t=\zeta_{6},\zeta_{6}^{5}$.

If $p=3$ and $t=-1$, then $\Gamma_{t}$ contains ${\rm Sym}(\\{0,1,\infty\\})$
in ${\rm Sym}(B_{t})$, and so $\Gamma_{t}\simeq S_{4}$.

If $p\geq 5$ and $t\in\\{-1,2,1/2\\}$ then $\Gamma_{t}$ contains a $2$-cycle
and has index $3$ in ${\rm Sym}(B_{t})$, and so $\Gamma_{t}\simeq D_{4}$. ∎

###  2.2. Isomorphisms

There are different kinds of isomorphisms between Artin–Schreier objects and
we clarify our definitions of these. Define ${\rm AS}(k(x)):=\\{z^{p}-z\mid
z\in k(x)\\}$. For $i=1,2$, suppose $\pi_{i}:C_{i}\to{\mathbb{P}}^{1}$ is
given by the equation $y_{i}^{p}-y_{i}=u_{i}(x)$ with the automorphism
$\iota_{i}:(x,y_{i})\mapsto(x,y_{i}+1)$.

Artin–Schreier covers: We say that $\pi_{1}$ and $\pi_{2}$ are isomorphic over
$k$ as Artin–Schreier covers if there exists an automorphism $\varphi:C_{1}\to
C_{2}$ over $k$ such that $\pi_{2}\circ\varphi=\pi_{1}$. This is equivalent to
the condition $u_{2}(x)-u_{1}(x)=z^{p}-z\in{\rm AS}(k(x))$ for some $z\in
k(x)$; in this case, the isomorphism is given by the change of variables
$y_{2}=y_{1}+z$. In particular, $C$ is geometrically irreducible if and only
if $u(x)\notin{\rm AS}(k(x))$.

Artin–Schreier curves with automorphism: We say that the pairs
$(C_{1},\iota_{1})$ and $(C_{2},\iota_{2})$ are isomorphic over $k$ as
Artin–Schreier curves with automorphism if there exists an isomorphism
$\varphi:C_{1}\to C_{2}$ over $k$ such that
$\iota_{2}\circ\varphi=\varphi\circ\iota_{1}$. This is true if and only if
$\varphi$ descends to the quotient ${\mathbb{P}}^{1}$, meaning that there
exists $\gamma\in{\rm Aut}_{k}({\mathbb{P}}^{1})$ such that
$\pi_{2}\circ\varphi=\gamma\circ\pi_{1}$. Viewing $\gamma$ as a change of
coordinates on the variable $x$, then $(C_{1},\iota_{1})$ and
$(C_{2},\iota_{2})$ are isomorphic over $k$ if and only if
$u_{2}(x)\equiv\gamma\cdot u_{1}(x)\bmod{\rm AS}(k(x))$.

In particular, an automorphism of an Artin–Schreier curve $(C,\iota)$ with
automorphism is an isomorphism $\varphi:C\to C$ such that
$\iota\circ\varphi=\varphi\circ\iota$ or, equivalently,
$\varphi\in\rm{Cent}_{{\rm Aut}_{k}(C)}\langle\iota\rangle$.

An Artin–Schreier curve with automorphism determines an Artin–Schreier cover
$\pi:C\to{\mathbb{P}}^{1}$, but this is only well-defined up to a change of
variables on ${\mathbb{P}}^{1}$.

Artin–Schreier curves: We say that $C_{1}$ and $C_{2}$ are isomorphic over $k$
as Artin–Schreier curves if there exists an isomorphism $\varphi:C_{1}\to
C_{2}$ over $k$. This is the case if there exists $c\in{\mathbb{F}}_{p}^{*}$
and $\gamma\in{\rm Aut}_{k}({\mathbb{P}}^{1})$ such that $u_{2}(x)\equiv
c\gamma\cdot u_{1}(x)\bmod{\rm AS}(k(x))$.

For example, if $u_{1}(x)=x^{e}$ and $u_{2}(x)=cx^{e}$, then $C_{1}$ and
$C_{2}$ are isomorphic over $k$ as Artin–Schreier curves, where $\varphi$
identifies $y_{2}=cy_{1}$; but not as Artin–Schreier curves with automorphism
since $\varphi$ identifies $\iota_{2}=\iota_{1}^{c}$.

###  2.3. Remarks

Suppose $C$ is an Artin–Schreier curve.

When $p=2$, there is a unique choice for $\iota$, namely, the hyperelliptic
involution of $C$; by [10, Theorem 11.98], $\iota$ is in the center of ${\rm
Aut}_{k}(C)$. In other words, $\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle={\rm Aut}_{k}(C)$. Also, the data of an
Artin–Schreier curve is equivalent to the data of an Artin–Schreier curve with
automorphism. So the mass formula $Z_{g}(q)$ in (2) specializes to (1) when
$p=2$.

When $p$ is odd, there are two subtleties with the definition of the mass
formula. The first is that we weight by $\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle$; this is because some automorphisms of $C$
may not descend to the quotient projective line, as seen in the following
result.

######  Lemma 2.3.

[10, Theorem 11.93] Suppose $p$ is odd and $g>1$. Then ${\rm Aut}_{k}(C)$ is
an extension of $\langle\iota\rangle$ by a finite group of $k$-automorphisms
of $k(x)$ except in the following cases:

  1. (1)

$y^{p}-y=a/(x^{p}-x)$ for $a\in k^{*}$,

  2. (2)

$y^{3}-y=b/x(x-1)$ with $b^{2}=2$, or

  3. (3)

$y^{p}-y=1/x^{c}$ with $c\mid(p+1)$.

The second subtlety is that we consider the weighted sum over isomorphism
classes of Artin–Schreier curves with automorphism. The data of the
automorphism is useful for defining the ramification data as in Section 2.4.
Note that each Artin–Schreier curve has only finitely many possible choices
for $\iota$. So the geometric facts in Section 2.6 about the dimension of
${\mathcal{A}S}_{g}$ and the number of its components apply equally well in
the context of Artin–Schreier curves with automorphism.

###  2.4. Ramification Data

If $u(x)$ has a pole at a point $Q\in{\mathbb{P}}^{1}(\bar{k})$, let $e_{Q}$
be the order of the pole and $\epsilon_{Q}=e_{Q}+1$. Without loss of
generality, we can assume that $p\nmid e_{Q}$; this may require a change of
coordinates of the form $y\mapsto y+z$ and uses the fact that every element of
${\mathbb{F}}_{q}$ is a $p$th power. Then the branch locus of $\pi$ is the set
of poles of $u(x)$. We denote by $D$ the degree of the ramification divisor of
$\pi$.

######  Proposition 2.4.

(Wild Riemann-Hurwitz formula) see [20, Proposition 3.7.8] Let $\pi$ be an
Artin–Schreier cover with equation $y^{p}-y=u(x)$ where $u(x)\in
k(x)-\textup{AS}(k(x))$. Suppose ${\rm div}_{\infty}(u(x))=\sum e_{Q}Q$ where
each $e_{Q}$ is a positive prime-to-integer. Then

$2g=-2(p-1)+D,\ {\rm where}\ D=\sum(e_{Q}+1)(p-1).$

Let $r$ be the number of poles of $u(x)$. We label the branch points of $\pi$
by $Q_{1},\ldots,Q_{r}$. Let $e_{i}$ be the order of the pole of $u(x)$ at
$Q_{i}$ and $\epsilon_{i}=e_{i}+1$. The ramification data of $\pi$ is the
tuple $\vec{\epsilon}=(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{r})$;
without the labeling of the branch points, it can be viewed as a multiset
$R=\\{\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{r}\\}$.

For a fixed genus $g$, there can be several possibilities for the ramification
data $R$. By Proposition 2.4, these correspond to partitions of $2+2g/(p-1)$
into numbers $\epsilon_{i}\not\equiv 1\bmod p$.

######  Example 2.5.

If $g=2(p-1)$, then the possibilities for $R$ are $\\{6\\}$ (if $p\not=5$),
$\\{4,2\\}$ (if $p\not=3$), $\\{3,3\\}$ (if $p\not=2$), and $\\{2,2,2\\}$.

###  2.5. Splitting Behavior

Let ${\rm Fr}(\alpha)=\alpha^{q}$ denote the absolute Frobenius of $\bar{k}$
over $k$. If $u(x)\in k(x)$, the poles of $u(x)$ are not necessarily in $k$;
it is possible that the set of poles includes an orbit under Frobenius of
points defined over a finite extension of $k$. In this case, the orders of the
poles of $u(x)$ at these points are the same.

The splitting behavior $S$ is the data of the fields of definition of the
poles of $u(x)$. The branch locus is split if all the poles of $u(x)$ are in
$k$. If not, we need some notation to keep track of the field of definition of
the branch points. For example, when $r=2$, we denote the split case by
$(\epsilon,\epsilon)$ and the non-split case by $((\epsilon-\epsilon))$. A
point of degree $m$ is the Frobenius orbit of $m$ distinct points defined over
${\mathbb{F}}_{q^{m}}$; to denote this, we replace the $m$ values of
$\epsilon$ in the ramification data by $(\epsilon-\cdots-\epsilon)$.

Let $s$ be the number of orbits under Frobenius of the set of poles of $u(x)$.
Let $m_{1},\ldots,m_{s}$ denote the degrees of representatives of points in
the orbits; then $\sum_{i=1}^{s}m_{i}=r$.

######  Example 2.6.

For $3$ branch points with the same pole order, the splitting behavior can be:
(i) $(\epsilon,\epsilon,\epsilon)$, meaning 3 points of degree $1$;
(ii) $(\epsilon,(\epsilon-\epsilon))$, meaning $1$ point of degree $1$ and $1$
point of degree $2$;
(iii) or $((\epsilon-\epsilon-\epsilon))$ meaning $1$ point of degree $3$.

###  2.6. Dimension of $p$-rank strata of moduli space

Let $d\geq 1$. Let ${\mathcal{A}S}_{g}$ denote the moduli space of
Artin–Schreier curves of genus $g=d(p-1)/2$. There are geometric results about
${\mathcal{A}S}_{g}$; for example, the number of its irreducible components
and their dimensions are known.

The moduli space ${\mathcal{A}S}_{g}$ can be stratified by the number of
branch points $r$ of the Artin–Schreier cover or, equivalently, by the
$p$-rank. Here the $p$-rank of $C$ is the integer $\sigma$ such that
$p^{\sigma}$ is the number of $p$-torsion points on ${\rm Jac}(C)$;
equivalently, it is the number of slopes of $0$ in the Newton polygon of $C$.
By the Deuring-Shafarevich formula, $\sigma=(r-1)(p-1)$.

Consider the weighted count $Z_{g}(q):=a({\mathcal{A}S}_{g})_{0}\mid_{g}$ as
in (2). If $Z_{g}(q)$ is a polynomial in $q$, then its degree is the dimension
of ${\mathcal{A}S}_{g}$ and its leading coefficient is the number of
irreducible components of ${\mathcal{A}S}_{g}$ having that dimension.

######  Theorem 2.7.

[18, Theorem 1.1] Let $g=d(p-1)/2$ and $\sigma=(r-1)(p-1)$ for $d,r\geq 1$.
Let $\mathcal{AS}_{g,\sigma}$ denote the $p$-rank $\sigma$ stratum of the
moduli space of Artin–Schreier curves of genus $g$.

