# Orbits of Theta Characteristics

H. W. Braden  ID School of Mathematics and Maxwell Institute for Mathematical
Sciences
The University of Edinburgh
Edinburgh EH9 3FD, Scotland, U.K<EMAIL_ADDRESS>and  Linden Disney-Hogg  ID
School of Mathematics
The University of Leeds
Leeds LS2 9JT, U.K<EMAIL_ADDRESS>
###### Abstract.

The theta characteristics on a Riemann surface are permuted by the induced
action of the automorphism group, with the orbit structure being important for
the geometry of the curve and associated manifolds. We describe two new
methods for advancing the understanding of these orbits, generalising existing
results of Kallel & Sjerve, allowing us to establish the existence of
infinitely many curves possessing a unique invariant characteristic as well as
determine the number of invariant characteristics for all Hurwitz curves with
simple automorphism group. In addition, we compute orbit decompositions for a
substantial number of curves with genus $\leq 9$, allowing the identification
of where current theoretical understanding falls short and the potential
applications of machine learning techniques.

Acknowledgements. We are grateful to Vanya Cheltsov for discussions. The
research of LDH is supported by a UK Engineering and Physical Sciences
Research Council (EPSRC) doctoral prize fellowship.

Data Availability. The datasets generated during the current study and the
code for their creation/analysis are available from the second author upon
reasonable request.

##  1\. Introduction

Given a smooth, compact, connected Riemann surface $\mathcal{C}$, a theta
characteristic on $\mathcal{C}$ is a line bundle $L\to\mathcal{C}$ such that
$L^{2}:=L\otimes L$ is isomorphic to the canonical bundle $K_{\mathcal{C}}$.
Given that $\deg K_{\mathcal{C}}=2g-2$ is even, where $g=g(\mathcal{C})$ is
the genus of the curve, it is necessarily true that theta characteristics
exist and moreover that there are exactly $2^{2g}$ of them. The latter follows
for, given a theta characteristic $L$ and a 2-torsion degree-0 line bundle
$L^{\prime}\in\operatorname{Pic}^{0}(\mathcal{C})[2]$, necessarily $L\otimes
L^{\prime}$ is also a theta characteristic. As such the set of characteristics
$S(\mathcal{C})$ is an affine space over $\mathbb{Z}_{2}$ modelled on
$H^{1}(\mathcal{C},\mathbb{Z}_{2})$. Theta characteristics are further
distinguished by their parity: a theta characteristic $L$ is called odd/even
if $\dim H^{0}(\mathcal{C},L)$ is odd/even. There are exactly
$2^{g-1}(2^{g}-1)$ odd theta characteristics and $2^{g-1}(2^{g}+1)$ even theta
characteristics [Ati71, Fay73]. For a general review of theta characteristics
and their applications, see [Far12].

Theta characteristics are permuted by the action of the automorphism group
$\operatorname{Aut}\mathcal{C}$, and in his landmark paper [Ati71] Atiyah
showed that every $f\in\operatorname{Aut}(\mathcal{C})$ leaves at least one
characteristic invariant. Kallel and Sjerve [KS10] then greatly expanded upon
this result, showing how to compute the orbits of characteristics given
knowledge of the rational representation of their automorphism group, and
using that to give further results about the number of characteristics
invariant under a given automorphism. In §2-3 we will briefly review their
results, as we will then use this to give a group cohomology description of
invariant characteristics. This allows us to generalise the results of [KS10],
giving the two main new results of this paper

######  Proposition 1.1.

If there exists $f\in G\leq\operatorname{Aut}(\mathcal{C})$ such that $f$ has
odd order, $\left\langle f\right\rangle$ is subnormal in $G$, and
$g(\mathcal{C}/\left\langle f\right\rangle)=0$, then $\mathcal{C}$ has a
unique theta characteristic invariant under the action of $G$.

######  Theorem 1.2.

There are infinitely many curves, both non-hyperelliptic and hyperelliptic,
with a unique invariant characteristic.

In §4 we will describe tables of curves of genus $\leq 9$ and their orbit
decompositions, stratified by the automorphism group action of the curve,
given in Appendix A. This will serve two purposes: first, it will act as a
collation of plane curve models of Riemann surfaces with automorphisms; and
second, it will identify cases where the current theory fails to explain why a
curve has a unique invariant characteristic. Moreover, by synthesising the
computational tools in Sage with those of [BRR13] we will demonstrate the
applicability of machine learning tools in gaining insight.

In particular, this machine learning will lead us to the work in §5 where, by
extending work of Dolgachev on invariant bundles over modular curves [Dol97b],
we will calculate the number of invariant characteristics for all Hurwitz
curves with simple automorphism group. These we can compare with the
predictions of the machine learning method.

Part of this work was completed during the PhD thesis of LDH [DH23]. The code
that was used there is available at
https://github.com/DisneyHogg/Riemann_Surfaces_and_Monopoles.

##  2\. Action of the Rational Representation

Choosing a basis $\left\\{a_{i},b_{i}\right\\}_{i=1}^{g}$ of the homology
group $H_{1}(\mathcal{C},\mathbb{Z})$ with canonical intersection matrix
$J=\begin{pmatrix}0&I_{g}\\\ -I_{g}&0\end{pmatrix}$, i.e.

$a_{i}\circ b_{j}=\delta_{ij},\quad a_{i}\circ a_{j}=0=b_{i}\circ b_{j},$

yields the rational representation
$\rho_{r}:\operatorname{Aut}(\mathcal{C})\to\operatorname{Sp}_{2g}(\mathbb{Z})$.
The starting point for this paper is the work of [KS10], where the authors use
the identification between theta characteristic and spin structures on a
Riemann surface to provide a particular choice of isomorphism
$S(\mathcal{C})\cong H_{1}(\mathcal{C},\mathbb{Z}_{2})\cong\mathbb{Z}^{2g}$
such that ${x}\in\mathbb{Z}_{2}^{2g}$ transforms under
$f\in\operatorname{Aut}(\mathcal{C})$ as

(1) ${x}\mapsto R^{T}{x}+{v}{\mod 2},$

where $R=\rho_{r}(f)$ and the vector ${v}$ is computed as
$v_{i}=\sum_{j<j^{\prime}}R_{ji}R_{j^{\prime}i}J_{jj^{\prime}}$. One can think
of this action on ${x}$ as matrix multiplication

$\begin{pmatrix}{x}\\\ 1\end{pmatrix}\mapsto\begin{pmatrix}R^{T}&{v}\\\
0&1\end{pmatrix}\begin{pmatrix}{x}\\\ 1\end{pmatrix}.$

######  Example 2.1.

Suppose $R=I$ is the identity matrix, then

$v_{i}=\sum_{j<j^{\prime}}\delta_{ji}\delta_{j^{\prime}i}J_{jj^{\prime}}=0.$

As such, as a sense check we can see that any theta characteristic is
invariant under the identity transformation.

This affine action is a classical result (see, for example, [Igu72, §V.1]),
but one of the strengths of Kallel and Sjerve’s approach using spin structures
is that the following result is not difficult to prove.

######  Lemma 2.2 ([Joh80], §5).

Writing ${x}=({u},{v})$ for ${u},{v}\in\mathbb{Z}_{2}^{g}$, the parity of the
associated theta characteristic is $q({x}):={u}\cdot{v}$.

As a result of Lemma 2.2 we see the parity of a spin structure is given by a
quadratic form on $H_{1}(\mathcal{C},\mathbb{Z}_{2})$ such that the associated
bilinear form $H({x},{y}):=q({x}+{y})-q({x})-q({y})$ is the reduction mod 2 of
the intersection pairing on $H_{1}(\mathcal{C},\mathbb{Z})$.

Given explicitly the rational representation, Equation (1) and Lemma 2.2
provide a fast and exact method to compute the orbit decomposition of theta
characteristics (as opposed to slow and numerical methods using the analytical
representation and the Abel-Jacobi map, see [DH23, BDH22]), split by parity.
An implementation of this method may be seen in the code from [BDH22]
available at https://github.com/DisneyHogg/Brings_Curve, and this is how we
shall compute tables of orbit decomposition in §4.

We shall at this stage introduce another concept which shall be used later,
that of the signature of the rational representation.

######  Definition 2.3 ([MSSV02], §2).

Take $G\leq\operatorname{Aut}(\mathcal{C})$ acting on a curve and denote by
$Q_{i}$, $i=1,\dots,r$, the branch points of the quotient map
$\pi:\mathcal{C}\to\mathcal{C}/G$. Denote by $C_{i}$ the conjugacy class of
$\operatorname{Stab}(P_{i})$ for any $P_{i}\in\pi^{-1}(Q_{i})$, and let
$c_{i}$ denote the order of an element of $C_{i}$. As the $Q_{i}$ are branch
points, the $C_{i}$ are nontrivial conjugacy classes, and $c_{i}>1$. We call
the ramification type of the $G$ action on $\mathcal{C}$ the data of the tuple
$(g(\mathcal{C}),G,(C_{1},\dots,C_{r}))$. The vector
$\bm{c}=(c_{1},\dots,c_{r})$ (or sometimes $(g_{0};c_{1},\dots,c_{r})$ where
$g_{0}=g(\mathcal{C}/G)$ is the orbit genus) is called the signature. We will
use exponents to indicate how many times a value of $c_{i}$ is repeated as in
[Bro91], e.g. the signature $(0;2,2,2,3)$ is written as $(0;2^{3},3)$. A
generating vector for the action is
$\left\\{\alpha_{1},\beta_{1},\dots,\alpha_{g_{0}},\beta_{g_{0}},\gamma_{1},\dots,\gamma_{r}\right\\}\subset
G$ such that

$\gamma_{1}^{c_{1}}=\dots=\gamma_{r}^{c_{r}}=\prod_{i=1}^{g_{0}}(\alpha_{i}\beta_{i}\alpha_{i}^{-1}\beta_{i}^{-1})\prod_{j=1}^{r}\gamma_{j}=1.$

The Riemann-Hurwitz theorem immediately says that

(2) $2g-2=\left\lvert
G\right\rvert\left[(2g_{0}-2)+\sum_{i}\left(1-\frac{1}{c_{i}}\right)\right].$

The (character of the) rational representation completely determines the
corresponding signature [Rie93, p. 401]. Conversely, given a signature
satisfying Equation (2) and a corresponding generating vector in an abstract
group $G$, there exists a Riemann surface with associated $G$ action, and the
generating vector determines the (character of the) rational representation
via the Eichler trace formula [Bre00].

##  3\. Invariant Characteristics and Group Cohomology

Suppose we had parametrised our set of theta characteristics differently in
terms of ${x}^{\prime}\in\mathbb{Z}_{2}^{2g}$ where ${x}={x}^{\prime}+{y}$ for
some fixed ${y}\in\mathbb{Z}_{2}^{2g}$. This is reparametrisation by
translation. One can derive the corresponding action on ${x}^{\prime}$ by
noting

$\displaystyle{x}={x}^{\prime}+{y}$ $\displaystyle\mapsto
R^{T}({x}^{\prime}+{y})+{v},$
$\displaystyle=R^{T}{x}^{\prime}+\left[{v}+(R^{T}-I){y}\right]+{y},$

and so

${x}^{\prime}\mapsto R^{T}{x}^{\prime}+{v}_{y},$

where ${v}_{y}={v}+(R^{T}-I){y}$. If ${y}$ were fixed by the action (1) then
${v}_{y}=0$ and moreover the converse is true. As such, we have seen the
following proposition.