  1. (1)

The set of irreducible components of $\mathcal{AS}_{g,\sigma}$ is in bijection
with the set of partitions
$R=\\{\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{r}\\}$ of $d+2$ into $r$
positive integers such that each $\epsilon_{i}\not\equiv 1\bmod p$.

  2. (2)

The irreducible component of $\mathcal{AS}_{g,\sigma}$ for the partition
$R=\\{\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{r}\\}$ has dimension

$\delta_{R}=d-1-\sum_{i=1}^{r}\lfloor\frac{\epsilon_{i}-1}{p}\rfloor.$

######  Remark 2.8.

In particular, $C$ is ordinary (meaning that $\sigma=g$) if and only if
$R=\\{2,\ldots,2\\}$. It follows directly from Theorem 2.7 that the ordinary
locus is not open and dense in $\mathcal{AS}_{g}$ if $p\geq 3$ and $d\geq 2$
(with $d\geq 6$ if $p=3$). See [19, Corollary 3.5] for another perspective on
this.

######  Example 2.9.

If $g=2(p-1)$, then $d=4$ and here are the dimensions for each partition:

$R$ | $\\{2,2,2\\}$ | $\\{3,3\\},\ p\not=2$ | $\\{4,2\\},\ p\not=3$ | $\\{6\\},\ p\not=5$
---|---|---|---|---
$\delta_{R}$ | $3$ | $3$ | $3\ {\rm if}\ p\geq 5$ | $3\ {\rm if}\ p\geq 7$
|  |  | $2\ {\rm if}\ p=2$ | $2\ {\rm if}\ p=3$
|  |  |  | $1\ {\rm if}\ p=2$

###  2.7. Earlier Results when $p=2$

We describe some earlier results when $q$ is a power of $p=2$ and $g=2,3$. In
these results, $Z_{g}(q)$ was computed by separating into cases based on the
ramification data $R$ and splitting behavior $S$. (We combine the various
splitting behaviors for brevity). These results were generalized for any genus
$g$ when $p=2$ using a different method [1, Sections 8-10]. However, the case-
based method is still valuable since it illustrates the connection with the
geometry of the $p$-rank strata as discussed in Section 2.6.

######  Theorem 2.10.

[6, Theorem 18] If $k$ is a finite field of cardinality $q=2^{n}$ and $g=2$,
then

$Z_{2}(q)=\sum_{[C]\in{\mathcal{A}S}_{2}(k)/\simeq_{k}}|\textup{Aut}_{k}(C)|^{-1}=q^{3}.$

This result was obtained by combining the following data.

$R$ | $\delta_{R}$ | Mass formula
---|---|---
$\\{2,2,2\\}$ | 3 | $q^{3}-q^{2}$
$\\{2,4\\}$ | 2 | $q^{2}-q$
$\\{6\\}$ | 1 | $q$

######  Theorem 2.11.

[17, Theorem 8] If $k$ is a finite field of cardinality $q=2^{n}$ and $g=3$,
then

$Z_{3}(q)=\sum_{[C]\in{\mathcal{A}S}_{3}(k)/\simeq_{k}}|\textup{Aut}_{k}(C)|^{-1}=q^{5}.$

This result was obtained by combining the following data.

$R$ | $\delta_{R}$ | Mass formula
---|---|---
$\\{2,2,2,2\\}$ | $5$ | $q^{5}-q^{4}$
$\\{2,2,4\\}$ | 4 | $q^{4}-2q^{3}+q^{2}$
$\\{4,4\\}$ | 3 | $q^{3}-q^{2}$
$\\{2,6\\}$ | 3 | $q^{3}-q^{2}$
$\\{8\\}$ | 2 | $q^{2}$

##  3\. Counting Artin-Schreier equations

The main result of this section is Proposition 3.5, in which we determine the
number of equations for an Artin–Schreier cover with fixed ramification data,
splitting behavior and branch divisor. This relies on Lemma 3.1, which
provides a useful way to describe the equation for an Artin–Schreier cover
when the branch points are not rational.

###  3.1. Orbit of rational functions under Frobenius

Recall that ${\rm Fr}(\alpha)=\alpha^{q}$ is the absolute Frobenius of
$\bar{k}$ over $k$. If the branch points of $\pi$ are not defined over $k$,
the condition that $u(x)\in k(x)$ places constraints on the partial fraction
decomposition of $u(x)$ over $\overline{k}$. In particular, the terms in the
partial fraction decomposition must respect the action of ${\rm Fr}$.

For example, suppose $q\equiv 3\bmod 4$ and $u(x)=(x^{2}+1)^{-1}$. Let $i$
denote $\sqrt{-1}\in{\mathbb{F}}_{q^{2}}$. Then

$u(x)=\frac{i/2}{x+i}+\frac{-i/2}{x-i}=\frac{i/2}{x+i}+{\rm
Fr}(\frac{i/2}{x+i}).$

The next lemma generalizes this example.

######  Lemma 3.1.

Let $u(x)\in{\mathbb{F}}_{q}(x)$ be such that ${\rm div}_{\infty}(u(x))=eP$,
where $P$ is an ${\mathbb{F}}_{q}$-point of degree $m$. Then there exist a
constant $c_{0}\in{\mathbb{F}}_{q}$, a rational function
$v(x)\in{\mathbb{F}}_{q^{m}}(x)$, and an ${\mathbb{F}}_{q^{m}}$-point $P_{0}$
of degree one such that ${\rm div}_{\infty}(v(x))=eP_{0}$ and

$u(x)=c_{0}+\sum_{j=0}^{m-1}{\rm Fr}^{j}(v(x)).$

###### Proof.

The ${\mathbb{F}}_{q}$-point $P$ consists of an orbit $\\{\theta_{j}\\}_{0\leq
j\leq m-1}$ of ${\mathbb{F}}_{q^{m}}$-points under Frobenius, where ${\rm
Fr}(\theta_{j})=\theta_{j+1\bmod m}$. Let $P_{0}=\theta_{0}$. The partial
fraction decomposition of $u(x)$ over ${\mathbb{F}}_{q^{m}}$ has the form

$u(x)=c_{0}+\sum_{j=0}^{m-1}\frac{v_{j}(x)}{(x-\theta_{j})^{e}},$

where $v_{j}(x)\in{\mathbb{F}}_{q^{m}}[x-\theta_{j}]$ is a polynomial in
$x-\theta_{j}$ of degree at most $e-1$ with non-zero constant term. Thus

$u(x)={\rm Fr}(u(x))={\rm Fr}(c_{0})+\sum_{j=0}^{m-1}\frac{{\rm
Fr}(v_{j}(x))}{(x-\theta_{j+1})^{\epsilon}}.$

Matching up the terms shows that ${\rm Fr}(c_{0})=c_{0}$ and ${\rm
Fr}(v_{j}(x))=v_{j+1\bmod m}(x)$ for $0\leq j\leq m-1$. The result follows by
letting $v(x)=v_{0}(x)/(x-\theta_{0})^{e}$. ∎

###  3.2. The constant term

######  Lemma 3.2.

Fix $u(x)\in k(x)-{\rm AS}(k(x))$. For $c\in k$, let
$\pi_{c}:C_{c}\to{\mathbb{P}}^{1}$ be given by the equation $y^{p}-y=u(x)+c$.
Then there are exactly $p$ isomorphism classes of Artin–Schreier covers of the
form $\pi_{c}$.

###### Proof.

Two such covers $\pi_{c_{1}}$ and $\pi_{c_{2}}$ are isomorphic if and only if
$c_{1}-c_{2}=z^{p}-z\in{\rm AS}(k(x))$ for some $z\in k$. So the number of
isomorphism classes equals the number of cosets of ${\rm AS}(k)$ in $k$. Since
there are $q/p$ elements in ${\rm AS}(k)$, the number of cosets of
$\textup{AS}(k)$ in $k$ is $p$. ∎

###  3.3. Counting rational functions

Given a pole $\theta_{0}$ of $u(x)$, we find the number of possibilities for
its contribution to the partial fraction decomposition of $u(x)$. This number
does not depend on the location of the pole but only on the order $e$ of the
pole at $\theta_{0}$ and the degree $m$ of the point $\theta_{0}$.

Given $\theta_{0}\in{\mathbb{P}}^{1}({\mathbb{F}}_{q^{m}})$, write $\bar{x}=x$
if $\theta_{0}=\infty$ and write $\bar{x}=(x-\theta_{0})^{-1}$ otherwise.
Recall the notation in Lemma 3.1. We write
$v(x)=\sum_{i=1}^{e}a_{i}\bar{x}^{i}$ for some constants
$a_{i}\in{\mathbb{F}}_{q^{m}}$. Without loss of generality, we can adjust
$u(x)$ over ${\mathbb{F}}_{q}$ to remove any monomial $a_{i}\bar{x}^{i}$ in
$v(x)$ with $p\mid i$. We count the remaining number of possibilities for
$v(x)$.

######  Definition 3.3.

Given $m$ and $e$, let $V_{m,e}$ be the subset of
$\\{v\in\bar{x}{\mathbb{F}}_{q^{m}}[\bar{x}]\mid\deg(v)=e\\}$ consisting of
polynomials $v$ that contain no monomials $a_{i}\bar{x}^{i}$ with $p\mid i$.

######  Lemma 3.4.

Then $\\#V_{m,e}=(q^{m}-1)q^{m\left(e-1-\lfloor\frac{e}{p}\rfloor\right)}$.

###### Proof.

Write $v=\sum_{i=1}^{e}a_{i}\bar{x}^{i}$ where $a_{i}\in{\mathbb{F}}_{q^{m}}$.
Since $\deg(v)=e$, there are $q^{m}-1$ choices for the leading coefficient
$a_{e}$ and $q^{m}$ choices for the coefficient $a_{i}$ if $1\leq i<e$. Also,
the number of monomials is $\\#\\{1\leq i\leq e-1\mid p\nmid
i\\}=e-1-\lfloor\frac{e}{p}\rfloor$. ∎

###  3.4. Counting Artin–Schreier covers with fixed branch divisor

Fix a ramification data $R$ of length $r$ as in Section 2.4 and a splitting
behavior $S$ as in Section 2.5. We count the number of Artin–Schreier covers
of type $R,S$ and with fixed branch divisor.

Write $R:=\\{\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{r}\\}$. Recall that
$e_{i}=\epsilon_{i}-1$ for $1\leq i\leq r$. Define

$E:=E(R)=\sum_{i=1}^{r}\left(e_{i}-1-\left\lfloor\frac{e_{i}}{p}\right\rfloor\right).$
(4)

Fix a weighted branch divisor $W_{\circ}=\sum e_{Q}\cdot Q$ of type $R,S$. Let
$B_{\circ}$ be its support. Recall that $s$ is the number of orbits of the
points in $B_{\circ}$ under Frobenius. Recall that $m_{1},\ldots,m_{s}$ are
the degrees of these points and $\sum_{i=1}^{s}m_{i}=r$.

Define $N_{W_{\circ}}$ to be the set of $u(x)\in k(x)$ such that ${\rm
div}_{\infty}(u(x))=W_{\circ}$ modulo ${\rm AS}(k(x))$.

######  Proposition 3.5.

With notation as above, the number of Artin–Schreier covers $y^{p}-y=u(x)$
with ${\rm div}_{\infty}(u(x))=W_{\circ}$ is

$|N_{W_{\circ}}|=p\cdot q^{E(R)}\prod_{i=1}^{s}(q^{m_{i}}-1).$

###### Proof.

It suffices to count the number of $u(x)\in{\mathbb{F}}_{q}(x)$ of type $R,S$
such that ${\rm div}_{\infty}(u(x))=W_{\circ}$ up to Artin–Schreier
equivalence. The partial fraction decomposition of $u(x)$ has the form
$u(x)=\sum_{i=1}^{s}u_{i}(x)$ where $u_{i}(x)\in{\mathbb{F}}_{q}(x)$ and ${\rm
div}_{\infty}(u_{i}(x))=e_{i}P_{i}$ for some ${\mathbb{F}}_{q}$-point $P_{i}$
of degree $m_{i}$. Using Lemma 3.1, we write
$u_{i}(x)=c_{0,i}+\sum_{j=0}^{m-1}{\rm Fr}^{j}(v_{i})$ with $v_{i}\in
V_{m_{i},e_{i}}$. There are $q$ choices for the combined constant term
$\sum_{i=1}^{s}c_{0,i}$. By Lemma 3.2, the number of choices for the constant
term up to Artin–Schreier equivalence is $p$. So
$|N_{W_{\circ}}|=p\prod_{i=1}^{s}|V_{m_{i},e_{i}}|$. This simplifies to the
given formula by Lemma 3.4. ∎

##  4\. The mass formula as an average

Fix $g=d(p-1)/2$. Fix a ramification data $R$ of length $r$ as in Section 2.4
and a splitting behavior $S$ as in Section 2.5. Given an Artin–Schreier curve
$(C,\iota)$ with automorphism, the ramification data and splitting behavior of
the resulting Artin–Schreier cover are well-defined, because they do not
depend on the choice of the parameter on the quotient ${\mathbb{P}}^{1}$.

Define a mass formula

$Z_{R,S}(q):=\sum_{[C]\in{\mathcal{A}S}_{g}(k)/\simeq_{k},\ {\rm type}\
R,S}|\rm{Cent}_{{\rm Aut}_{k}(C)}\langle\iota\rangle|^{-1},$

where the sum is over all isomorphism classes $[C]=(C,\iota)$ of
Artin–Schreier curves with automorphism over $k$ such that the cover has type
$R,S$. The main result in this section is Theorem 4.7, in which we re-
interpret this mass formula as an average over conjugacy classes in the
symmetric group $S_{r}$. This result was inspired by [9, Corollary 5.3].

###  4.1. Stabilizers

Suppose $W=\sum e_{Q}Q$ is a weighted branch divisor of type $R,S$.

######  Definition 4.1.

Let $\Gamma_{W}=\\{\gamma\in{\rm
Aut}_{k}({\mathbb{P}}^{1})\mid\gamma(W)=W\\}$.

For example, consider the case that $r\leq 3$. Then we can fix a convenient
choice of branch locus $B$ because the action of ${\rm PGL}_{2}(k)$ is triply
transitive on the points of degree $1$, and is transitive on the points of
degree $2$ (resp. 3). We fix a point $\omega_{2}$ of degree $2$ and a point
$\omega_{3}$ of degree $3$. We compute the order of the stabilizer
$\Gamma_{W}$ in these cases.

Let $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ be distinct positive integers
with $p\nmid e_{i}=\epsilon_{i}-1$. Write $\epsilon=\epsilon_{1}$.

######  Lemma 4.2.

The following table provides the order of the stabilizer $\Gamma_{W}$ in each
case.

$r$ |  $R$ and $S$ | $B$ | $|\Gamma_{W}|$
---|---|---|---
1 | $(\epsilon)$ | $\\{\infty\\}$ | $q(q-1)$
2 | $(\epsilon,\epsilon_{2})$ | $\\{0,\infty\\}$ | $q-1$
2 | $(\epsilon,\epsilon)$ | $\\{0,\infty\\}$ | $2(q-1)$
2 | $((\epsilon-\epsilon))$ | $\\{\omega_{2}\\}$ | $2(q+1)$
3 | $(\epsilon,\epsilon_{2},\epsilon_{3})$ | $\\{0,1,\infty\\}$ | 1
3 | $(\epsilon,\epsilon_{2},\epsilon_{2})$ | $\\{0,1,\infty\\}$ | 2
3 | $(\epsilon,\epsilon,\epsilon)$ | $\\{0,1,\infty\\}$ | 6
3 |  $(\epsilon,(\epsilon_{2}-\epsilon_{2}))$ or $(\epsilon,(\epsilon-\epsilon))$ | $\\{\infty,\omega_{2}\\}$ | 2
3 | $((\epsilon-\epsilon-\epsilon))$ | $\\{\omega_{3}\\}$ | 3

###### Proof.

For $r=1$, then $W=e\\{\infty\\}$ and so
$\Gamma_{W}=\Gamma_{\infty}:=\\{x\mapsto ax+b\mid a\in k^{*},b\in k\\}$.

For $r=2$, $(\epsilon,\epsilon_{2})$, then $W=e\\{0\\}+e_{2}\\{\infty\\}$ and
so $\Gamma_{W}=\\{x\mapsto ax\mid a\in k^{*}\\}$.

For $r=2$, $(\epsilon,\epsilon)$, then $W=e\\{0\\}+e\\{\infty\\}$ and so

$\Gamma_{W}=\\{x\mapsto ax\mid a\in k^{*}\\}\cup\\{x\mapsto b/x\mid b\in
k^{*}\\}.$

For $r=2$, $((\epsilon-\epsilon))$, then $W=e\\{\omega_{2}\\}$. Let $T_{2}$ be
the set of points of degree $2$ over $k$. Then $|T_{2}|=(q^{2}-q)/2$. Since
the action of ${\rm PGL}_{2}(k)$ on $T_{2}$ is transitive, $|\Gamma_{W}|=|{\rm
PGL}_{2}(k)|/\\#T_{2}=2(q+1)$, by the orbit-stabilizer theorem.

For $r=3$, in the split cases, then $B=\\{0,1,\infty\\}$. For
$(\epsilon,\epsilon_{2},\epsilon_{3})$, then $\Gamma_{W}=\rm{Id}$. For
$(\epsilon,\epsilon_{2},\epsilon_{2})$, let
$W=e\\{1\\}+e_{2}\\{0\\}+e_{2}\\{\infty\\}$. Then $\Gamma_{W}$ also includes
$\\{x\mapsto 1/x\\}$. For $(\epsilon,\epsilon,\epsilon)$, there are 6
fractional linear transformations fixing $W=e\\{0\\}+e\\{1\\}+e\\{\infty\\}$.

For the two quadratic cases with $r=3$, consider $B=\\{\infty,\theta_{2}\\}$,
for an arbitrary point $\theta_{2}$ of degree $2$. There are exactly two
automorphisms $\gamma\in{\rm PGL}_{2}({\mathbb{F}}_{q^{2}})$ that fix $\infty$
and stabilize $\theta_{2}$; these are defined over ${\mathbb{F}}_{q}$ and so
are contained in $\Gamma_{\infty}$. Furthermore, because
$|\Gamma_{\infty}|=2|T_{2}|$, the action of $\Gamma_{\infty}$ on the set of
branch divisors $\\{\infty,\theta_{2}\\}$ is transitive.

For the cubic case with $r=3$, then $W=e\\{\omega_{3}\\}$. Let $T_{3}$ be the
set of points of degree $3$ over $k$. Then $|T_{3}|=(q^{3}-q)/3$. Since the
action of ${\rm PGL}_{2}(k)$ on $T_{3}$ is transitive,
$|\Gamma_{\omega_{3}}|=|{\rm PGL}_{2}(k)|/|T_{3}|=3$, by the orbit-stabilizer
theorem. ∎

###  4.2. The action on rational functions

If $\gamma\in{\rm PGL}_{2}(k)$, then $\gamma$ acts on rational functions in
$k(x)$. Specifically, $\gamma\cdot f(x)=f(\gamma(x))$.

######  Definition 4.3.

If $u(x)\in k(x)$, let $\Gamma_{u(x)}=\\{\gamma\in{\rm
Aut}_{k}({\mathbb{P}}^{1})\mid u(\gamma(x))=u(x)\\}$.

If ${\rm div}_{\infty}(u(x))=W$, then $\Gamma_{u(x)}\subset\Gamma_{W}$.

######  Lemma 4.4.

Suppose $\pi:C\to\mathbb{P}^{1}$ is an Artin–Schreier cover with equation
$y^{p}-y=u(x)$. If $\varphi\in\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle$, then there exists an automorphism
$\gamma:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ such that
$\pi\circ\varphi=\gamma\circ\pi$. The map $\psi:\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle\to\textup{Aut}(\mathbb{P}^{1})$ is a
homomorphism with image $\Gamma_{u(x)}$. There is a short exact sequence of
groups:

$1\rightarrow\langle\iota\rangle\overset{I}{\rightarrow}\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle\overset{\psi}{\rightarrow}\Gamma_{u(x)}\to 1.$
(5)

###### Proof.

Suppose $\varphi\in\rm{Cent}_{{\rm Aut}_{k}(C)}\langle\iota\rangle$. For
$x\in\mathbb{P}^{1}$, choose $y\in\pi^{-1}(x)$, and define
$\gamma(x)=\pi(\varphi(y))$. We check that $\gamma$ is well-defined: if
$y^{\prime}\in\pi^{-1}(x)$, then $y^{\prime}=\iota^{e}(y)$ for some $0\leq
e\leq p-1$; then
$\varphi(y^{\prime})=\varphi(\iota^{e}(y))=\iota^{e}(\varphi(y))$ so
$\pi(\varphi(y^{\prime}))=\pi(\varphi(y))$. One can check that $\gamma$ is an
automorphism and $\psi$ is a homomorphism.

By Section 2.2, the condition $\varphi\in\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle$ implies that $u(\gamma(x))=u(x)$, so
$\gamma\in\Gamma_{u(x)}$. Conversely, if $\gamma\in\Gamma_{u(x)}$, then the
map $(x,y)\mapsto(\gamma(x),y)$ defines an automorphism
$\varphi\in\textup{Aut}(\mathbb{P}^{1})$ and $\varphi\in\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle$. Thus $\textup{Im}(\psi)=\Gamma_{u(x)}$. The
sequence in (5) is exact because $\varphi\in\ker(\psi)$ if and only if
$\pi(\varphi(y))=\pi(y)$ if and only if $\varphi\in\langle\iota\rangle$. ∎

By Lemma 4.4,

$|\rm{Cent}_{{\rm Aut}_{k}(C)}\langle\iota\rangle|=p|\Gamma_{u(x)}|.$ (6)

###  4.3. Orbits of weighted branch divisors

Fix a ramification data $R$ of length $r$ and a splitting behavior $S$. An
Artin–Schreier cover $\pi:C\to{\mathbb{P}}^{1}$ of type $R,S$ determines a
weighted branch divisor $W=\sum e_{Q}Q$ of type $R,S$, namely $W={\rm
div}_{\infty}(u(x))$ where $y^{p}-y=u(x)$ is the equation for $\pi$.