######  Proposition 3.1.

There is an invariant characteristic on a curve $\mathcal{C}$ if and only if
the corresponding affine representation on $\mathbb{Z}_{2}^{2g}$

$\begin{pmatrix}{x}\\\ 1\end{pmatrix}\mapsto\begin{pmatrix}R^{T}&{v}\\\
0&1\end{pmatrix}\begin{pmatrix}{x}\\\ 1\end{pmatrix}$

is equivalent by a translation ${x}\to{x}^{\prime}:={x}-{y}$ to the linear
action

$\begin{pmatrix}{x}^{\prime}\\\ 1\end{pmatrix}\mapsto\begin{pmatrix}R^{T}&0\\\
0&1\end{pmatrix}\begin{pmatrix}{x}^{\prime}\\\ 1\end{pmatrix}$

for some ${y}\in\mathbb{Z}_{2}^{2g}$.

To utilize Proposition 3.1, we note that the question of whether an affine
representation can be reduced to a linear one may be presented as a cohomology
problem111LDH is grateful to Andrew Beckett for highlighting this to him.. In
particular, given a group $G$ and a (left) $G$-representation
$\rho:G\to\operatorname{GL}(V)$ we have the following results.

######  Proposition 3.2.

An affine representation of $G$ on $V$ which acts multiplicatively via $\rho$
determines a 1-cocycle in the group cohomology222For an introduction to group
cohomology with necessary definitions see [Wei95, §6]. We shall from here on
in drop the $\operatorname{Grp}$ subscript as through context it shall not
cause confusion with any other cohomologies in this work.
$H^{1}_{\operatorname{Grp}}(G,V)$ (making $V$ into a $G$-module in the natural
way with $\rho$) with the standard linear representation $G\times V\to V$,
$(g,x)=\rho(g)x$, corresponding to the zero 1-cocycle.

###### Proof.

An affine representation acting multiplicatively by $\rho$ is defined by

$\displaystyle G\times V$ $\displaystyle\to V,$ $\displaystyle(g,x)$
$\displaystyle\mapsto g\cdot x:=\rho(g)x+v(g).$

for some set map $v:G\to V$. By definition the set map $v$ is exactly a
1-cochain in group cohomology, with the linear representation giving the zero
1-cochain. Moreover, to truly get an action we require $\forall\,g,h\in
G,\,x\in V$, $g\cdot(h\cdot x)=(gh)\cdot x$. We can compute from the
definition

$\displaystyle g\cdot(h\cdot x)$ $\displaystyle=\rho(g)(h\cdot x)+v(g),$
$\displaystyle=\rho(g)[\rho(h)x+v(h)]+v(g),$
$\displaystyle=\rho(gh)x+\rho(g)v(h)+v(g),$ $\displaystyle=(gh)\cdot
x+\rho(g)v(h)-v({gh})+v(g),$

and so we must have

(3) $\forall\,g,h\in G,\,0=\rho(g)v(h)-v({gh})+v(g).$

In particular, setting $g=e$ the identity element in Equation (3) shows
$v(e)=0$. Equation (3) is exactly the condition that the 1-cochain $v$ is in
fact a 1-cocycle [Wei95, Example 6.5.6]. ∎

######  Proposition 3.3.

Two affine representations as defined in Proposition 3.2 are equivalent under
a translation of $V$ if and only if the associated 1-cocycle is a
1-coboundary.

###### Proof.

Fixing $y\in V$ and $v:G\to V$ defining an affine representation we have that

$\displaystyle g\cdot(x+y)$ $\displaystyle=\rho(g)(x+y)+v(g),$
$\displaystyle=\left\\{\rho(g)x+\left[v(g)+(\rho(g)-I)y\right]\right\\}+y.$

This defines a different affine action on $V$ given by 1-cocycle
$v_{y}(g):=v(g)+(\rho(g)-I)y$. This new affine action is actually linear if
and only if

(4) $\forall g\in G,\,(\rho(g)-I)y+v(g)=0\Leftrightarrow\forall g\in
G,\,g\cdot y=y\Leftrightarrow y\in V^{G},$

where we have used $V^{G}$ to denote the subset of $V$ invariant under $G$.
The condition that $v(g)=\rho(g)y-y$ for some $y\in V$ is exactly the
condition that $v$ is a 1-coboundary [Wei95, Example 6.5.6]. ∎

In the case at hand of considering the group action of the automorphism group
on theta characteristics the representation will be the reduction mod 2 (with
the mod 2 reduction of $R$ denoted by $\overline{R}$, following [KS10]) of the
transpose of the rational representation $\rho=\overline{\rho}_{r}^{T}$ acting
on $V=H_{1}(\mathcal{C},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}^{2g}$. Moreover, we
can count the number of invariant characteristics as the size of $H^{0}(G,V)$,
as $H^{0}(G,V)=V^{G}$ is exactly the submodule of invariants. This fact gives
us an immediate refinement of [KS10, Corollary 1.3].

######  Proposition 3.4.

The number of characteristics invariant under the action of the whole group is
either 0 or $2^{k}$, where $k=\dim H^{0}(G,V)$ is the dimension333When writing
$\dim$ for the dimension of the group cohomology of a $\mathbb{Z}_{2}$ vector
space, we naturally mean the dimension over $\mathbb{Z}_{2}$. of the subspace
of invariants.

###### Proof.

This is immediate from the fact $H^{0}(G,V)$ is a vector space over
$\mathbb{Z}_{2}$. ∎

We shall now want to consider two simple examples, for which we require the
(proof of the) following lemma.

######  Lemma 3.5 ([Ati71], Lemma 5.1).

Let $V$ be a finite-dimensional $\mathbb{Z}_{2}$ vector space and
$q:V\to\mathbb{Z}_{2}$ a quadratic function fixed under an affine
transformation $x\mapsto Ax+b$ whose associated bilinear $H$ defined by

$H(x,y)=q(x+y)-q(x)-q(y)$

is non-degenerate. Then the affine transformation has a fixed point.

###### Proof.

We shall recall the proof of [Ati71]. As the transform preserves $q$ we get

$q(x)=q(Ax+b)=q(Ax)+q(b)+H(Ax,b).$

Setting $x=0$ gives $q(b)=0$ and hence $q(x)=q(Ax)+H(Ax,b)$. We can thus say

$\displaystyle H(x,y)$ $\displaystyle=q(x+y)-q(x)-q(y),$
$\displaystyle=\left[q(A(x+y))+H(A(x+y),b)\right]-\left[q(Ax)+H(Ax,b)\right]$
$\displaystyle\phantom{=}\quad-\left[q(Ay)+H(Ay,b)\right],$
$\displaystyle=\left[q(Ax+Ay)-q(Ax)-q(Ay)\right],$ $\displaystyle=H(Ax,Ay),$

and so defining $A^{\ast}$ to be the dual of $A$ with respect to the non-
degenerate inner product $H$ we have $A^{\ast}A=I$. Suppose we have
$x\in\operatorname{Ker}(A-I)^{\ast}$, then

$\displaystyle A^{\ast}x=x\Rightarrow Ax=x\Rightarrow H(x,b)=0$

and hence we know $b\perp\operatorname{Ker}(A-I)^{\ast}$. Now
$(\operatorname{Ker}(A-I)^{\ast})^{\perp}=\operatorname{Im}(A-I)$, and so

$b\in\operatorname{Im}(A-I)\Rightarrow\exists y\in
V,\,b=(A-I)y\Rightarrow\exists y\in V,\,Ay+b=y.$

∎

We are now ready to consider two examples.

######  Example 3.6 (Hyperelliptic Involution).

Suppose we just have $G=C_{2}=\left\\{\pm 1\right\\}$, where the generator of
the $C_{2}$ is the hyperelliptic involution, for which the rational
representation is given by $R=-I$. For a cochain $v:G\to V$ to be closed it
must satisfy $v(1)=0\in V$, and then $v(-1):=x$ is arbitrary. It is a
coboundary if in addition there exists $y\in V$, $v(-1)=y-y=0$, and so with
the hyperelliptic representation the cohomology group is given by
$H^{1}(C_{2},V)=V$. Note this is what we would expect: if the representation
is trivial, then for all ${y}\in V$ we have ${v}_{y}={v}$, meaning that the
action is always given by the specific shift ${v}$. We may now use Lemma 3.5.
The quadratic function on $H_{1}(\mathcal{C},\mathbb{Z}_{2})$ given by the
parity is preserved under the affine action of
$\operatorname{Aut}(\mathcal{C})$, so the affine transformation given by the
hyperelliptic involution has a fixed point. The vector $b$ in the proof of
Lemma 3.5 is the value $v(-1)$, and so we must have
$v(-1)\in\operatorname{Im}0=0$. Hence we have an invariant spin structure, and
moreover we have $2^{2g}=\left\lvert V\right\rvert$ of them.

######  Example 3.7 (Cyclic groups).

Let us now try and understand [KS10, Theorem 1.1] in this language of group
cohomology. Suppose we take a cyclic automorphism group $\left\langle
f\right\rangle=G$, and let $n$ be the order of $f$. A closed cochain must have
$v(1)=0$ as before, and then it is specified by $v(f)$, which is subject only
to the condition that

(5) $(I+A+\dots+A^{n-1})v(f)=v(f^{n})=v(1)=0,$

where $A=\bar{R}^{T}=\rho(f)$ (already reduced mod 2). Provided $A\neq I$,
given that we have $(A-I)(\sum_{k=0}^{n-1}A^{k})=A^{n}-I=0$, the sum
$\sum_{k=0}^{n-1}A^{k}$ has nontrivial kernel in which some $v(f)\neq 0$ must
lie. A coboundary is given by $v(f)=(A-I)y$ for some $y\in V$, and hence

$H^{1}(G,V)\cong\operatorname{Ker}(I+\dots+A^{n-1})/\operatorname{Im}(A-I).$

Note this result is contained in [Wei95, Theorem 6.2.2].

Certainly if $A-I$ is invertible, then there is necessarily an invariant spin
structure as $H^{1}=0$, and it is unique as then
$H^{0}(G,V)=\operatorname{Ker}(A-I)=0$. Moreover the converse is true that if
there is a unique invariant characteristic then $A-I$ is invertible, as
$\operatorname{Ker}(A-I)=0$ is exactly the condition for invertibility. By the
argument required in [BN12, p. 17], we can see that for $A-I$ to be
invertible, it is necessary that $\sum_{k=0}^{n-1}A^{k}=0$, and in fact this
is sufficient when $n$ is odd as we can simply write down the inverse which is
$n^{-1}\sum_{k=1}^{n}kA^{k-1}$. Note also certainly if $n=2$ then $A-I$ cannot
be invertible as

$(A-I)^{2}=A^{2}-2A+I=0.$

This in fact generalises; if $n=2^{k}$ then $A-I$ cannot be invertible as

$(A-I)^{n}=\sum_{i=0}^{n}\binom{n}{i}A^{i}=A^{n}-I=0$

using Lucas’ theorem for the binomial coefficient of a prime power mod that
prime (see for example [Fin47, Theorem 3], or alternatively prove this with
induction).

Now, as in Example 3.6, the quadratic function given by the parity is
preserved under the affine transformation generating the group action on
$H_{1}(\mathcal{C},\mathbb{Z}_{2})$, so the transformation has a fixed point.
As such, the remaining question is just how many invariant characteristics
there are. This is given by $\dim\ker(A-I)$ and Proposition 3.4, and [KS10]
shows how to compute whether this is zero using cyclotomic polynomials.
Importantly they show that a necessary and sufficient condition when $n$ is
odd is that the genus of the quotient $g=g(\mathcal{C}/\left\langle
f\right\rangle)$, related to $f$ by
$\operatorname{rank}_{\mathbb{Z}}\ker(\rho_{r}(f)-I)=2g$, is 0. By combining
the work of [Rie92], we see that when $n$ is even the necessary and sufficient
conditions are that $g(\mathcal{C}/\left\langle f^{2^{l}}\right\rangle)=0$ for
all $l$ such that $2^{l}|n$.

######  Remark 3.8.

We can also use [Sco77, Theorem 1] to say more. Namely, suppose that $G$ is
generated by elements $\gamma_{1},\dots,\gamma_{r}$ such that
$\prod_{i}\gamma_{i}=1$, then

$\displaystyle\sum_{i=1}^{r}\left[2g-\dim
H^{0}(\left\langle\gamma_{i}\right\rangle,V)\right]$
$\displaystyle\geq\left[2g-\dim H^{0}(G,V)\right]+\left[2g-\dim
H^{0}(G,V^{\ast})\right],$ $\displaystyle\Rightarrow\dim H^{0}(G,V)$
$\displaystyle\geq\sum_{i}\dim
H^{0}(\left\langle\gamma_{i}\right\rangle,V)-2g(r-2)-\dim H^{0}(G,V^{\ast}),$

where $V^{\ast}=\operatorname{Hom}(V,\mathbb{Z}_{2})$ is the dual $G$-module.
Scott defines the homology invariant $h$ to be the nonnegative integer which
is the deficit in the inequality, as well as $\tilde{H}^{1}(G,V)$ to be the
subgroup of $H^{1}(G,V)$ which restricts to $0\in
H^{1}(\left\langle\gamma_{i}\right\rangle,V)$ for each $i$, showing
$h\geq\dim\tilde{H}^{1}(G,V)$. As such, we can use knowledge of the number of
characteristics invariant under the $\left\langle\gamma_{i}\right\rangle$ to
give information about how many possible invariant characteristics there may
be.

A key question we shall want to address using group cohomology is when the
action of $G$ has a Unique Invariant Characteristic (UIC). To make more
progress, we use the inflation-restriction exact sequence. That is for normal
subgroup $N\triangleleft G$ and abelian group $V$ with $G$ action we have
[Wei95, p. 196]

$0\to H^{1}(G/N,V^{N})\to H^{1}(G,V)\to H^{1}(N,V)^{G/N}\to
H^{2}(G/N,V^{N})\to H^{2}(G,V),$

so named because the 3 inner maps are inflation, restriction, and
transgression. Suppose we know $H^{1}(N,V)=0$ and $H^{0}(N,V)=V^{N}=0$ (as we
have when $N$ is an odd-order cyclic group quotienting to $\mathbb{P}^{1}$),
then denoting $K=G/N$ we have

$0\to H^{1}(K,0)\to H^{1}(G,V)\to 0\to H^{2}(K,0)\to H^{2}(G,V).$

This trivially gives $H^{1}(G,V)=0$, and moreover because we have
$(V^{N})^{(G/N)}\cong V^{G}$, we get $H^{0}(G,V)=0$.

In fact we do not need the restrictive condition that $H^{1}(N,V)=0$ in the
situation we care about, as we want the case where there is a unique
characteristic invariant under $N$, which is the case when
$H^{0}(N,V)=V^{N}=0$ but also when the specific affine representation of $G$
as an element in $H^{1}(G,V)$ is the zero class when restricted to the action
as an element in $H^{1}(N,V)$. This means that the class in $H^{1}(G,V)$ is in
the kernel of the restriction map, and as we have said that the image of the
inflation map is trivial, this means the class in $H^{1}(G,V)$ must be the
zero class. Read together these tell us that if we have a normal subgroup
given with a UIC, then the action of the whole group has a UIC. Moreover, this
extends to if we have a subgroup which is subnormal in the original group
(that is $H\leq G$ such that there exist $H_{i}$ with $H\triangleleft
H_{1}\triangleleft\dots\triangleleft H_{k}\triangleleft G$). In summation this
gives the following proposition.

######  Proposition 3.9.

If there exists $f\in G\leq\operatorname{Aut}(\mathcal{C})$ such that $f$ has
odd order, $\left\langle f\right\rangle$ is subnormal in $G$, and
$g(\mathcal{C}/\left\langle f\right\rangle)=0$, then $\mathcal{C}$ has a
unique theta characteristic invariant under the action of $G$. We will call a
curve with this property Subnormal Odd Cyclic (SOC).

This condition captures the existence of unique invariant characteristics in
the case of even-order cyclic group actions seen earlier in Example 3.7. In
that case, writing $n=2^{k}m$ for some odd integer $m$, the necessary and
sufficient conditions previously described are equivalent to the subnormal
subgroup $C_{m}$ having quotient genus 0. In the case of SOC curves, we are
also able to determine the parity of the invariant characteristic from the
signature.

######  Proposition 3.10.

Given a SOC curve when the subnormal cyclic group $\left\langle
f\right\rangle$ has odd order $n$ and signature $(0;c_{1},\dots,c_{r})$, the
parity is

$n\sum_{i=1}^{r}\frac{c_{i}-c_{i}^{-1}}{8}\mod 2.$

###### Proof.

Note that the $c_{i}$ must be odd because they divide $n$. Hence, using
[Ser90, Equation 16] we have that the parity is

$\displaystyle\sum_{P\in\mathcal{C}}\frac{\left\lvert\operatorname{Stab}(P)\right\rvert^{2}-1}{8}$
$\displaystyle=\sum_{Q\in\mathcal{C}/G}\sum_{\begin{subarray}{c}{P\in\mathcal{C}}\\\
{\pi(P)=Q}\end{subarray}}\frac{\left\lvert\operatorname{Stab}(P)\right\rvert^{2}-1}{8}\mod
2,$
$\displaystyle=n\sum_{\pi(P)\in\mathcal{C}/G}\frac{\left\lvert\operatorname{Stab}(P)\right\rvert-\left\lvert\operatorname{Stab}(P)\right\rvert^{-1}}{8}\mod
2,$ $\displaystyle=n\sum_{i=1}^{r}\frac{c_{i}-c_{i}^{-1}}{8}\mod 2.$

∎

##  4\. Computation of Orbit Tables

In Appendix A we shall provide tables of orbit decompositions for many
different curves of genera $\leq 9$, giving those for all possible curves of
genus 2, 3, and 4. In this section we shall make some comments on the
computation of these tables and their results.

The tables are computed using the representation in terms of binary vectors
via spin structures as in [KS10]· We restrict to curves for which we can
represent the curve in plane form as $f(x,y)=0$, whereby we may use the method
of [BSZ19] implemented in Sage to compute the rational representation. We will
also want to compute the signature of the action where possible, defined as
follows. Throughout signatures are computed with the help of the LMFDB
[LMF23], and in cases of ambiguity were verified by comparing the character of
the rational representation found using this signature and the Eichler trace
formula [Bre00, p. 41] to that found by computing directly with Sage, or by
determining the signature from the rational representation as in [Rie93].

In doing the orbit decomposition computations, we are aided by the fact that
we need only choose one representative curve from each class because the
decomposition is uniquely determined by the rational representation and two
curves in the interior of an equisymmetric family (in the sense of [Bro90])
have equivalent rational representations [RCR22, p. 896]. One can understand
this intuitively: if the coefficients of the curve are varied only slightly
(and generically) then as the rational representation is given in terms of
integer valued matrices these would not be expected to change.

It is shown in [KS10, BGS07] that the hyperelliptic involution (if it exists)
is the only nonidentity automorphism $\iota$ for which $\rho_{r}(\iota)=I\mod
2$, and so is the only nonidentity automorphism of a curve that fixes every
theta characteristic, hence we know that the reduced automorphism group
[Rau70, Pop72]

$\operatorname{\overline{\operatorname{Aut}}}(\mathcal{C}):=\left\\{\begin{array}[]{cc}\operatorname{Aut}(\mathcal{C})/\left\langle\iota\right\rangle,&\mathcal{C}\text{
is hyperelliptic},\\\
\operatorname{Aut}(\mathcal{C}),&\text{otherwise},\end{array}\right.$

acts faithfully on the theta characteristics. There is no a priori reason to
expect $\operatorname{\overline{\operatorname{Aut}}}(\mathcal{C})$ to have an
action on $\mathcal{C}$ when $\mathcal{C}$ is hyperelliptic [Rau70].

As such we shall give tables of curves of a given genus as a plane curve
$f(x,y)=0$, their reduced automorphism group (with the GAP group ID if the
presentation given of the group is not specific [GAP22]), the quotient genus
$g_{0}$ and signature $\bm{c}$ of the
$\operatorname{\overline{\operatorname{Aut}}}$ action on the curve (recall
Definition 2.3) when it exists and is known, the
$\operatorname{\overline{\operatorname{Aut}}}$ orbits of the odd and even
characteristics respectively presented as a list of values $a_{b}$ indicating
$b$ orbits of size $a$, and the total number of characteristics $I$ invariant
under the group action. When giving $f$ we shall leave in free parameters
where possible, not specifying values that must be avoided. The code to
recreate this data (except for the signature) is given in the Sage notebooks
list_of_plane_curves.ipynb and theta_characteristic_orbit.ipynb. Table
LABEL:tab:_orbit_decomposition,_genus_2 shows the table constructed in the
case of genus-2 curves.

######  Remark 4.1.

We will use the convention that the dihedral group $D_{n}$ be of size $2n$.

Table 1. Orbit decomposition, all genus-2 curves $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$y^{2}-(x^{2}-1)(x^{2}-a)(x^{2}-b)$ |  $C_{2}$, $(1;2^{2})$ |  $2_{3}$ |  $1_{4},2_{3}$ |  4
$y^{2}-(x^{2}-1)(x^{2}-a)(x^{2}-a^{-1})$ |  $V_{4}$, $(0;2^{5})$ |  $2_{1},4_{1}$ |  $1_{2},2_{2},4_{1}$ |  2
$y^{2}-(x^{5}-1)$ |  $C_{5}$, $(0;5^{3})$ |  $1_{1},5_{1}$ |  $5_{2}$ |  1
$y^{2}-(x^{6}-ax^{3}+1)$ |  $S_{3}$, $(0;2^{2},3^{2})$ |  $6_{1}$ |  $1_{1},3_{3}$ |  1
$y^{2}-(x^{6}-1)$ |  $D_{6}$, $(0;2^{3},3)$ |  $6_{1}$ |  $1_{1},3_{1},6_{1}$ |  1
$y^{2}-x(x^{4}-1)$ |  $S_{4}$ |  $6_{1}$ |  $4_{1},6_{1}$ |  0

######  Remark 4.2.

The code provided may easily be adapted to compute the orbit decomposition for
all subgroups of the automorphism group.