Fix one weighted branch divisor $W_{\circ}$ of type $R,S$. Let
$\Gamma_{W_{\circ}}$ be the stabilizer of $W_{\circ}$ in ${\rm PGL}_{2}(k)$.
Recall that $N_{W_{\circ}}$ is the set of $u(x)\in k(x)$ such that ${\rm
div}_{\infty}(u(x))=W_{\circ}$ modulo ${\rm AS}(k(x))$. Note that
$\Gamma_{W_{\circ}}$ acts on $N_{W_{\circ}}$. By the orbit-stabilizer theorem,

$|\textup{Orb}_{\Gamma_{W_{\circ}}}(u(x))||\Gamma_{u(x)}|=|\Gamma_{W_{\circ}}|.$
(7)

If two Artin–Schreier curves with automorphism are isomorphic over $k$, then
their weighted branch divisors $W_{1}$ and $W_{2}$ are in the same orbit under
${\rm PGL}_{2}(k)$. Thus the stabilizers $\Gamma_{W_{1}}$ and $\Gamma_{W_{2}}$
are conjugate in the symmetric group $S_{r}$.

######  Definition 4.5.

Consider the set of weighted branch divisors $W$ over $k$ of type $R,S$. Let
$\theta$ be the set of orbits of $W$ under ${\rm PGL}_{2}(k)$. Given $H\subset
S_{r}$, let $\theta_{H}$ be the set of orbits for $W$ under ${\rm PGL}_{2}(k)$
having the property that $\Gamma_{W}$ is conjugate to $H$. Let
$T=\sum_{H}\frac{|\theta_{H}|}{|H|}$, where the sum ranges over a set of
representatives of conjugacy classes of subgroups $H\subset S_{r}$.

######  Example 4.6.

If $r\leq 3$, for given $R$ and $S$, recall the choice of weighted branch
divisor $W$ and $\Gamma_{W}$ from (the proof of) Lemma 4.2. Note that
$\theta_{H}$ is empty unless $H$ is conjugate to $\Gamma_{W}$, in which case
$|\theta_{H}|=1$. Thus $T=|\Gamma_{W}|^{-1}$ if $r\leq 3$.

###  4.4. Main Theorem

######  Theorem 4.7.

As $[C]=(C,\iota)$ ranges over the $k$-isomorphism classes of Artin–Schreier
curves with automorphism having type $R,S$, then

$\sum_{[C]/\simeq k,\ {\rm type}\ R,S}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1}=p^{-1}|N_{W_{\circ}}|\cdot T.$

###### Proof.

We specify the information needed to define an Artin–Schreier curve with
automorphism of type $R,S$. First, for each orbit in $\theta$, we choose a
representative weighted branch divisor $W$ of type $R,S$ in that orbit. Then
we choose $u(x)\in k(x)$ such that ${\rm div}_{\infty}(u(x))=W$ up to
Artin–Schreier equivalence. Then we divide by
$|\textup{Orb}_{\Gamma_{W}}(u(x))|$ to avoid over-counting. So

$\displaystyle\sum_{[C]/\simeq k,\ {\rm type}\ R,S}\frac{1}{|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|}$ $\displaystyle=$
$\displaystyle\sum_{W\in\theta}\sum_{u(x)}\frac{1}{|\textup{Orb}_{\Gamma_{W}}(u(x))|\cdot|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|}$ $\displaystyle=$
$\displaystyle\sum_{W\in\theta}\sum_{u(x)}\frac{1}{|\textup{Orb}_{\Gamma_{W}}(u(x))|\cdot
p|\Gamma_{u(x)}|}$ $\displaystyle=$
$\displaystyle\frac{1}{p}\sum_{W\in\theta}\frac{1}{|\Gamma_{W}|}\sum_{u(x)}1,$

by Lemma 4.4 and the orbit-stabilizer theorem.

The number of $u(x)$ such that ${\rm div}_{\infty}(u(x))=W$ is independent of
$W$ when $R$ and $S$ are fixed. Up to Artin–Schreier equivalence, this
cardinality equals $|N_{W_{\circ}}|$. Using a set of representatives of
conjugacy classes of subgroups $H\subset S_{r}$, we re-organize the sum by
combining the contributions from all $W$ such that $\Gamma_{W}$ is conjugate
to $H$. Since $|\Gamma_{W}|=|H|$, this yields

$\sum_{[C]/\simeq k,\ {\rm type}\ R,S}\frac{1}{|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|}=\frac{1}{p}\sum_{H}\frac{|\theta_{H}|}{|H|}|N_{W_{\circ}}|=p^{-1}|N_{W_{\circ}}|\cdot
T.$

∎

##  5\. New results for arbitrary prime $p$ and $r\leq 3$ branch points

In this section, we suppose that the number of branch points of $\pi$
satisfies $r\leq 3$.

###  5.1. Arbitrary genus with $r\leq 3$ branch points

Recall the definition of $E(R)$ from (4).

######  Proposition 5.1.

Given $R$ with length $1\leq r\leq 3$, this table gives the mass formula

$Z_{R}(q)=\sum_{[C]/\simeq k,\ {\rm type}\ R}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1}$

for $k$-isomorphism classes of Artin–Schreier curves with automorphism with
ramification data $R$.

$R$ | $Z_{R}(q)$
---|---
$\\{\epsilon\\}$ | $q^{E(R)-1}$
$\\{\epsilon_{1},\epsilon_{2}\\}$ | $(q-1)q^{E(R)}$
$\\{\epsilon,\epsilon\\}$ | $(q-1)q^{E(R)}$
$\\{\epsilon_{1},\epsilon_{2},\epsilon_{3}\\}$ | $(q-1)^{3}q^{E(R)}$
$\\{\epsilon_{1},\epsilon_{2},\epsilon_{2}\\}$ | $(q-1)^{2}q^{E(R)+1}$
$\\{\epsilon,\epsilon,\epsilon\\}$ | $(q-1)q^{E(R)+2}$

###### Proof.

For fixed $R$ and $S$, define the branch locus $B$ as in Lemma 4.2 and
consider a weighted branch divisor $W$ supported on $B$ as in the proof of
that result. By Theorem 4.7 and Example 4.6,
$Z_{R,S}(q)=p^{-1}|N_{W}||\Gamma_{W}|^{-1}$. The formula for $|N_{W}|$ is in
Proposition 3.5. This yields the table below.

$R,S$ | $|N_{W}|$ | $|\Gamma_{W}|$
---|---|---
$(\epsilon)$ | $p(q-1)q^{E(R)}$ | $q(q-1)$
$(\epsilon_{1},\epsilon_{2})$ | $p(q-1)^{2}q^{E(R)}$ | $q-1$
$(\epsilon,\epsilon)$ | $p(q-1)^{2}q^{E(R)}$ | $2(q-1)$
$((\epsilon-\epsilon))$ | $p(q^{2}-1)q^{E(R)}$ | $2(q+1)$
$(\epsilon_{1},\epsilon_{2},\epsilon_{3})$ | $p(q-1)^{3}q^{E(R)}$ | $1$
$(\epsilon_{1},\epsilon_{2},\epsilon_{2})$ | $p(q-1)^{3}q^{E(R)}$ | $2$
$(\epsilon,\epsilon,\epsilon)$ | $p(q-1)^{3}q^{E(R)}$ | $6$
$(\epsilon_{1},(\epsilon_{2}-\epsilon_{2}))$ | $p(q-1)(q^{2}-1)q^{E(R)}$ | $2$
$(\epsilon,(\epsilon-\epsilon))$ | $p(q-1)(q^{2}-1)q^{E(R)}$ | $2$
$((\epsilon-\epsilon-\epsilon))$ | $p(q^{3}-1)q^{E(R)}$ | $3$

If some values in $R$ repeat, we combine contributions from different
splitting behaviors:

If $R=\\{\epsilon,\epsilon\\}$, the split and non-split cases sum to
$(q-1)q^{E(R)}$.

If $R=\\{\epsilon_{1},\epsilon_{2},\epsilon_{2}\\}$, the split and split-
quadratic case sum to $(q-1)^{2}q^{E(R)+1}$.

If $R=\\{\epsilon,\epsilon,\epsilon\\}$, the split, split/quadratic, and cubic
cases sum to $(q-1)q^{E(R)+2}$.

∎

###  5.2. Results for moduli of Artin–Schreier curves with $r\leq 3$

We compute the mass formula when the Artin–Schreier curve has low genus $g$ by
allowing the ramification data $R$ and splitting behavior $S$ to vary. The
leading terms of the formulas are compatible with the information about
irreducible components of ${\mathcal{A}S}_{g}$ from Section 2.6. The lower
degree terms provide new and more subtle information about the cohomology of
${\mathcal{A}S}_{g}$.

######  Theorem 5.2.

Let $p=3$. As $[C]=(C,\iota)$ ranges over the $k$-isomorphism classes of
Artin–Schreier curves with automorphism such that $C$ has genus $g$, then

$Z_{g}(q):=\sum_{[C]\in{\mathcal{A}S}_{g}(k)/\simeq k}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1}=\begin{cases}1&\text{if }g=1\\\
q-1&\text{if }g=2\\\ q^{2}&\text{if }g=3\\\ 2q^{3}-q^{2}&\text{if }g=4\\\
q^{4}-q^{3}+q^{2}&\text{if }g=5.\\\ \end{cases}$

This is immediate by combining the following data.

$g=d$ | $R$ | $\delta_{R}$ | Mass formula
---|---|---|---
1 |  {3} | 0 | 1
2 |  {2,2} | 1 | $q-1$
3 |  {2,3} | 2 | $q^{2}-q$
|  {5} | 1 | $q$
4 |  {2,2,2} | 3 | $q^{3}-q^{2}$
|  {3,3} | 3 | $q^{3}-q^{2}$
|  {6} | 2 | $q^{2}$
5 |  {2,2,3} | 4 | $q^{4}-2q^{3}+q^{2}$
|  {2,5} | 3 | $q^{3}-q^{2}$
|  {7} | 2 | $q^{2}$

###### Proof.

The proof follows from Proposition 5.1. Some lines include several choices of
$S$. ∎

######  Theorem 5.3.

Let $p\geq 5$. As $[C]=(C,\iota)$ ranges over the $k$-isomorphism classes of
Artin–Schreier curves with automorphism such that $C$ has genus $g$, then

$Z_{g}(q):=\sum_{[C]\in{\mathcal{A}S}_{g}(k)/\simeq k}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1}=\begin{cases}1&\text{if }g=1(p-1)/2\\\
2q-1&\text{if }g=2(p-1)/2\\\ 2q^{2}-q&\text{if }g=3(p-1)/2\\\
3q^{3}-3q^{2}&\text{if }g=4(p-1)/2,\ p=5\\\ 4q^{3}-3q^{2}&\text{if
}g=4(p-1)/2,\ p\geq 7\\\ 3q^{4}-3q^{3}+q^{2}&\text{if }g=5(p-1)/2,\ p=5\\\
4q^{4}-4q^{3}+q^{2}&\text{if }g=5(p-1)/2,\ p\geq 7.\\\ \end{cases}$

This is immediate by combining the following data.