Every genus-2 curve with a unique invariant characteristic satisfied the
conditions of Proposition 3.9, that is it is SOC.

We shall now make some remarks about Tables
LABEL:tab:_orbit_decomposition,_genus_3,
LABEL:tab:_orbit_decomposition,_genus_4,
LABEL:tab:_orbit_decomposition,_genus_5,
LABEL:tab:_orbit_decomposition,_genus_6,
LABEL:tab:_orbit_decomposition,_genus_7,
LABEL:tab:_orbit_decomposition,_genus_8, and
LABEL:tab:_orbit_decomposition,_genus_9.

  * •

At genus 3, not every curve with a unique invariant characteristic is SOC.
There is a single exception, Klein’s curve, whose automorphism group
$\operatorname{PSL}_{3}(\mathbb{F}_{2})$ is simple and so cannot have a
nontrivial subnormal cyclic group. There is a $C_{7}$ subgroup of the
automorphism group quotienting to $\mathbb{P}^{1}$ (as clearly seen by writing
Klein’s curve in Lefschetz form [Lef21a, Lef21b, Zom10]), but it is not
subnormal.

  * •

The non-hyperelliptic genus-3 curve with
$\operatorname{\overline{\operatorname{Aut}}}=S_{4}$ is the first example of a
curve with large automorphism group without $I\leq 1$.

  * •

At genus 4 one sees the first example of curves of the same genus with the
same pair $(\operatorname{\overline{\operatorname{Aut}}},\bm{c})$, but
different orbit decompositions. This highlights the fact that the signature
does not fully the determine the rational representation, one must also pick a
generating vector [Bro91, Definition 2.2]. The converse, that the (character
of the) rational representation determines the signature, is true [Rie93].

  * •

At genus 4 not all curves with a unique invariant characteristic are SOC. The
exceptions are

    1. (1)

the curve with
$(\operatorname{\overline{\operatorname{Aut}}},\bm{c})=(A_{4},(0;2,3^{3}))$,

    2. (2)

the curve with
$(\operatorname{\overline{\operatorname{Aut}}},\bm{c})=(S_{5},(0;2,4,5))$,
Bring’s curve.

More about the orbit decomposition on Bring’s curve is said in [BDH22]; here
we only note that similarly to Klein’s curve there is an odd order cyclic
group quotienting to $\mathbb{P}^{1}$ (here a $C_{5}$) that is not subnormal.
On the $A_{4}$ curve, we note that the quotient of the curve by the $C_{3}$
action has genus 1, and hence presents the first case where the existence of a
UIC is not clearly governed by an odd cyclic quotient to $\mathbb{P}^{1}$.

  * •

At genus 5 the Wiman octic is not SOC; there is a unique characteristic
invariant under the $((C_{4}\times C_{2})\rtimes C_{4})\rtimes C_{3}$ normal
subgroup, but not under any subnormal cyclic group. Moreover, again we find
that the quotient by $C_{3}$ has genus 1, and so the Wiman octic presents the
second case where the existence of a UIC is not clearly governed by an odd
cyclic quotient to $\mathbb{P}^{1}$. The curve with automorphism group
$C_{3}\times D_{5}$ is clearly SOC.

  * •

At genus 6 every curve written down with a UIC is SOC.

  * •

Edge describes the orbits of some of the odd characteristics of the genus-7
Fricke-Macbeath curve in terms of tangent hyperplanes [Edg67].

  * •

All the genus-7 curves written with a unique invariant characteristic are SOC.

  * •

At genus 8 the two curves which have a UIC are SOC.

  * •

Not all the genus-9 curves written here with a UIC are SOC; the hyperelliptic
curve with $\operatorname{\overline{\operatorname{Aut}}}\cong A_{5}$ cannot
have a subnormal cyclic group.

At genera greater than 8 the computations were becoming prohibitively slow,
with the calculation of the symplectic automorphism group of the Fricke
octavic curve in Sage taking just under three hours on a Intel Core i5-8350U
CPU at 1.70GHz. Leaving behind the criteria of requiring a plane model of the
curve, one can compute additional examples of theta characteristic
decompositions using the code from [BRR13], available at
https://github.com/rojas-ani/sage-routines,444LDH is grateful to Anita Rojas
for her correspondence on the workings of this code. from the data of a group,
signature, and choice of generating vector provided the quotient genus is 0.
The Sage notebook genus_order_invariants_data.ipynb shows how this can be
done.

Having computed now many examples in low genera, we can pick out some families
of curves with unique invariant characteristics, giving us the following
theorem.

######  Theorem 4.3.

There are infinitely many curves, both non-hyperelliptic and hyperelliptic,
with a unique invariant characteristic.

###### Proof.

It is sufficient to consider only curves of Lefschetz type, in particular the
two families we shall consider are one of the Wiman hyperelliptic curves
$y^{2}=x^{2g+1}-1$ and the Lefschetz curves of the form
$x^{m}y^{n}+y^{m}+x^{n}=0$ for coprime $m,n$ where $p:=m^{2}-mn+n^{2}>7$ is a
prime congruent to 1 mod 3.

The former has automorphism group $C_{2g+1}\times C_{2}$, which contains the
normal subgroup $C_{2g+1}$ of odd order which quotients to $\mathbb{P}^{1}$.
The latter has automorphism group of the form $C_{p}\rtimes C_{3}$, which
contains the normal subgroup $C_{p}$ of odd order which quotients to
$\mathbb{P}^{1}$, which is easiest seen by writing the curve in the from
$\tilde{y}^{p}+\tilde{x}^{a}(1+\tilde{x})=0$ as can always be done for some
$a$ [Lef21b, p. 464]. ∎

######  Remark 4.4.

For the hyperelliptic family in the proof of Theorem 4.3 we can say more in
the case that $2g+1$ is a prime $p$, as all characteristics that are not
invariant are in orbits of order $p$.

Using Riemann-Hurwitz we know that the $C_{p}$ subgroup acts with signature
$(0;p^{r})$ where

$2\times\frac{p-1}{2}-2=p\left[-2+r\left(1-\frac{1}{p}\right)\right]\Rightarrow
r=3.$

Using Proposition 3.10 we know the parity of the UIC is $3p(p-p^{-1})/8$, and
hence is determined entirely by whether $p=\pm 1$ or $p=\pm 3$ mod 8.

##  5\. Hurwitz Curves and the Method of Dolgachev

###  5.1. Machine Learning Predictions

One use of the code for computing orbit decomposition united with the method
of [BRR13] for computing rational representation from signatures is that we
can investigate the application of machine learning techniques in illuminating
the behaviour of orbits. Machine classification (namely a pipeline using a
Standard Scaler followed by a Random Forest classifier555LDH is very grateful
to Jacob Bradley for showing him how to do this.) using features built from
the group data and the signature alone achieved an accuracy of approximately
93% in cross-validation when predicting if the corresponding group action gave
a unique invariant characteristic when trained on data of over 1000 group
actions, a far higher accuracy than the approximately $54\%$ one would expect
if choosing randomly with prior knowledge of the frequency of actions with a
UIC in the dataset. Specifically, we provided the features of genus, group
order, whether the group action was large (in the sense $\left\lvert
G\right\rvert>4(g-1)$), the maximum power of 2 dividing the group order, the
number of involutions in the group, the number of involutions up to conjugacy,
the number of entries in the signature, the number of even entries in the
signature, the maximum entry of the signature, and the dimension of the
corresponding family for 1326 group actions in genus $12$ or less. This
suggests that from very basic heuristics alone one should be able to get
strong results understanding the behaviour of UICs. Moreover, estimating
feature importance using these methods showed that whether or not a group
action was large for a given genus was unimportant in determining whether a
given action led to a UIC.

In order to lay a benchmark to guide future research we computed in Table
LABEL:tab:_invariants_prediction the prediction of the classifier given the
corresponding data for all the simple Hurwitz group with order $<10^{6}$,
provided in [Con87] (the $J_{i}$ are the first two Janko groups). These
results are correct for the two Hurwitz curves known, though removing their
data from the training set makes the classifier less accurate. Running a
similar pipeline which instead predicts whether a group action has 0, 1, or
many invariants characteristics confirms the predictions of Table
LABEL:tab:_invariants_prediction, predicting those without a unique invariant
characteristic have none.

Table 2. Machine prediction of whether $I=1$, all simple Hurwitz groups of order $<10^{6}$ $G$ |  $g$ |  $I=1$
---|---|---
$\operatorname{PSL}_{2}(7)$ |  3 |  True
$\operatorname{PSL}_{2}(8)$ |  7 |  False
$\operatorname{PSL}_{2}(13)$ |  14 |  True
$\operatorname{PSL}_{2}(27)$ |  118 |  True
$\operatorname{PSL}_{2}(29)$ |  146 |  True
$\operatorname{PSL}_{2}(41)$ |  411 |  False
$\operatorname{PSL}_{2}(43)$ |  474 |  True
$J_{1}$ |  2091 |  False
$\operatorname{PSL}_{2}(71)$ |  2131 |  False
$\operatorname{PSL}_{2}(83)$ |  3404 |  True
$\operatorname{PSL}_{2}(97)$ |  5433 |  False
$J_{2}$ |  7201 |  False
$\operatorname{PSL}_{2}(113)$ |  8589 |  False
$\operatorname{PSL}_{2}(125)$ |  11626 |  True

In §5.2-5.6 we will now show how to compute number of invariant
characteristics for the curves in Table LABEL:tab:_invariants_prediction
analytically using a new method, and in §5.7, Table
LABEL:tab:_invariants_prediction_answers, we shall compare the answers.

###  5.2. The Method of Dolgachev

Approximately four months after constructing these machine estimates we were
alerted by Vanya Cheltsov to the paper [Dol97b], which allows one to get
analytic solutions to the questions raised in Table
LABEL:tab:_invariants_prediction, which we shall describe now.

Denote with $\operatorname{Pic}(\mathcal{C})^{G}$ the group of $G$-invariant
isomorphism classes of line bundles on $\mathcal{C}$, that is line bundles
$L\to\mathcal{C}$ that admit a collection of isomorphisms
$\phi_{g}:g^{\ast}L\to L$ for each $g\in G$. Moreover, let
$\operatorname{Pic}(G;\mathcal{C})$ denote the group of $G$-linearised line
bundles on $\mathcal{C}$ and corresponding linearisations, that is
$G$-invariant line bundles that admit a collection of isomorphisms such that

$\forall g,h\in G,\quad\phi_{g\circ h}=\phi_{h}\circ h^{\ast}\phi_{g}.$

Thinking in terms of divisors $\operatorname{Div}(\mathcal{C})$, principal
divisors $\operatorname{PDiv}(\mathcal{C})$, and nonzero meromorphic functions
$\mathcal{M}(\mathcal{C})^{\times}$ instead of line bundles,
$\operatorname{Pic}(\mathcal{C})^{G}$ corresponds to
$(\operatorname{Div}(\mathcal{C})/\operatorname{PDiv}(\mathcal{C}))^{G}$,
while $\operatorname{Pic}(G;\mathcal{C})$ corresponds to
$\operatorname{Div}(\mathcal{C})^{G}/[\mathcal{M}(\mathcal{C})^{\times}]^{G}$
[Dol97b, Corollary 2.3]. The key results of Dolgachev are the following.

######  Proposition 5.1 ([Dol97b], Proposition 2.2).

We have a short exact sequence of abelian groups

(6)
$0\to\operatorname{Hom}(G,\mathbb{C}^{\times})\to\operatorname{Pic}(G;\mathcal{C})\to\operatorname{Pic}(\mathcal{C})^{G}\to
H^{2}(G,\mathbb{C}^{\times})\to 0.$

Here $H^{2}(G,\mathbb{C}^{\times})$ is the group cohomology viewing
$\mathbb{C}^{\times}$ as a trivial $G$-module, which is known as the Schur
multiplier of the group $G$.

###### Proof.

Dolgachev proves this using the spectral sequence of [Gro57, §5.2], but we
shall give a more elementary approach here following [FV89].

Observe that the Short Exact Sequence (SES)

$0\to\operatorname{PDiv}(\mathcal{C})\to\operatorname{Div}(\mathcal{C})\to\operatorname{Pic}(\mathcal{C})\to
0,$

gives rise to the Long Exact Sequence (LES)

$0\to\operatorname{PDiv}(\mathcal{C})^{G}\to\operatorname{Div}(\mathcal{C})^{G}\to\operatorname{Pic}(\mathcal{C})^{G}\to
H^{1}(G,\operatorname{PDiv}(\mathcal{C}))\to
H^{1}(G,\operatorname{Div}(\mathcal{C})).$

We can use [FV89, Lemma 4.1] to say

  * •

$H^{1}(G,\operatorname{Div}(\mathcal{C}))=0$, and

  * •

$H^{1}(G,\operatorname{PDiv}(\mathcal{C}))\cong H^{2}(G,\mathbb{C}^{\times})$.

Moreover, from the SES

$0\to\mathbb{C}^{\times}\to\mathcal{M}(\mathcal{C})^{\times}\to\operatorname{PDiv}(\mathcal{C})\to
0,$

where $\mathcal{M}(\mathcal{C})$ are the meromorphic functions on the curve,
we get the portion of the LES

$\displaystyle 0$
$\displaystyle\to\mathbb{C}^{\times}\to[\mathcal{M}(\mathcal{C})^{\times}]^{G}\to\operatorname{PDiv}(\mathcal{C})^{G}\to\operatorname{Hom}(G,\mathbb{C}^{\times})\to
0,$

where we have used [Wei95, Exercise 6.1.6] to equate
$H^{1}(G,\mathbb{C}^{\times})=\operatorname{Hom}(G,\mathbb{C}^{\times})$. As
such, quotienting by $[\mathcal{M}(\mathcal{C})^{\times}]^{G}$ we are done. ∎

######  Remark 5.2.

The map
$\operatorname{Pic}(G;\mathcal{C})\to\operatorname{Pic}(\mathcal{C})^{G}$
sends any $G$-linearised line bundle $L$ and its choice of linearisation to
the isomorphism class of $L$. In this way, we think of
$\operatorname{Hom}(G,\mathbb{C}^{\times})$ as parametrising the possible
$G$-linearisations of the trivial bundle $\mathcal{O}_{\mathcal{C}}$.

######  Proposition 5.3 ([Dol97b]).

Suppose that $G$ acts on $\mathcal{C}$ with signature $(0;c_{1},\dots,c_{r})$.
Then we have

(7)
$\operatorname{Pic}(G;\mathcal{C})\cong\mathbb{Z}\oplus\left[\bigoplus_{i=1}^{r-1}\mathbb{Z}/(d_{i}/d_{i-1})\mathbb{Z}\right],$

where

$d_{i}=\gcd\left\\{\prod_{j\in
S}c_{j}\,|\,S\subset\left\\{1,\dots,r\right\\},\left\lvert
S\right\rvert=i\right\\},\quad\text{taking}\quad d_{0}=1.$

The generator of the $\mathbb{Z}$ factor $\gamma$ has underlying $G$-invariant
line bundle (isomorphism class) $\Gamma$ determined by

$K_{\mathcal{C}}=N\left(r-2-\sum_{i=1}^{r}\frac{1}{c_{i}}\right)\Gamma,$

where $N=\operatorname{lcm}(c_{i})$.

###### Proof.

We shall not recreate the entire proof but note that, as it shall be helpful
later, letting $Q_{1},\dots,Q_{r}\in\mathbb{P}^{1}$ be the branch points of
the projection $\pi:\mathcal{C}\to\mathcal{C}/G$, then
$\operatorname{Pic}(G;\mathcal{C})$ is generated by the $G$-invariant divisors

$D_{i}=\pi^{-1}(Q_{i})=\sum_{\begin{subarray}{c}{P\in\mathcal{C}}\\\
{\pi(P)=Q_{i}}\end{subarray}}P.$

These are subject to the equivalence relation
$c_{i}D_{i}-c_{j}D_{j}=\pi^{\ast}(Q_{i}-Q_{j})\sim 0$. Using Smith normal form
to compute the abelian invariants yields the $d_{i}$. ∎

######  Corollary 5.4.

In the situation of Proposition 5.3, letting $e=\left\lvert\left\\{\text{even
$c_{i}$}\right\\}\right\rvert$,

$\displaystyle\operatorname{Pic}(G;\mathcal{C})/2\operatorname{Pic}(G;\mathcal{C})$
$\displaystyle=\left\\{\begin{array}[]{cc}\mathbb{Z}_{2},&e=0,\\\
\mathbb{Z}_{2}^{e},&e\neq 0.\end{array}\right.,$
$\displaystyle\operatorname{Pic}(G;\mathcal{C})_{2}$
$\displaystyle=\left\\{\begin{array}[]{cc}0,&e=0,\\\
\mathbb{Z}_{2}^{e-1},&e\neq 0.\end{array}\right..$

###### Proof.

The first equality from writing

$\operatorname{Pic}(G;\mathcal{C})/2\operatorname{Pic}(G;\mathcal{C})=\left\langle
D_{i},\,i=1,\dots,r\,|\,\forall
i,j,\,c_{i}D_{i}-c_{j}D_{j}=0=2D_{i}\right\rangle.$

If all $c_{i}$ are odd, then we may use the second relation to reduce the
first relation to $D_{i}=D_{j}$, and then the cokernel has presentation
$\left\langle D_{1}\,|\,2D_{1}=0\right\rangle$. If there are $c_{i}$ which are
even, say $c_{1},\dots,c_{e}$ then they will drop out of the first relation to
just give $D_{i}=0$ for $i>e$ and then the cokernel has presentation
$\left\langle D_{i},\,i=1,\dots,e\,|\,\forall i,\,2D_{i}=0\right\rangle$.

The second follows from observing that

(8)
$\mathbb{Z}/N\mathbb{Z}\oplus\left[\bigoplus_{i=1}^{r-1}\mathbb{Z}/(d_{i}/d_{i-1})\mathbb{Z}\right]=\bigoplus_{i=1}^{r}\mathbb{Z}/(d_{i}/d_{i-1})\mathbb{Z}=\bigoplus_{i=1}^{r}\mathbb{Z}/c_{i}\mathbb{Z}.$

The 2-torsion part of the left hand side is
$\mathbb{Z}_{\gcd(2,N)}\oplus\operatorname{Pic}(G;\mathcal{C})_{2}$, while
2-torsion of the right hand side is $\mathbb{Z}_{2}^{e}$. The result follows
using

$\mathbb{Z}_{\gcd(2,N)}=\left\\{\begin{array}[]{cc}0,&e=0,\\\
\mathbb{Z}_{2}&e\neq 0.\end{array}\right.$

∎

As invariant theta characteristics would give elements of
$\operatorname{Pic}(\mathcal{C})^{G}$ which square to $K_{\mathcal{C}}$, we
can hope that in particular circumstances Equation (7) may give us enough
information to determine the number of invariant characteristics. We have the
following characterisations.

######  Lemma 5.5.

Suppose we are in the situation of Proposition 5.3. Then

  * •

an invariant characteristic exists if and only if the image of
$K_{\mathcal{C}}\in\operatorname{Pic}(G;\mathcal{C})$ under the map
$\operatorname{Pic}(G;\mathcal{C})\to\operatorname{Pic}(\mathcal{C})^{G}$ is a
square, and

  * •

if exactly $m>0$ invariant characteristic exists, then the 2-torsion subgroup
$\operatorname{Pic}(\mathcal{C})^{G}_{2}$ has rank $m-1$.

As computing the 2-torsion subgroup of $\operatorname{Pic}(\mathcal{C})^{G}$
will be important, we recall briefly a quick fact from homological algebra

######  Lemma 5.6.

Given a SES $0\to A\to B\to C\to 0$ there is an associated LES

(9) $0\to A_{2}\to B_{2}\to C_{2}\to A/2A\to B/2B\to C/2C\to 0$

where $A_{2}$ is the 2-torsion part of $A$ (and likewise for $B,C$).

######  Example 5.7 (Cyclic Groups).

Consider the case $G=C_{n}$ with quotient genus 0, which gives

$\operatorname{Hom}(G,\mathbb{C}^{\times})\cong C_{n},\quad
H^{2}(G,\mathbb{C}^{\times})\cong 0,$

and

$\operatorname{Pic}(G;\mathcal{C})\cong\mathbb{Z}\oplus\left[\bigoplus_{i=1}^{r-1}\mathbb{Z}/(d_{i}/d_{i-1})\mathbb{Z}\right],$

with

$K_{\mathcal{C}}=N\left(r-2-\sum_{i}\frac{1}{c_{i}}\right)\Gamma.$

Equation (6) thus becomes

(10)
$0\to\mathbb{Z}_{n}\to\operatorname{Pic}(G;\mathcal{C})\to\operatorname{Pic}(\mathcal{C})^{G}\to
0.$

In the case of a cyclic group action which quotients to $\mathbb{P}^{1}$
[Har66] shows that $N=n$. Then Riemann-Hurwitz gives that
$K_{\mathcal{C}}=2(g-1)\Gamma$ which shows that $(g-1)\Gamma$ is an invariant
theta characteristic. We now need to count how many exist.