$g$ | $R$ | $\delta_{R}$ | Mass formula
---|---|---|---
$1(p-1)/2$ |  {3} | 0 | 1
$2(p-1)/2$ |  {2,2} | 1 | $q-1$
|  {4} | 1 | $q$
$3(p-1)/2$ |  {2,3} | 2 | $q^{2}-q$
|  {5} | 2 | $q^{2}$
$4(p-1)/2$ |  {2,2,2} | 3 | $q^{3}-q^{2}$
|  {2,4} | 3 | $q^{3}-q^{2}$
|  {3,3} | 3 | $q^{3}-q^{2}$
|  {6} | 3 |  $q^{3}$ if $p\geq 7$
$5(p-1)/2$ |  {2,2,3} | 4 | $q^{4}-2q^{3}+q^{2}$
|  {2,5} | 4 | $q^{4}-q^{3}$
|  {3,4} | 4 | $q^{4}-q^{3}$
|  {7} | 3 |  $q^{3}$ if $p=5$
|  | 4 |  $q^{4}$ if $p\geq 7$

###### Proof.

The proof follows from Proposition 5.1. For example, for the $R=\\{2,2,3\\}$
line, we combine the contributions from the split and the quadratic case to
obtain

$(1/2)((q-1)^{3}q+(q-1)(q^{2}-1)q)=q^{2}(q-1)^{2}.$

∎

######  Remark 5.4.

In Theorem 5.3, when $g=4(p-1)$, the case $p=5$ is different because
${\mathcal{A}S}_{g}$ has one additional component of dimension $3$ when $p\geq
7$, namely that for $R=\\{6\\}$.

######  Remark 5.5.

The ordinary locus (curves with $p$-rank $g$) corresponds to the ramification
data $R=\\{2,2,\ldots,2\\}$. When the leading coefficient of $Z_{g}(q)$ is
bigger than $1$, this reflects the fact that ${\mathcal{A}S}_{g}$ has more
than $1$ irreducible component. This means that the ordinary locus is not open
and dense in ${\mathcal{A}S}_{g}$.

For example, when $p=3$ and $g=4$, then the leading coefficient in Theorem 5.2
is $2$; the $3$-rank is $4$ for the Artin-Schreier curves with $R=\\{2,2,2\\}$
but is only $2$ for $R=\\{3,3\\}$.

##  6\. Orbits of $4$-sets

Recall that $k$ is a finite field of cardinality $q$. A 4-set is a set of $4$
distinct points of ${\mathbb{P}}^{1}(\overline{k})$ which is stabilized by the
Frobenius action over $k$. In this section, we study the action of ${\rm
PGL}_{2}(k)$ on 4-sets. The main result is Proposition 6.4, which gives a
formula for the number of orbits of $4$-sets with fixed splitting behavior. We
expect this material is well-understood by experts in finite geometry, but we
did not find a reference that applied directly. In particular, we extend
results found in [14], [15], and [17].

In Section 7, we use Proposition 6.4 to determine the mass formula when the
Artin-Schreier cover has $r=4$ branch points, after determining additional
information about the sizes of the stabilizers of representatives of the
orbits.

###  6.1. The total number of orbits

Let $N_{orbit}$ be the total number of orbits of 4-sets of
$\mathbb{P}^{1}(\overline{k})$ under ${\rm PGL}_{2}(k)$. Let

$\binom{-3}{q}=\begin{cases}1&\text{ if }q\equiv 1\bmod 3\\\ -1&\text{ if
}q\equiv-1\bmod 3\\\ 0&\text{ if }3\mid q.\end{cases}$

######  Theorem 6.1.

[14, Theorem 2.2] The total number of orbits is

$N_{orbit}=\begin{cases}2q+2+\binom{-3}{q}&\text{ if }p\geq 5\\\ 2q+2&\text{
if }p=3\text{ and }q\equiv 1\bmod 4\\\ 2q+1&\text{ if }p=3\text{ and }q\equiv
3\bmod 4\\\ 2q+\binom{-3}{q}&\text{ if }p=2.\end{cases}$

###  6.2. The splitting behavior when $r=4$

We name the cases as follows.

$S$ | Description | Notation
---|---|---
Split | four points of degree $1$ | $(\cdot,\cdot,\cdot,\cdot)$
Split/Quad | two points of degree $1$ and one of degree $2$ | $(\cdot,\cdot,(\cdot-\cdot))$
Quad | two points of degree $2$ | $((\cdot-\cdot),(\cdot-\cdot))$
Cubic | one point of degree $1$ and one of degree $3$ | $(\cdot,(\cdot-\cdot-\cdot))$
Quartic | one point of degree $4$ | $((\cdot-\cdot-\cdot-\cdot))$

Given $S$, let $N_{orbit}(S)$ denote the number of orbits of $4$-sets of
$\mathbb{P}^{1}(\overline{k})$ with splitting behavior $S$ under ${\rm
PGL}_{2}(k)$.

###  6.3. The case $p=2$

######  Proposition 6.2.

[17, Proposition 3] If $p=2$, then $N_{orbit}(S)$ is given by:

$S$ | Split | Split/Quad | Quad | Cubic | Quartic
---|---|---|---|---|---
$N_{orbit}(S)$ | $(q+2\binom{-3}{q})/6$ | $q/2$ | $(q-2)/2$ | $(q+3+2\binom{-3}{q})/3$ | $q/2$

Note that these sum to the quantity $N_{orbit}$ when $p=2$ in Theorem 6.1.

###  6.4. Number of split orbits

Let $N_{orbit}(split)$ be the number of split orbits, meaning the number of
orbits of $4$ distinct points of $\mathbb{P}^{1}(k)$ under the action of ${\rm
PGL}_{2}(k)$.

######  Theorem 6.3.

[15, Theorem C] If $p$ is odd, then $N_{orbit}(split)=(q+3+2\binom{-3}{q})/6$.

###  6.5. The number of orbits in the non-split cases when $p$ is odd

We now determine the number of orbits $N_{orbit}(S)$ when $p$ is odd and $S$
is not split.

Recall that $\binom{-1}{q}$ equals $1$ if $q\equiv 1\bmod 4$ and equals $-1$
if $q\equiv 3\bmod 4$.

######  Proposition 6.4.

Let $p$ be odd. For each non-split $S$, the number $N_{orbit}(S)$ of orbits of
$4$-sets of $\mathbb{P}^{1}(\overline{k})$ with splitting behavior $S$ under
${\rm PGL}_{2}(k)$ is as follows:

  1. (1)

($S$ split/quadratic) $N_{orbit}(S)=(q+1)/2$.

  2. (2)

($S$ quadratic) $N_{orbit}(S)=(q-1)/2$.

  3. (3)

($S$ cubic) $N_{orbit}(S)=(q+3+2\binom{-3}{q})/3$.

  4. (4)

($S$ quartic) $N_{orbit}(S)=(q+\binom{-1}{q})/2$ if $p=3$ and
$N_{orbit}(S)=(q+1)/2$ if $p\geq 5$.

Note that $N_{orbit}$ equals the sum of $N_{orbit}(S)$ over the different
splitting behaviors $S$ (including the split one from Theorem 6.3).

The proof of Proposition 6.4 is contained in the following subsections. Recall
Burnside’s Lemma: if $\Gamma$ is a group acting on a set $I$, then the number
of orbits of $I$ under $\Gamma$ is

$|I/\Gamma|=|\Gamma|^{-1}\sum_{\gamma\in\Gamma}|{\rm Fix}_{\gamma}|.$

###  6.6. Split/quadratic case

######  Lemma 6.5.

If $p$ is odd and $S$ is split/quadratic then $N_{orbit}(S)=(q+1)/2$.

###### Proof.

Let $I$ be the set of irreducible monic degree $2$ polynomials $f(x)\in k[x]$.
If $\theta,\theta^{\prime}\in{\mathbb{F}}_{q^{2}}\backslash{\mathbb{F}}_{q}$
are the roots of some $f(x)\in I$, then
$W=\\{0,\infty,\theta,\theta^{\prime}\\}$ is a split/quadratic $4$-set.
Conversely, the orbit of every split/quadratic 4-set contains a representative
of the form $W=\\{\infty,0,\theta,\theta^{\prime}\\}$. So it suffices to
determine the number of different orbits among $4$-sets of the form $W$. For
this, it suffices to determine the number of orbits of $f(x)\in I$ under
$\Gamma_{W}$.

If $\gamma\in\Gamma_{W}$, then either $\gamma$ fixes $0$ and $\infty$ or
$\gamma$ transposes $0$ and $\infty$. So
$\Gamma_{W}=\Gamma_{0}\cup\Gamma_{\infty}$ where
$\Gamma_{0}=\\{\gamma_{a}:x\mapsto ax\mid a\in k^{*}\\}$ and
$\Gamma_{\infty}=\\{\bar{\gamma}_{b}:x\mapsto b/x\mid b\in k^{*}\\}$.

The action of $\Gamma_{W}$ on $I$ is given as follows. If
$\gamma_{a}\in\Gamma_{0}$, then $\gamma_{a}(f(x))=f(ax)/a^{2}$. To define the
action of $\bar{\gamma}_{b}\in\Gamma_{\infty}$ on $I$, we think of $I$ as the
set of irreducible (possibly not monic) degree $2$ polynomials $f(x)\in k[x]$,
up to equivalence $\sim$, where $f_{1}(x)\sim f_{2}(x)$ if and only if
$f_{1}(x)=\lambda f_{2}(x)$ for some $\lambda\in k^{*}$. Then
$\bar{\gamma}_{b}(f(x))=x^{2}f(\frac{b}{x})$. Writing $f(x)=A_{0}x^{2}+Ax+B$,
then $\bar{\gamma}_{b}(f(x))=A_{0}b^{2}+Abx+Bx^{2}$.

Claim 1: Let $I_{0}=\\{f(x)=x^{2}+B\mid-B\in k^{*}\text{ is a quadratic non-
residue}\\}$. Then $I_{0}\subset I$ and $\\#I_{0}=(q-1)/2$. The elements of
$I_{0}$ are in one orbit under the action of $\Gamma_{W}$.

Proof of Claim 1: It suffices to show that the elements of $I_{0}$ are in one
orbit under the action of $\Gamma_{0}$. If $f(x)\in I_{0}$ is in
$\textup{Fix}_{\gamma_{a}}$ then

$\displaystyle x^{2}+B=f(x)=\gamma_{a}(f(x))=f(ax)/a^{2}=x^{2}+B/a^{2}.$

Thus, $\textup{Fix}_{\gamma_{a}}=\emptyset$ unless $a=\pm 1$ and
$\textup{Fix}_{\gamma_{1}}=\textup{Fix}_{\gamma_{-1}}=I_{0}$. By Burnside’s
lemma, $|I_{0}/\Gamma_{0}|=(q-1)^{-1}2(q-1)/2=1$.

Claim 2: Let $I_{\not=0}=I-I_{0}$. Then $\\#I_{\not=0}=(q-1)^{2}/2$. The
elements of $I_{\not=0}$ form $(q-1)/2$ orbits under the action of
$\Gamma_{W}$.

Proof of Claim 2: Note that
$\\#I_{\not=0}=\\#I-\\#I_{0}=(q^{2}-q)/2-(q-1)/2=(q-1)^{2}/2$. We apply
Burnside’s Theorem to the action of $\Gamma_{W}$ on $I_{\not=0}$. If
$f(x)\in\textup{Fix}_{\gamma_{a}}$, then

$x^{2}+Ax+B=f(x)=\gamma_{a}(f(x))=f(ax)/a^{2}=x^{2}+Ax/a+B/a^{2}.$

So $\textup{Fix}_{\gamma_{a}}$ is trivial unless $a=1$ and
$\textup{Fix}_{\gamma_{1}}=I_{\not=0}$. If
$f(x)\in\textup{Fix}_{\bar{\gamma}_{b}}$, then

$\displaystyle A_{0}x^{2}+Ax+B\sim x^{2}f(\frac{b}{x})=A_{0}b^{2}+Abx+Bx^{2},$

which is true if and only if $b=B/A_{0}$. So each $f(x)\in I_{\not=0}$ is
fixed by exactly one $\bar{\gamma}_{b}$. Thus,

$|I_{\not=0}/\Gamma_{W}|=(2(q-1))^{-1}(2(q-1)^{2}/2)=(q-1)/2.$

∎

###  6.7. Quadratic case

The goal of this section is to prove Lemma 6.9, which states that the number
of orbits of quadratic $4$-sets of $\mathbb{P}^{1}(\overline{k})$ under ${\rm
PGL}_{2}(k)$ is $(q-1)/2$ when $p$ is odd.

Fix a quadratic non-residue $s\in k$. Let $f_{1}(x)=x^{2}-s$. The orbit of
every quadratic $4$-set contains a $4$-set of the form
$W_{s}=\\{\pm\sqrt{s},\tau,\tau^{\prime}\\}$ where $\\{\tau,\tau^{\prime}\\}$
are the roots of an irreducible monic degree $2$ polynomial $f_{2}(x)$ that is
not equal to $f_{1}(x)$.

We investigate the action of $\Gamma_{W_{s}}$ on the set of such polynomials
$f_{2}(x)$.

######  Lemma 6.6.

Let $W_{s}=\\{\pm\sqrt{s}\\}$, then $|\Gamma_{W_{s}}|=2(q+1)$. The elements of
$\Gamma_{W_{s}}$ are represented by the equivalence classes in ${\rm
PGL}_{2}(k)$ of these matrices (for $a,c\in k$ not both equal to 0):

$\gamma_{1}(a,c)=\left(\begin{array}[]{ll}a&cs\\\
c&a\end{array}\right){\text{a}nd\
}\gamma_{2}(a,c)=\left(\begin{array}[]{ll}a&-cs\\\ c&-a\end{array}\right).$
(8)

###### Proof.

If $\gamma\in\Gamma_{W_{s}}$, then $\gamma(f_{1}(x))\sim f_{1}(x)$. This
implies that $x^{2}-s$ is equivalent to:

$\displaystyle(cx+d)^{2}f_{1}\left(\frac{ax+b}{cx+d}\right)$ $\displaystyle=$
$\displaystyle(a^{2}x^{2}+2abx+b^{2})-s(c^{2}x^{2}+2cdx+d^{2})$
$\displaystyle=$ $\displaystyle x^{2}(a^{2}-sc^{2})+x(2ab-2cds)+b^{2}-sd^{2}.$

So $ab=cds$ and $(b^{2}-sd^{2})/(a^{2}-sc^{2})=-s$.

If $a=0$, then $cds=0$ which implies $d=0$ since $c\neq 0$. Then $b=\pm cs$.

If $a\neq 0$, then $b=cds/a$ and one computes that
$s(a^{2}-sc^{2})(a^{2}-d^{2})=0$. Since $s$ is a quadratic non-residue, this
implies that $a=\pm d$.

Thus $\gamma$ is either $\gamma_{1}(a,c)$ or $\gamma_{2}(a,c)$ for some pair
$(a,c)\in k^{2}$, with $a,c$ not both $0$. There are $2(q^{2}-1)/(q-1)$
equivalence classes of $\gamma$ in ${\rm PGL}_{2}(k)$. Representatives of
these are $\gamma_{1}(a,c)$ and $\gamma_{2}(a,c)$ where either $(a,c)=(0,1)$
or $a=1$ with $c\in k$. ∎

Let $\Gamma_{1}$ be the subgroup of ${\rm PGL}_{2}(k)$ of elements $\gamma$
that fix each of $\sqrt{s}$ and $-\sqrt{s}$. Note that $\gamma\in\Gamma_{1}$
if and only if $\gamma=\gamma_{1}(a,c)$ for some pair $(a,c)$ not both zero.

Let $\Gamma_{2}$ be the non-trivial coset of $\Gamma_{1}$ in $\Gamma_{W_{s}}$.
Note that every $\gamma\in\Gamma_{2}$ exchanges $\sqrt{s}$ and $-\sqrt{s}$.
Also $\gamma\in\Gamma_{2}$ if and only if $\gamma=\gamma_{2}(a,c)$ for some
pair $(a,c)$ not both zero.

######  Lemma 6.7.

Then $\Gamma_{1}\simeq C_{q+1}$ and $\Gamma_{W_{s}}\simeq D_{q+1}$, the
dihedral group of order $2(q+1)$.

###### Proof.

Consider the subgroup $\tilde{\Gamma}_{0,\infty}$ of elements of ${\rm
PGL}_{2}({\mathbb{F}}_{q^{2}})$ that fix each of $0$ and $\infty$; it is
cyclic of order $q^{2}-1$. The subgroup $\tilde{\Gamma}_{1}$ of elements of
${\rm PGL}_{2}({\mathbb{F}}_{q^{2}})$ that fix each of $\sqrt{s}$ and
$-\sqrt{s}$ is conjugate to $\tilde{\Gamma}_{0,\infty}$, and thus is also
cyclic of order $q^{2}-1$. Then $\Gamma_{1}=\tilde{\Gamma}_{1}\cap{\rm
PGL}_{2}({\mathbb{F}}_{q})$ which must also be cyclic. By Lemma 6.6,
$|\Gamma_{1}|=q+1$.

The non-identity coset $\Gamma_{2}$ of $\Gamma_{1}$ in $\Gamma_{W_{s}}$ is
represented by $\bar{\gamma}_{2}:x\mapsto-x$. The conjugation action of
$\bar{\gamma}_{2}$ on $\Gamma_{1}$ is inversion:
$\bar{\gamma}_{2}\gamma_{1}\bar{\gamma}_{2}^{-1}=\gamma_{1}^{-1}$. Thus
$\Gamma_{W_{s}}\simeq D_{q+1}$. ∎

######  Lemma 6.8.

Let $\gamma_{1}=\gamma_{1}(a,c)\in\Gamma_{1}$ as in (8). Let
$f_{2}(x)=x^{2}+Ax+B$ be an irreducible monic degree $2$ polynomial in $k[x]$
with $f_{2}(x)\not=f_{1}(x)$. Then:

  1. (1)

$\gamma_{1}(f_{2}(x))=x^{2}+x\frac{A(a^{2}+c^{2}s)+2ac(B+s)}{a^{2}+Aac+Bc^{2}}+\frac{c^{2}s^{2}+Aacs+Ba^{2}}{a^{2}+Aac+Bc^{2}}.$
(9)

  2. (2)

If $\gamma_{1}\not={\rm Id}$, then $f_{2}(x)\in{\rm Fix}_{\gamma_{1}}$ if and
only if $a=0$ and $B=s$.

###### Proof.

  1. (1)

The polynomial on the right of (9) equals $(cx+a)^{2}f_{2}((ax+cs)/(cx+a))$.

  2. (2)

Suppose $f_{2}(x)\in{\rm Fix}_{\gamma_{1}}$. Then
$f_{2}(x)=\gamma_{1}(f_{2}(x))$. By part(1), this implies

$\displaystyle A^{2}ac+ABc^{2}-2acs-Ac^{2}s-2Bac$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle ABac+B^{2}c^{2}-c^{2}s^{2}-Aacs$
$\displaystyle=$ $\displaystyle 0.$

Since $\gamma_{1}\not={\rm Id}$, we can suppose $c\not=0$. If $a=0$, without
loss of generality, take $c=1$; then $A(B-s)=0$ and $B^{2}-s^{2}=0$, so $B=\pm
s$. If $B=-s$, then $A=0$, which contradicts the fact $f_{2}(x)\not=f_{1}(x)$.
Thus $B=s$.

If $ac\not=0$; without loss of generality, take $a=1$. We will show that the
hypothesis $f_{2}(x)\in\textup{Fix}_{\gamma_{1}}$ leads to a contradiction.
Dividing the equations above by $c$ yields

$\displaystyle Ac(B-s)-2(B+s)+A^{2}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle c(B^{2}-s^{2})+A(B-s)$ $\displaystyle=$ $\displaystyle 0.$

If $B\neq s$, some arithmetic shows $A=-c(B+s)$ and then $(B+s)(c^{2}s-1)=0$.
Thus $B=-s$, which contradicts the fact $f_{2}(x)\not=f_{1}(x)$. If $B=s$,
some arithmetic shows that $A^{2}-4B=0$, which contradicts the fact that
$f_{2}(x)$ is irreducible.

∎

######  Lemma 6.9.

If $p$ is odd and $S$ is quadratic, then $N_{orbit}(S)=(q-1)/2$.

###### Proof.

Every quadratic orbit contains a $4$-set of the form
$W=\\{\pm\sqrt{s},\tau,\tau^{\prime}\\}$ where $\\{\tau,\tau^{\prime}\\}$ are
the roots of an irreducible monic quadratic polynomial $f_{2}(x)$ (not equal
to $f_{1}(x)$). Let

$I=\\{f_{2}(x)=x^{2}+Ax+B\mid A,B\in k,\ f_{2}(x)\text{ irreducible},\
f_{2}(x)\neq x^{2}-s\\}.$

Then $\\#I=(q^{2}-q)/2-1=(q+1)(q-2)/2$.

Claim 3: The number of orbits of $I$ under $\Gamma_{1}$ is $(q-1)/2$.

The reason for this is that $|\textup{Fix}_{\rm Id}|=\\#I$ and
$\textup{Fix}_{\gamma_{1}}$ has size $(q+1)/2$ if $(a,c)=(0,1)$ and is trivial
otherwise by Lemma 6.8. By Burnside’s Lemma,

$|I/\Gamma_{1}|=(q+1)^{-1}\left((q+1)(q-2)/2)+(q+1)/2\right)=(q-1)/2.$

To finish the proof of Lemma 6.9, note that $\Gamma_{2}$ is a coset of
$\Gamma_{1}$ in $\Gamma_{W_{s}}$. A representative of this coset is the map
$\bar{\gamma}_{2}(x)=-x$.

Claim 4: Given $f_{2}(x)=x^{2}+Ax+B\in I$, there exists
$\gamma_{1}=\gamma_{1}(a,c)\in\Gamma_{1}$ such that $\gamma_{1}\cdot
f_{2}(x)=\bar{\gamma}_{2}\cdot f_{2}(x)$.

To see this, note that $\bar{\gamma}_{2}\cdot f_{2}(x)=f_{2}(-x)=x^{2}-Ax+B$.
If $A=0$, the claim is true, taking $\gamma_{1}={\rm Id}$. Let $A\not=0$. By
(9), the condition $\gamma_{1}\cdot f_{2}(x)=x^{2}-Ax+B$ is equivalent to

$B=\frac{c^{2}s^{2}+Aacs+Ba^{2}}{a^{2}+Aac+Bc^{2}},\ {\rm and}\
-A=\frac{A(a^{2}+c^{2}s)+2ac(B+s)}{a^{2}+Aac+Bc^{2}}.$ (10)

If $a^{2}+Aac+Bc^{2}\not=0$, this is equivalent to

$0=c^{2}(B^{2}-s^{2})+Aac(B-s),\ {\rm and}\
0=a^{2}(2A)+a(2cs+2Bc+A^{2}c)+Ac^{2}(B+s).$

The two equations are satisfied when $c=1$ and $a=-(B+s)/A$.