Applying Lemma 5.6 to Equation (10) we get

(11)
$0\to\mathbb{Z}_{\gcd(2,n)}\to\operatorname{Pic}(G;\mathcal{C})_{2}\to\operatorname{Pic}(\mathcal{C})^{G}_{2}\to\mathbb{Z}_{n}/2\mathbb{Z}_{n}\to\operatorname{Pic}(G;\mathcal{C})/2\operatorname{Pic}(G;\mathcal{C}),$

We analyse this in the case of $n$ odd and $n$ even. Let $e$ be the number of
$c_{i}$ which are even. We will use Corollary 5.4.

  * •

When $n$ is odd, $e=0$, and so Equation (11) becomes

$0\to 0\to 0\to\operatorname{Pic}(\mathcal{C})^{G}_{2}\to 0\to\mathbb{Z}_{2},$

and immediately we see $\operatorname{Pic}(\mathcal{C})^{G}_{2}=0$, i.e. the
invariant characteristic is unique.

  * •

When $n$ is even Equation (11) becomes

$0\to\mathbb{Z}_{2}\to\mathbb{Z}_{2}^{e-1}\to\operatorname{Pic}(\mathcal{C})^{G}_{2}\to\mathbb{Z}_{2}\to\mathbb{Z}_{2}^{e}.$

At this stage we claim that the map
$\operatorname{Pic}(\mathcal{C})^{G}_{2}\to\mathbb{Z}_{2}$ is the zero map. To
see this, suppose we have $[L]\in\operatorname{Pic}(\mathcal{C})^{G}_{2}$.
Because $H^{2}(G,\mathbb{C}^{\times})=0$ we know $[L]=[L^{\prime}]$ where
$L^{\prime}$ is some $G$-linearisable line bundle. Pick a linearisation
$\left\\{\phi_{g}\right\\}$ of $L^{\prime}$, of which there are $n$ choices.
We know $[L^{2}]=[L]^{2}=[\mathcal{O}_{\mathcal{C}}]$, so $L^{2}$ is the image
of a linearisation $\left\\{\psi_{g}\right\\}$ of the trivial line bundle
(there are $n$ many of them), and it is this trivialisation the connecting map
sends $[L]$ to. We have $\phi_{g}^{2}\sim\psi_{g}$, and then
$\left\\{\psi_{g}\right\\}$ is necessarily trivial in the cokernel. Suppose we
had instead chosen $\left\\{\tilde{\phi}_{g}\right\\}$ as the linearisation of
$L^{\prime}$. Then the two linearisations would differ by an element of
$\operatorname{Hom}(G,\mathbb{C}^{\times})$, and so $\phi_{g}^{2}$ differs
from $\tilde{\phi}_{g}$ by an element of
$2\operatorname{Hom}(G,\mathbb{C}^{\times})$, so they are equal in the
cokernel.

###  5.3. Hurwitz Curves

When $\mathcal{C}$ is a Hurwitz curve and $G$ is the full automorphism group,
we know $\left\lvert G\right\rvert=84(g-1)$ and the signature is $(0;2,3,7)$,
which means that simply $\operatorname{Pic}(G;\mathcal{C})=\mathbb{Z}\gamma$,
and moreover that $\Gamma=K_{\mathcal{C}}$. Moreover, all Hurwitz groups are
perfect (that is $\left[G,G\right]=G$) [Con90], and so
$\operatorname{Hom}(G,\mathbb{C}^{\times})=0$. This reduces Equation (7) to

(12) $0\to\mathbb{Z}K_{\mathcal{C}}\to\operatorname{Pic}(\mathcal{C})^{G}\to
H^{2}(G,\mathbb{C}^{\times})\to 0.$

Obviously when $H^{2}(G,\mathbb{C}^{\times})=0$, Equation (12) is just

$0\to\mathbb{Z}K_{\mathcal{C}}\to\operatorname{Pic}(\mathcal{C})^{G}\to 0,$

and no invariant characteristics can exist, so we assume that the multiplier
group is nontrivial from hereon in.

We can immediately apply Lemma 5.6 to get

$0\to\operatorname{Pic}(G;\mathcal{C})_{2}\to\operatorname{Pic}(\mathcal{C})_{2}^{G}\to
H^{2}(G,\mathbb{C}^{\times})_{2}\to\operatorname{Pic}(G;\mathcal{C})/2\operatorname{Pic}(G;\mathcal{C}),$

which can be rewritten as

(13) $0\to 0\to\operatorname{Pic}(\mathcal{C})_{2}^{G}\to
H^{2}(G,\mathbb{C}^{\times})_{2}\to\mathbb{Z}_{2}.$

Dolgachev provides the following helpful interpretation.

######  Lemma 5.8 ([Dol97a]).

Let $G$ be a perfect group acting with signature containing only a single even
element $c_{1}=2$. The map
$H^{2}(G,\mathbb{C}^{\times})_{2}\to\operatorname{Pic}(G;\mathcal{C})/2\operatorname{Pic}(G;\mathcal{C})$
is the restriction map $H^{2}(G,C_{2})\to H^{2}(G_{P_{1}},C_{2})$ where
$P_{1}\in\mathcal{C}$ is a ramification point whose isotropy subgroup
$G_{P_{1}}\cong C_{2}$.

When we restrict to looking at simple Hurwitz groups, we may use the fact that
the Schur multipliers have been computed in the Atlas [Con85] and are cyclic
groups. In particular, for the finite simple groups which are Hurwitz groups
[Con90, Wil93, Wil01] we give the relevant multipliers in Table
LABEL:tab:_finite_simple_Hurwitz_groups.

Table 3. All finite simple Hurwitz groups and their Schur multiplier $G$ |  $H^{2}(G,\mathbb{C}^{\times})$ |  Comments
---|---|---
$\operatorname{PSL}_{2}(q)$ |  $C_{\gcd(2,q-1)}$ |  $q=7$, $q=p$ for prime $p=\pm 1\mod 7$, or $q=p^{3}$ for prime $p=\pm 2,\pm 3\mod 7$.
$A_{n}$ |  $C_{2}$ |  $A_{n}$ Hurwitz for all but finitely many $n$
${}^{\leavevmode{2}}_{\mathchoice{\makebox[5.00002pt][c]{$\displaystyle$}}{\makebox[5.00002pt][c]{$\textstyle$}}{\makebox[3.5pt][c]{$\scriptstyle$}}{\makebox[2.5pt][c]{$\scriptscriptstyle$}}}G_{2}(3^{p})$ |  trivial |  $p$ a prime $>3$
$J_{1}$ |  trivial |
$J_{2}$ |  $C_{2}$ |
$Co_{3}$ |  trivial |
$He$ |  trivial |
$Ru$ |  $C_{2}$ |
$HN$ |  trivial |
$Ly$ |  trivial |
$Fi_{24}^{\prime}$ |  $C_{3}$ |
$Th$ |  trivial |
$Fi_{22}$ |  $C_{6}$ |
$J_{4}$ |  trivial |
$M$ |  trivial |

Our earlier results show that the Hurwitz curves with automorphism group
$\operatorname{PSL}_{2}(8)$,
${}^{\leavevmode{2}}_{\mathchoice{\makebox[5.00002pt][c]{$\displaystyle$}}{\makebox[5.00002pt][c]{$\textstyle$}}{\makebox[3.5pt][c]{$\scriptstyle$}}{\makebox[2.5pt][c]{$\scriptscriptstyle$}}}G_{2}(3^{p})$,
$J_{1}$, $Co_{3}$, $He$, $HN$, $Ly$, $Th$, $J_{4}$, or $M$ have no invariant
theta characteristics as the corresponding multiplier group is trivial.

Now because $\operatorname{Hom}(G,\mathbb{C}^{\times})=0$ we have that
$\operatorname{Pic}(G;\mathcal{C})\cong\mathbb{Z}\gamma$ is a subgroup of
$\operatorname{Pic}(\mathcal{C})^{G}$ of index $n:=\left\lvert
H^{2}(G,\mathbb{C}^{\times})\right\rvert$. Set
$A=\operatorname{Pic}(G;\mathcal{C})\cong\mathbb{Z}$,
$B=\operatorname{Pic}(\mathcal{C})^{G}\cong\mathbb{Z}\oplus T$, where $T$ is
the torsion subgroup, $C=H^{2}(G,\mathbb{C}^{\times})\cong C_{n}$, and write
our exact sequence as

$0\to\mathbb{Z}\overset{\phi}{\to}\mathbb{Z}\oplus
T\overset{\psi}{\to}C_{n}\to 0,\qquad\phi(1)=(k,l).$

Now if $0\rightarrow A\rightarrow B\rightarrow C$ is exact then so is
$0\rightarrow\operatorname{Tor}(A)\rightarrow\operatorname{Tor}(B)\rightarrow\operatorname{Tor}(C)$.
Thus $T$ injects into the cyclic group $C_{n}$ and so $T\cong C_{c}$ for some
$c|n$. The matrix of generators and relations of the quotient
$B/A=\operatorname{Pic}(\mathcal{C})^{G}/\operatorname{Pic}(G;\mathcal{C})$ is
then $\begin{pmatrix}0&c\\\ k&l\end{pmatrix}$. The Smith normal form of this
will be $C_{d_{1}}\times C_{d_{2}/d_{1}}$ where $d_{1}=\gcd(c,k,l)$,
$d_{2}=ck$. For this quotient to be isomorphic to $C_{n}$, it certainly must
be true that $ck=n$.

Now an invariant theta characteristic exists if and only if $k$ is even and
there exists some $m$ such that $l=2m\mod c$, which is to say that an
invariant characteristic can never exist when $n$ is odd as then necessarily
$k$ is odd. This tells us that Hurwitz curves with automorphism group
$Fi^{\prime}_{24}$ have no invariant theta characteristics, and hereon we
assume $n$ is even. Looking at Table LABEL:tab:_finite_simple_Hurwitz_groups
we see that when $n$ is even, the maximum power of 2 dividing $n$ is just 2,
and so we have

$k\text{ even}\Leftrightarrow c\text{
odd}\Leftrightarrow\operatorname{Pic}(\mathcal{C})^{G}_{2}=0.$

As $c$ being odd is sufficient to be able to solve $l=2m\mod c$ for some $m$,
we see that we have an invariant characteristics, and moreover a unique one,
when $\operatorname{Pic}(\mathcal{C})^{G}_{2}=0$. By using Lemma 5.8 with
Equation (13) this is equivalent to having the map $H^{2}(G,C_{2})\to
H^{2}(G_{P_{1}},C_{2})$ being injective, equivalently surjective.

######  Proposition 5.9.

Let $\mathcal{C}$ be a Hurwitz curve with simple automorphism group $G$. Then
$\mathcal{C}$ has a unique invariant characteristic exactly when
$H^{2}(G,\mathbb{C}^{\times})\cong C_{n}$ has $n>0$ even and the map
$H^{2}(G,C_{2})\to H^{2}(G_{P_{1}},C_{2})$ is surjective. Otherwise,
$\mathcal{C}$ has no invariant characteristics.