Claim 4 implies that the number of orbits under $\Gamma_{W_{s}}$ is the same
as the number of orbits under $\Gamma_{1}$, namely $(q-1)/2$. ∎

###  6.8. Cubic case

######  Lemma 6.10.

If $p$ is odd and $S$ is cubic, then $N_{orbit}(S)$ equals $(q+5)/3$ if
$q\equiv 1\bmod 3$ and equals $(q+1)/3$ if $q\equiv-1\bmod 3$.

###### Proof.

Let $I$ be the set of irreducible monic degree $3$ polynomials $f(x)\in k[x]$.
If
$\theta,\theta^{\prime},\theta^{\prime\prime}\in{\mathbb{F}}_{q^{3}}\backslash{\mathbb{F}}_{q}$
are the roots of some $f(x)\in I$, then
$W=\\{\infty,\theta,\theta^{\prime},\theta^{\prime\prime}\\}$ is a cubic
$4$-set. Conversely, the orbit of every cubic 4-set contains a representative
of the form $W=\\{\infty,\theta,\theta^{\prime},\theta^{\prime\prime}\\}$.

It suffices to find the number of orbits among $4$-sets of the form $W$. For
this, it suffices to find the number of orbits of $f(x)\in I$ under
$\Gamma_{W}$. Here $\Gamma_{W}=\\{\gamma:x\rightarrow ax+b,\ a\in k^{*},b\in
k\\}$ and $|\Gamma_{W}|=q(q-1)$.

If $f(x)\in I$, write $f(x)=x^{3}+Ax^{2}+Bx+C$. Then
$f(x)\in\textup{Fix}_{\gamma}$ if and only if $f(ax+b)/a^{3}=f(x)$, which is
equivalent to

$A=(3b+A)/a,\ B=(3b^{2}+2Ab+B)/a^{2},\ C=(b^{3}+Ab^{2}+Bb+C)/a^{3},$

or

$(a-1)A=3b,\ (a^{2}-1)B=3b^{2}+2Ab,\ (a^{3}-1)C=b^{3}+Ab^{2}+Bb.$ (11)

Case 1: If $a=1$, then $b=0$ and $\gamma={\rm Id}$, with $|\textup{Fix}_{\rm
Id}|=|I|=(q^{3}-q)/3$.

Case 2: If $a=-1$, then $\gamma(x)=-x+b$ has order $2$, which cannot stabilize
a cubic point, so $\textup{Fix}_{\gamma}=\emptyset$.

Case 3: If $a^{3}=1$ with $a\not=1$, then $f(x)\in\textup{Fix}_{\gamma}$ if
and only if

$A=3b/(a-1)\ {\rm and}\ B=3b^{2}/(a-1)^{2}.$

Let $x=y-A/3$ then $f(y-A/3)=y^{3}+C-\left(\frac{b}{a-1}\right)^{3}$, which is
irreducible if and only if $(\frac{b}{a-1})^{3}-C$ is not a cube in $k^{*}$.
If $q\equiv-1\bmod 3$, this is impossible so
$\textup{Fix}_{\gamma}=\emptyset$. If $q\equiv 1\bmod 3$, there are $2(q-1)/3$
non-cubes and $2q$ choices of $\gamma(x)=ax+b$ for each of these, which
contributes in total $4q(q-1)/3$.

Case 4: If $a^{3}\neq 1,a\neq-1$, we compute that

$A=3b/(a-1),\ B=3b^{2}/(a-1)^{2},\ C=b^{3}/(a-1)^{3}.$

Let $x=y-A/3$. Then $f(y-A/3)=y^{3}$, which is reducible. So this case does
not occur.

Hence, the total number of orbits is

$|I/\Gamma_{W}|=\left\\{\begin{array}[]{lr}(q(q-1))^{-1}\left((q^{3}-q)/3+4q(q-1)/3\right)=(q+5)/3&{\rm
if}\ q\equiv 1\bmod 3\\\ (q(q-1))^{-1}\left((q^{3}-q)/3\right)=(q+1)/3&{\rm
if}\ q\equiv-1\bmod 3.\end{array}\right.$

∎

Now consider the cubic case when $k$ has characteristic $3$. The following
fact about $f(x)=x^{3}+Bx+C\in k[x]$ will be useful: $f(x)$ is irreducible in
$k[x]$ if and only if $-B=w^{2}$ for some $w\in k$ and $\text{tr}(C/w^{3})\neq
0$ [23, Theorem 2].

######  Lemma 6.11.

If $p=3$ and $S$ is cubic, then $N_{orbit}(S)=(q+3)/3$.

###### Proof.

From (11) in the previous proof, $f(x)\in\textup{Fix}_{\gamma}$ if and only if

$(a-1)A=0,\ (a^{2}-1)B=2Ab,\ (a^{3}-1)C=b^{3}+Ab^{2}+Bb.$

Suppose $f(x)\in\textup{Fix}_{\gamma}$ with $\gamma(x)=ax+b$.

If $a\not=\pm 1$, then $A=B=0$. Since $f(x)=x^{3}+C$ factors in $k[x]$, this
implies $\textup{Fix}_{\gamma}=\emptyset$.

If $a=1$ and $b=0$, then $|\textup{Fix}_{\rm Id}|=(q^{3}-q)/3$.

If $a=1$ and $b\neq 0$, then $A=0$ and $B=-b^{2}$. So $f(x)=x^{3}-b^{2}x+C$,
which is irreducible if and only if $\text{tr}(C/b^{3})\neq 0$. Let
$\kappa={\rm Fr}^{-1}(C)$; then $C=\kappa^{3}$. Then
$\text{tr}(C/b^{3})=\text{tr}({\rm Fr}(\kappa/b))=\text{tr}(\kappa/b)$. Recall
that $\text{tr}:{\mathbb{F}}_{q}\to{\mathbb{F}}_{3}$ is a surjective group
homomorphism. So, for each $b\in{\mathbb{F}}_{q}^{*}$, the number of
$C\in{\mathbb{F}}_{q}$ such that $f(x)$ is irreducible is $2q/3$ and
$|\textup{Fix}_{\gamma}|=2q/3$.

If $a=-1$, then $A=0$ and $C=b(b^{2}+B)$. If $-B=w^{2}$ for some $w\in k^{*}$,
let $z=b/w$. Then $\text{tr}(b(b^{2}-w^{2})/w^{3})=\text{tr}(z(z-1)(z-2))=0$
so $f(x)$ factors in $k[x]$. This implies $\textup{Fix}_{\gamma}=\emptyset$.

Thus

$|I/\Gamma|=(q(q-1))^{-1}((q^{3}-q)/3+(q-1)2q/3)=(q+3)/3.$

∎

###  6.9. Quartic case

######  Lemma 6.12.

(Quartic Case) If $S$ is quartic, then $N_{orbit}(S)$ equals $(q-1)/2$ if
$p=3$ and $q\equiv-1\bmod 4$ and equals $(q+1)/2$ otherwise.

###### Proof.

The total number of orbits is found in Theorem 6.1 ([14, Theorem 2.2]). From
this, we subtract the number of orbits from the non-quartic splitting
behaviors found in Theorem 6.3 ([15, Theorem C]), and Lemmas 6.5, 6.9, and
6.11. ∎

##  7\. The mass formula for $4$ branch points

In this section, we consider the mass formula for Artin-Schreier curves with
automorphism when the number of branch points of the cover is $4$. For a
ramification data $R$ of length $r=4$, we provide formulas for

$Z_{R,S}(q):=\sum_{[C],\text{ type }R,S/\simeq k}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1},$

when $S$ is split in Section 7.2, when $S$ is non-split and $p=2$ in Section
7.3, and some cases when $S$ is non-split and $p$ is odd in Section 7.4.

From a moduli-theoretic perspective, it is natural to fix the ramification
data $R$ and allow the splitting behavior $S$ to vary. By combining the
results in this section, we obtain the following conclusions about

$Z_{R}(q):=\sum_{[C]\text{ type }R/\simeq k}|\rm{Cent}_{{\rm
Aut}_{k}(C)}\langle\iota\rangle|^{-1}.$

###  7.1. Conclusions

Consider ramification data
$R=\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}\\}$ with
$\epsilon_{i}\not\equiv 1\bmod p$. Let $e_{i}=\epsilon_{i}-1$ and define

$E:=E(R)=\sum_{i=1}^{4}\left(e_{i}-1-\left\lfloor\frac{e_{i}}{p}\right\rfloor\right).$

######  Corollary 7.1.

The mass formula $Z_{R}(q)$ has the following formulas:

  1. (1)

If $\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}$ are distinct, then
$Z_{R}(q)=(q-1)^{4}q^{E(R)}T$ where $T=(q+3+2\binom{-3}{q})/6$ if $p\geq 5$
and $T=(q+2\binom{-3}{q})/6$ if $p=2$.

  2. (2)

If $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ are distinct and
$\epsilon_{3}=\epsilon_{4}$, then $Z_{R}(q)=(q-1)^{3}q^{E(R)}T^{\prime}$,
where $T^{\prime}=(1/3)(2q^{2}+q+(q-1)\binom{-3}{q})$.

  3. (3)

If $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}$ and
$\epsilon_{4}\not=\epsilon_{3}$, then
$Z_{R}(q)=(q-1)^{2}q^{E(R)}(q^{3}+q+\gamma)$, where $\gamma=(2/3)(q^{2}+q+1)$
if $p=3$ and $\gamma=(-2/3)(q^{2}-2q+1)$ if $p\geq 5$ and $q\equiv 1\bmod 6$
and $\gamma=0$ otherwise.

###### Proof.

  1. (1)

Since all pole orders are distinct, then $S$ is split. The result follows from
Case 1 of Proposition 7.2.

  2. (2)

The condition on the pole orders implies that $S$ is either split or
split/quadratic. The result follows by adding Case 2 of Proposition 7.2 and
the case of having different pole orders from Propositions 7.3 and 7.4.

  3. (3)

The condition on the pole orders implies that $S$ is either split,
split/quadratic with different pole orders, or cubic. The result follows by
adding Case 3 of Proposition 7.2 and the appropriate cases from Propositions
7.3 and 7.4.

∎

###  7.2. The split case with $r=4$ points

######  Proposition 7.2.

When $S$ is split, then $Z_{R,S}(q)=(q-1)^{4}q^{E(R)}T$, where $T$ is as
follows:

Case |  Condition on $R$ | $T$
---|---|---
1 |  $\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}$ distinct |  $(q+3+2\binom{-3}{q})/6$, $p$ odd; $(q+2\binom{-3}{q})/6$, $p=2$
2 |  $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ distinct, $\epsilon_{3}=\epsilon_{4}$ | $(q+2\binom{-3}{q})/6$
3 | $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}\not=\epsilon_{4}$ | $(q-2)/6$
4 | $\epsilon_{1}=\epsilon_{2}\not=\epsilon_{3}=\epsilon_{4}$ | $(q+2\binom{-3}{q})/12$
5 | $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=\epsilon_{4}$ | $(q-2)/24$

###### Proof.

The action of ${\rm PGL}_{2}(k)$ on ${\mathbb{P}}^{1}(k)$ is triply
transitive. So each split orbit contains a representative of the form
$B_{t}=\\{0,1,\infty,t\\}$ for some $t\in k-\\{0,1\\}$. Let $\Gamma_{t}={\rm
Stab}(B_{t})\subset S_{4}$. Recall the information about $\Gamma_{t}$ from
Lemma 2.2.

Let $W_{\circ}$ be a weighted branch divisor with ramification data $R$
supported on $B_{t}$. By Theorem 4.7, $Z_{R,S}(q)=p^{-1}|N_{W_{\circ}}|\cdot
T$ where $T=\sum_{H}\frac{|\theta_{H}|}{|H|}$. By Proposition 3.5,
$|N_{W_{\circ}}|=p(q-1)^{4}q^{E(R)}$.

Let $\Gamma_{W_{\circ}}={\rm Stab}(W_{\circ})\subset{\rm PGL}_{2}(k)$. This
subgroup depends on the values in $R$ because if the pole orders of two branch
points are different, then no automorphism in $\Gamma_{W_{\circ}}$ can map one
of them to the other. For each representative $H$ of a conjugacy class in
$S_{4}$, we consider the set $\theta_{H}$ of orbits of $t$ for which
$\Gamma_{W_{\circ}}$ is conjugate to $H$. We need to compute $T$.

The proof is then a case-by-case analysis using Lemmas 2.1 and 2.2 and Theorem
6.3.

Case 1: then $\theta_{H}$ is empty unless $H=\\{\rm id\\}$, so
$T=N_{orbit}(split)$.

Case 2: Without loss of generality, the two branch points with pole order
$\epsilon_{3}-1$ are $0$ and $\infty$. Then $|H_{t}|=1$ unless $t=-1$. If
$p=2$, then this does not happen and $T=N_{orbit}(split)$. If $p$ is odd, then
there is one orbit where $|H_{t}|=2$, so $T=N_{orbit}(split)-1+1/2$.

Case 3: Set the three branch points with pole order $\epsilon_{1}-1$ to be
$0,1,\infty$.

If $p=2$, then $\zeta_{3},\zeta_{3}^{2}\in k$ if and only if $q\equiv 1\bmod
3$; this orbit has $|H_{t}|=3$, so $T=(1/3)+(q+2)/6-1$ if $q\equiv 1\bmod 3$
and $T=(q-2)/6$ if $q\equiv-1\bmod 3$.

If $p=3$, then the orbit of $-1$ has $|H_{t}|=6$ so $T=(1/6)+(q+3)/6-1$.

If $p=5$, then the orbit of $-1$ has $|H_{t}|=2$ and $\zeta_{6}\in k$ if and
only if $q\equiv 1\bmod 3$, in which case it has $|H_{t}|=3$; so
$T=((q+5)/6-2)+(1/2)+(1/3)=(q-2)/6$ if $q\equiv 1\bmod 3$ and
$T=((q+1)/6-1)+(1/2)=(q-2)/6$ if $q\equiv-1\bmod 3$.

Case 4: Set the two branch points with pole order $\epsilon_{1}-1$ to be
$0,\infty$ and the two branch points with pole order $\epsilon_{3}-1$ to be
$1,t$. Then $\gamma_{1,t}=(0,\infty)(1,t)\in H_{t}$ for all $t$. The only
other elements of $H_{t}$ are automorphisms $x\mapsto 1/x$ and $x\mapsto-x$
when $t=-1$. So when $p=2$, then $|H_{t}|=2$ for all orbits. When $p$ is odd
then $|H_{t}|=2$ for all but one orbit, for which $|H_{t}|=4$. So
$T=(N_{orbit}(split)-1)/2+(1/4)$.

Case 5: then $|H_{t}|=\Gamma_{t}$ is as in Lemma 2.2. So $|H_{t}|=4$ except in
the special cases.

When $p=2$, $q\equiv 1\bmod 3$, then the orbit of $\zeta_{3}$ has
$|H_{t}|=12$; so $T=(q-2)/24$ when $q\equiv-1\bmod 3$ and
$T=((q+2)/6-1)/4+(1/12)$ when $q\equiv 1\bmod 3$.

When $p=3$, then the orbit of $-1$ has $|H_{t}|=24$; so
$T=((q+3)/6-1)/4+(1/24)$.

When $p\geq 5$, then the orbit of $-1$ has $|H_{t}|=8$ and when $q\equiv
1\bmod 3$ then the orbit of $\zeta_{6}$ has $|H_{t}|=12$. So
$T=((q+1)/6-1)/4+(1/8)$ when $q\equiv-1\bmod 3$ and
$T=((q+5)/6-2)/4+(1/8)+(1/12)$ when $q\equiv 1\bmod 3$. ∎

###  7.3. The case when $p=2$

In the next result, we generalize the case $R=\\{2,2,2,2\\}$ when $p=2$ found
in [17, Section 3.1].

######  Proposition 7.3.

Let $p=2$. Let $R=\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}\\}$
with $\epsilon_{i}$ even. If $S$ is non-split, then $Z_{R,S}(q)$ is given by:

$S$ | $Z_{R,S}(q)$
---|---
split/quad if split points have different pole orders | $(q-1)^{2}(q^{2}-1)q^{E+1}/2$
split/quad if split points have the same pole order | $(q-1)^{2}(q^{2}-1)q^{E+1}/4$
quadratic | $(q^{2}-1)^{2}q^{E}(q-2)/8$
cubic | $(q-1)(q^{3}-1)q^{E}(q+1)/3$
quartic | $(q^{4}-1)q^{E+1}/4$

As a check, when $R=\\{2,2,2,2\\}$, these values sum to $q^{5}-q^{4}$; when
combined with the $r\leq 3$ data, the sum is $q^{5}$ as in Theorem 2.11.

###### Proof.

By Theorem 4.7, $Z_{R,S}(q)=p^{-1}|N_{W_{\circ}}|\cdot T$ where
$T=\sum_{H}\frac{|\theta_{H}|}{|H|}$. In this case, $|N_{W_{\circ}}|$ depends
on $S$ as in Proposition 3.5. The computation of $T$ follows from the
following table, which is mostly contained in [17, Section 3.1].

Case | $H$ | Number of Orbits
---|---|---
split/quad | {id} if split points have different pole orders | $q/2$
split/quad |  $C_{2}$ if split points have same pole order | $q/2$
quadratic | $C_{2}\times C_{2}$ | $q/2-1$
cubic |  $\mu_{3}$ if $t=\infty$ when $q\equiv 1\bmod 3$ | 1
|  $\mu_{3}$ if $t=0$ when $q\equiv 1\bmod 3$ | 1
| $\\{\rm id\\}$ | $(q+1)/3$
quartic | $C_{2}$ | $q/2$

∎

###  7.4. Some non-split cases when $p$ is odd

######  Proposition 7.4.

If $p$ is odd and $S$ is non-split, then $Z_{R,S}(q)$ is given by:

$S$ | $Z_{R,S}(q)$
---|---
split/quad if split points have different pole orders | $(q-1)^{2}(q^{2}-1)q^{E+1}/2$
split/quad if split points have same pole orders | $(q-1)^{2}(q^{2}-1)q^{E+1}/4$
cubic if $p\neq 3$ | $(q-1)(q^{3}-1)(q+1)q^{E}/3$
cubic if $p=3$ | $(q-1)(q^{3}-1)(q+3)q^{E}/3$

###### Proof.

By Theorem 4.7, $Z_{R,S}(q)=p^{-1}|N_{W_{\circ}}|\cdot T$ where
$T=\sum_{H}\frac{|\theta_{H}|}{|H|}$. In this case, $|N_{W_{\circ}}|$ depends
on $S$ as in Proposition 3.5.

Consider the split/quadratic case. The orbit of every split/quadratic $4$-set
contains a representative $W_{\circ}=\\{0,\infty,\theta,\theta^{\prime}\\}$
where $\theta,\theta^{\prime}$ are the roots of an irreducible polynomial of
the form $f(x)=x^{2}+Ax+B\in k[x]$. Note that $\theta\theta^{\prime}=B$. Then
$|N_{W_{\circ}}|=p(q-1)^{2}(q^{2}-1)q^{E}$.

Consider the cubic case. The orbit of every cubic $4$-set contains a
representative $W=\\{\infty,\theta,\theta^{\prime},\theta^{\prime\prime}\\}$
where $\theta,\theta^{\prime},\theta^{\prime\prime}$ are the roots of an
irreducible polynomial of the form $f(x)=x^{3}+Ax^{2}+Bx+C\in k[x]$. In this
case, $|N_{W_{\circ}}|=p(q-1)(q^{3}-1)q^{E}$.

To determine $T$, for each representative $H$ of a conjugacy class in $S_{4}$,
we find the number of orbits for which $\Gamma_{W}$ is conjugate to $H$.

Case | $H$ | Number of Orbits
---|---|---
split/quad when |  $\\{\pm x\\}$ when $\theta^{\prime}=-\theta$ | 1
split points have different pole orders | $\\{x\\}$ | $(q-1)/2$
split/quad when |  $\\{\pm x,\pm B/x\\}$ when $\theta^{\prime}=-\theta$ | 1
split points have same pole orders | $\\{x,B/x\\}$ | $(q-1)/2$
cubic when $p\neq 3$ |  $\\{x,\zeta_{3}x,\zeta_{3}^{2}x\\}$ when $q\equiv 1\bmod 3$ | 2
|  $\\{x\\}$ when $q\equiv 1\bmod 3$ | $(q-1)/3$
|  $\\{x\\}$ when $q\equiv-1\bmod 3$ | $(q+1)/3$
cubic when $p=3$ | $\\{x\\}$ | $(q+3)/3$

We give the details of the proof in two cases.

Case: split/quad when split points have same pole orders.

By Claim 1 of Lemma 6.5, the set $I_{0}$ of irreducible polynomials of the
form $f(x)=x^{2}+B$ forms one orbit under the action of $\Gamma_{W_{\circ}}$.
Then $\Gamma_{W_{\circ}}=\\{\pm x,\pm B/x\\}$ and $|\Gamma_{W_{\circ}}|=4$.

By Lemma 6.5, there are $(q-1)/2$ other orbits. For these,
$\Gamma_{W_{\circ}}=\\{x,B/x\\}$ and $|\Gamma_{W_{\circ}}|=2$.

Thus $T=(1/4)+((q-1)/2)/2$ and $p^{-1}|N_{W_{\circ}}|\cdot
T=(q-1)^{2}(q^{2}-1)q^{E}q/4$.

Case: cubic when $p\neq 3$ and $q\equiv 1\bmod 3$.

There are $2(q-1)/3$ irreducible polynomials of the form $f(x)=x^{3}+C$. The
group $k^{*}$ acts on these by $\gamma\cdot f(x)=f(ax)/a^{3}$ for $a\in
k^{*}$. This scales $C$ by a cube. Each is stabilized by $\mu_{3}\subset
k^{*}$. So these polynomials form two orbits, each with $|H|=3$. There are
$(q+5)/3-2$ orbits for which $H=\\{x\\}$ and $|H|=1$. Thus
$T=2/3+(q+5)/3-2=(q+1)/3$ and $p^{-1}|N_{W_{\circ}}|\cdot
T=(q-1)(q^{3}-1)q^{E}(q+1)/3$. ∎

######  Remark 7.5.

Here is a problem for someone interested in finite geometry. For each orbit in
the quadratic and quartic $4$-branch points cases, determine the size of the
stabilizers of the orbits. Specifically, for each $H\subset S_{4}$, the goal
is to find the number of orbits of the branch locus $W$ for which $\Gamma_{W}$
is conjugate to $H$. With this data, it would be possible to extend the
results in this paper to the case of Artin–Schreier covers with genus
$g=6(p-1)/2$.

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