One way to understand the map $H^{2}(G,C_{2})\to H^{2}(G_{P_{1}},C_{2})$ is
using the fact that isomorphism classes of central extensions of $G$ by an
abelian group $A$, $0\to A\overset{i}{\to}E\overset{\pi}{\to}G\to 0$, are in
bijection with elements of $H^{2}(G,A)$ where $A$ is a trivial $G$-module.
Given then inclusion $\imath:H\hookrightarrow G$, the restriction morphism
$H^{2}(G,A)\to H^{2}(H,A)$ works via the diagram [Wei95, Exercise 6.6.4]

${0}$${A}$${E^{\prime}}$${H}$${0}$${0}$${A}$${E}$${G}$${0,}$$\scriptstyle{=}$$\scriptstyle{\imath}$$\scriptstyle{\pi}$

sending the cocycle corresponding to the extension $E$ to the cocycle
corresponding to the extension

$E^{\prime}:=E\times_{G}H=\left\\{(e,h)\in E\times
H\,|\,\pi(e)=\imath(h)\right\\}.$

There are two central extensions of $G=C_{2}$ by $A=C_{2}$, these are
$E=V_{4}$ and $E=C_{4}$, with the latter corresponding to the generator of
$H^{2}(C_{2},C_{2})$.

Restricting to the case where
$H^{2}(G,\mathbb{C}^{\times})_{2}=H^{2}(G,C_{2})\cong C_{2}$, suppose
$g=\gamma_{1}\in G$ generates the subgroup $H=G_{P_{1}}$ and $\tilde{g}$ is
one of the elements of $\pi^{-1}(g)\in E$ where $E$ is the unique nontrivial
$C_{2}$ central extension of $G$, often denoted $2\cdot G$. We can define
likewise $\tilde{e}$ to be the order-2 lift of the identity $e\in G$. We then
have

$E^{\prime}=\left\\{(e,e),(\tilde{e},e),(\tilde{g},g),(\tilde{e}\tilde{g},g)\right\\}.$

We define the lifting order of $g$ to be the order of $\tilde{g}$ in $E$, and
it depends only on the conjugacy class of $g$ [Con85]. There are two possible
cases:

  1. (1)

the lifting order is 2, so $\tilde{g}^{2}=e$, and $E^{\prime}\cong V_{4}$, or

  2. (2)

the lifting order is 4, to $\tilde{g}^{2}=\tilde{e}$, and $E^{\prime}\cong
C_{4}$.

The latter is equivalent to the map $H^{2}(G,C_{2})\to H^{2}(G_{P_{1}},C_{2})$
being surjective. As such, we have proven the following result.

######  Proposition 5.10.

Given a Hurwitz curve $\mathcal{C}$ with simple automorphism group $G$, and
$g\in G$ generating the stabiliser group $G_{P_{1}}\cong C_{2}$, either:

  1. (1)

$H^{2}(G,\mathbb{C}^{\times})$ is cyclic of odd order, and $\mathcal{C}$ has
no invariant characteristics,

  2. (2)

$H^{2}(G,\mathbb{C}^{\times})$ is cyclic of even order, the lifting order of
$g$ in $2\cdot G$ is 2, and $\mathcal{C}$ has no invariant characteristics, or

  3. (3)

$H^{2}(G,\mathbb{C}^{\times})$ is cyclic of even order, the lifting order of
$g$ in $2\cdot G$ is 4, and $\mathcal{C}$ has exactly one invariant
characteristic.

We will now consider distinguish between the latter two cases for the curves
in Table LABEL:tab:_finite_simple_Hurwitz_groups.

###  5.4. $\operatorname{PSL}_{2}(q)$

First consider the case of $G=\operatorname{PSL}_{2}(q)$. The only even value
of $q$ which gives a Hurwitz group is $q=8$, corresponding to the Fricke-
Macbeath curve. In this case the Schur multiplier group is trivial, and so we
have analytically shown that the Fricke-Macbeath curve has 0 invariant
characteristics.

Restrict then to the case when $q$ is odd. The generator of the group

$H^{2}(\operatorname{PSL}_{2}(q),C_{2})\cong C_{2}$

corresponds to the extension

$1\to
C_{2}\to\operatorname{SL}_{2}(q)\overset{\pi}{\to}\operatorname{PSL}_{2}(q)\to
1.$

######  Proposition 5.11.

The lifting order of any involution in $\operatorname{PSL}_{2}(q)$ to
$\operatorname{SL}_{2}(q)$ is 4.

###### Proof.

One can show easily by exhaustively writing out entries that involutions in
$\operatorname{SL}_{2}(q)$ are diagonal of the form
$\operatorname{diag}(a,a^{-1})$ where $a^{2}=1$. There are $n$th roots of
unity in $\operatorname{GF}(q)$ if and only if $n|q-1$, and in that event
there are $\gcd(n,q-1)$ many roots. We are focusing on square roots of units,
which will exist when $q$ is odd as we have, and then the $\gcd(2,q-1)=2$
square roots must be $\pm 1$. As such $\tilde{g}^{2}=1\Rightarrow\tilde{g}=\pm
1\Rightarrow g$ is the identity in $\operatorname{PSL}_{2}(q)$. Hence any
involution must have lifting order 4. ∎

######  Corollary 5.12.

Hurwitz curves with automorphism group $\operatorname{PSL}_{2}(q)$, $q$ odd,
have a unique invariant characteristic.

###  5.5. $A_{n}$

The alternating group $A_{n}$ is a Hurwitz group for all $n$ except 64 values
in the range $1\leq n\leq 167$ [Con80]. The group $S_{n}$ has the
$n$-dimensional permutation representation which splits into a sum of a
1-dimensional acting on vectors with equal entries and a $(n-1)$-dimensional
representation acting on vectors whose entries sum to 0 (that is on an $n-1$
simplex), and so this gives an embedding
$A_{n}\hookrightarrow\operatorname{SO}(n-1)$. Restricting to $n>4$, the
nontrivial double cover of $\operatorname{SO}(n-1)$ is the well known group
$\operatorname{Spin}(n-1)$, and so we get a nontrivial double cover of $A_{n}$
denoted $2\cdot A_{n}$ as

${0}$${C_{2}}$${2\cdot
A_{n}}$${A_{n}}$${0}$${0}$${C_{2}}$${\operatorname{Spin}(n-1)}$${\operatorname{SO}(n-1)}$${0.}$$\scriptstyle{=}$$\scriptstyle{\imath}$$\scriptstyle{\pi}$

This is the nontrivial element of $H^{2}(A_{n},C_{2})\cong C_{2}$.

We can then reduce the question to asking about whether, given nontrivial
$g\in\operatorname{SO}(n-1)$ such that $g^{2}=1$, under what conditions can we
have $\tilde{g}^{2}=1$ where $\tilde{g}\in\operatorname{Spin}(n-1)$ is a lift
of $g$.

######  Lemma 5.13.

$\tilde{g}^{2}=1$ if and and only if $g$ is a product of $4k$ disjoint
transpositions for some integer $k$.

###### Proof.

This is proven in [BF02, Proposition 5.10], though we shall take a different
approach here. Using the geometric interpretation of multiplication in
$\operatorname{Spin}(n-1)$ as coming from lifts of paths in
$\operatorname{SO}(n-1)$, we see that if we take a path
$\gamma:[0,1]\to\operatorname{SO}(n-1)$ such that $\gamma(0)=1$,
$\gamma(1)=g$, then $\gamma^{2}:[0,1]\to\operatorname{SO}(n-1)$ is a path with
$\gamma^{2}(0)=1=g^{2}=\gamma^{2}(1)$, such that when we lift this to a path
$\widetilde{\gamma^{2}}:[0,1]\to\operatorname{Spin}(n-1)$ with
$\widetilde{\gamma^{2}}(0)=1$,
$\tilde{g}^{2}:=\widetilde{\gamma^{2}}(1)\in\pi^{-1}(1)$. This tells us that
$\tilde{g}^{2}=1$ precisely when the path $\gamma^{2}$ is contractible. Now we
may diagonalise $g$ as

$g=\operatorname{diag}(\underbrace{-1}_{\times 2l},\underbrace{1}_{\times
m}),$

where we must have $2l+m=n-1$ and $m-2l=\operatorname{Tr}(g)$, which together
determine $4l=n-1-\operatorname{Tr}(g)$ and $2m=n-1+\operatorname{Tr}(g)$. We
can work out $\operatorname{Tr}(g)$ using some representation theory when $g$
corresponds to an element of $A_{n}$. Recall that in the permutation
representation $\chi(g)=\left\lvert\operatorname{Fix}(g)\right\rvert$ is the
number of fixed points of $g$. Because the permutation representation
decomposes as $\underline{n-1}\oplus\underline{1}$ we must have
$\operatorname{Tr}(g)=\left\lvert\operatorname{Fix}(g)\right\rvert-1$. Thus
$\left\lvert\operatorname{Fix}(g)\right\rvert=n-4l$. As $g$ is an involution,
its cycle type will consist of just transpositions, and so
$\left\lvert\operatorname{Fix}(g)\right\rvert=n-2(\\#\text{transpositions})$;
indeed, as $g\in A_{n}$ it must be a product of an even number of
transpositions. We have therefore that $g$ consists of $2l$ transpositions.

Now each $\operatorname{diag}(-1,-1)$ block in $g$ corresponds to a rotation
of a disjoint plane by $\pi$ radians, and the path in $\operatorname{SO}(n-1)$
corresponding to the rotation about this axis by an angle increasing from $0$
to $2\pi$ is non-contractible. The fundamental group of
$\operatorname{SO}(n-1)$ is $\pi_{1}(\operatorname{SO}(n-1))=\mathbb{Z}_{2}$,
so a composition of non-contractible loops will be contractible if there are
an even number of them. Hence the overall path $\gamma^{2}$ will be
contractible if and only if $l$ is even. ∎

######  Corollary 5.14.

The lifting order of $g$ is 4, and hence the corresponding $A_{n}$ Hurwitz
curve has a UIC, if and only if $l$ is odd, when $g$ is written as the product
of $2l$ disjoint $2$-cycles.

We may give a relatively simple criterion to identify the two cases. In
particular, when $A_{n}$ is a Hurwitz group let $\gamma_{i}$, $i=1,2,3$, be
the generators of order $2,3,7$ respectively in the corresponding generating
vector, and denote
$f_{i}=\left\lvert\operatorname{Fix}(\gamma_{i})\right\rvert$. Then [Con80,
Con84] there exists $h\in\mathbb{Z}_{\geq 0}$ such that

$n=84(h-1)+21f_{1}+28f_{2}+36f_{3}.$

Using that $n-f_{1}=4l$, we get a simple lemma.

######  Lemma 5.15.

The parity of $l$ is determined by

$2|l\Leftrightarrow h-1+f_{1}+f_{2}+f_{3}\equiv 0\mod 2.$

The five smallest values of $n$ for which we get that $A_{n}$ is a Hurwitz
group are found in [EGM05] and yield Table LABEL:tab:_UIC_for_An giving the
number of invariant characteristics $I$.

Table 4. First five $A_{n}$ Hurwitz groups and the number of invariant characteristics $(n;h,f_{1},f_{2},f_{3})$ | $I=h-1+f_{1}+f_{2}+f_{3}\mod 2$
---|---
$(15;0,3,0,1)$ | 1
$(21;0,1,3,0)$ | 1
$(22;0,2,1,1)$ | 1
$(28;0,4,1,0)$ | 0
$(29;0,1,2,1)$ | 1

This shows that the question of whether an $A_{n}$ Hurwitz curve has a UIC
depends on $n$ in a way that is not immediately clear. More research is
required to determine the structure of the involution which generates $A_{n}$
as a Hurwitz group.

###  5.6. Sporadic Examples

It remains to determine the number of invariant characteristics for the
sporadic simple groups $J_{2}$, $Ru$ and $Fi_{22}$. The Atlas [Con85] records
the lifting orders of conjugacy classes of $G$ to the double cover $2\cdot G$,
and so we need only to identify the conjugacy class of the generator of
$G_{P_{1}}$.

  * •

$J_{2}$ has two conjugacy classes of involutions, $2A$ and $2B$, with lifting
order 2 and 4 respectively in $2\cdot J_{2}$. It is known that the $2B$
conjugacy class is that which generated $J_{2}$ as a Hurwitz group [FR73,
Bro91], and so a $J_{2}$ Hurwitz curve has a UIC.

  * •

$Ru$ has two conjugacy classes of involutions, $2A$ and $2B$, which have
lifting order 2 and 4 respectively in $2\cdot Ru$. It is shown in [DA03] that
$Ru$ is $(2B,3,7)$-generated, but they do not answer the question of whether
it is $(2A,3,7)$-generated. Using the “Brauer trick” argument of [Bro87] with
respect to the character $\chi_{4}$ of [Con85] shows that in fact $Ru$ is not
$(2A,3,7)$-generated, and so any Hurwitz curve with automorphism group $Ru$
has a UIC.

  * •

$Fi_{22}$ has three conjugacy classes of involutions, but all of them have
lifting order $2$ in $2\cdot Fi_{22}$, and so any Hurwitz curve with
automorphism group $Fi_{22}$ has no invariant theta characteristics.

######  Remark 5.16.

Had the lifting orders not been computed in the Atlas, we would have been able
to work them out by taking a presentation of the group in an appropriate
alternating group, and then applying the method of §5.5.

###  5.7. Comparison to Machine Prediction

We can now make a copy of Table LABEL:tab:_invariants_prediction which gives
also the true value of $I$. We see that the machine estimation had an accuracy
of $9/14\approx 64\%$, with zero false positives. It is not surprising that
the machine classification performed poorly on the subset of simple Hurwitz
groups, as these are acting in a range of genera and signatures poorly
represented in the training dataset.

Table 5. Machine prediction of whether $I=1$, all simple Hurwitz groups of order $<10^{6}$, and true value of $I$ $G$ |  $g$ |  $I=1$ prediction |  $I$
---|---|---|---
$\operatorname{PSL}_{2}(7)$ |  3 |  True |  1
$\operatorname{PSL}_{2}(8)$ |  7 |  False |  0
$\operatorname{PSL}_{2}(13)$ |  14 |  True |  1
$\operatorname{PSL}_{2}(27)$ |  118 |  True |  1
$\operatorname{PSL}_{2}(29)$ |  146 |  True |  1
$\operatorname{PSL}_{2}(41)$ |  411 |  False |  1
$\operatorname{PSL}_{2}(43)$ |  474 |  True |  1
$J_{1}$ |  2091 |  False |  0
$\operatorname{PSL}_{2}(71)$ |  2131 |  False |  1
$\operatorname{PSL}_{2}(83)$ |  3404 |  True |  1
$\operatorname{PSL}_{2}(97)$ |  5433 |  False |  1
$J_{2}$ |  7201 |  False |  1
$\operatorname{PSL}_{2}(113)$ |  8589 |  False |  1
$\operatorname{PSL}_{2}(125)$ |  11626 |  True |  1

A surprising fact that we have encountered so far is that the number of
invariant characteristics on a Hurwitz curve just depends on the group and not
its generating vector, i.e. the topological type of its action. It is known
(see for example [Con87]) that the Hurwitz groups can have topologically
distinct generating vectors.

##  6\. Outlook

In this work we’ve laid the groundwork for a more developed theory of
invariant characteristics, moving beyond the work of [Ati71, KS10, Bea13]
which considered only those invariant under a single automorphism. We have
approached this from two separate directions: by rephrasing the question of
invariant characteristics in terms of group cohomology of an affine action;
and by understanding the whole group of invariant line bundles. At present, it
is not clear how the two may be linked, and clarifying this will undoubtedly
serve to help the development of theory.

At present, sufficient conditions for unique invariant characteristics have
been found, but not any strong necessary conditions. In our current use of the
group cohomology method, we have not utilised much of the theory of the
rational representation, and it seems likely that better understanding this
will be helpful. This is especially pressing as we provide examples through
our computation of curves for which the SOC property does not explain the
existence of a UIC, though the property is enough to cover the majority of
curves seen.

With regards to Hurwitz curves with a simple automorphism group, though we
have provided a criterion for checking whether certain generating vectors of
$A_{n}$ Hurwitz curves yield a UIC, it is not clear whether a general
statement can be made. Further research should go into investigating what
generating vectors of $A_{n}$ as a Hurwitz group can occur.

For future work, one could also generalise to consider $r$-spin structures,
which are $r$th-roots of the canonical bundle. These exist when $r|2g-2$, and
in principle our methods can be extended to study the orbits of these.

##  Appendix A Tables of Orbits

In this appendix we provide all the tables of orbit decompositions described
in §4. We shall use the notation of [Bar12, BB16] that $L_{d}(x,y,\dots)$ is a
generic homogeneous degree-$d$ polynomial in the arguments. Moreover, we shall
make some comment about the completeness of the data.

  * •

The complete list of genus-2 curves with nontrivial reduced automorphism group
comes from [Bol87].

  * •

The complete list of non-hyperelliptic genus-3 curves with nontrivial
(reduced) automorphism group comes from [Bar12]. Bars attributes the first
work completing this to Henn in 1976, but Wiman appears to have completed the
calculation earlier in [Wim95a]. Bars and Dolgachev disagree on the
automorphism group of the curve given by
$f=1+y^{4}+x^{4}+a(x^{2}+y^{2})+bx^{2}y^{2}$; using Sage we find agreement
with Bars.

  * •

The list of hyperelliptic curves with many automorphisms comes from [Sha07,
MP21]. The latter reference will also be used for higher genera.

  * •

[Sha07, Table 4] was used to verify the signatures of the non-hyperelliptic
actions.

  * •

The complete list of non-hyperelliptic genus-4 curves with nontrivial
(reduced) automorphism group comes from [Wim95b]. Wiman distinguishes his
curves by whether they lie on the nonsingular quadric or the cone, and we have
followed this putting those that lie on the cone in the first portion of the
table. For the curves which lie on the nonsingular quadric Wiman described the
curve by providing the quadric and cubic, hence one must use the resultant to
get a single plane equation, and this may require a projective linear
transformation to find a nondegenerate coordinate system. We find a typo in
Wiman’s curve (8) with octahedral symmetry.

  * •

Above genus 4 we are unaware of any complete lists classifying curves by their
automorphism groups and giving plane models of the curves. Curves of genus
$(d-1)(d-2)/2$ for $d\geq 3$ stand out because of the Plücker formula. In
order to implement the numerical method for computing the rational
representation it is also necessary to have a model of the curve with
coefficients in $\mathbb{Q}$, and so this further limits the possible curves
we may investigate.

  * •

As the LMFDB does not contain the data of signatures with quotient genus $>0$
at genera $>4$, we shall sometimes omit the signature of the action where it
is unknown.

  * •

The first non-hyperelliptic curve of genus 5 used in Table
LABEL:tab:_orbit_decomposition,_genus_5 is the family of Humbert curves given
in [KR89, (5.9)], the second from [Swi16], and the third comes from taking
resultants of the polynomials provided in [Wim95b], hence one may choose to
call it the Wiman octic for want of a better name.

  * •

The non-hyperelliptic genus-6 curves come from a list of all nonsingular plane
quintics in [BB16], with the exception of the curve with automorphism group
$(48,15)$ which is from [Swi16]. The curve with automorphism group of order
150 is the Fermat quintic curve which has maximal automorphism group for a
genus-6 curve [MSSV02], the curve with automorphism group $S_{5}$ is the Wiman
sextic [Wim95b, Edg81], and the curve with automorphism group of order 72 is
given in the LMFDB with label 6.72-15.0.2-4-9.1. The curve with automorphism
group of order 39 is attributed to Snyder in [Lef21b, p. 464], where it is
constructed in a manner similar to Klein’s curve.

  * •

The curve with automorphism group $\operatorname{PSL}_{2}(\mathbb{F}_{8})$ is
the Fricke-Macbeath curve [Fri99, Mac65], the unique Hurwitz curve of genus 7,
the rational plane model of which is a attributed to Brock in [Hid17]. The
remaining curves come from [Zom10, Table 2, Table 5]. Table 5 in Zomorrodian
gives all possible automorphism groups of non-hyperelliptic genus-7 curves
where $\left\lvert\operatorname{Aut}\right\rvert<65$, and a plane curve form
for each; this list contains some errors, for example a typo in curve 4 and
the fact that curve 8 is hyperelliptic (as checked with Maple [Map22]).

  * •

We are unaware of any plane models of non-hyperelliptic curves of genus 8 and
so examples in this genus are sadly missing; one can in principle get such
models from the methods of [Shi95], using Sage’s modular symbol functionality,
but in practice the process of going from a canonical embedding to a plane
model becomes infeasible. Likewise, one could use the methods of [Swi16] to
get the canonical embedding, but the problem of finding a plane form from this
remains.

  * •

The genus-9 curve with $\operatorname{\overline{\operatorname{Aut}}}=S_{5}$ is
the Fricke octavic curve, defined in [Edg84], constructed similarly to Bring’s
curve in $\mathbb{P}^{3}$ and so a plane form of the curve is found using
resultants and a judicious choice of projective transformation to find a well
conditioned coordinate system.

  * •

The genus-9 curve with automorphism group of order 57 is a generalisation of
Klein’s curve and Snyder’s curve [Lef21b, p. 464].

Table 6. Orbit decomposition, all non-hyperelliptic and hyperelliptic genus-3 curves $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$1+L_{2}(x,y)+L_{4}(x,y)$ |  $C_{2}$, $(1;2^{4})$ |  $1_{4}$, $2_{12}$ |  $1_{12}$, $2_{12}$ |  16
$L_{1}(x,y)+L_{3}(x,y)$ |  $C_{3}$, $(0;3^{5})$ |  $1_{1}$, $3_{9}$ |  $3_{12}$ |  1
$1+y^{4}+x^{4}+(ay^{2}+bx^{2})+cx^{2}y^{2}$ |  $V_{4}$, $(0;2^{6})$ |  $2_{6},4_{4}$ |  $1_{8},2_{6},4_{4}$ |  8
$bx^{2}y^{2}+x^{3}+y^{3}+axy+1$ |  $S_{3}$, $(0;2^{4},3)$ |  $1_{1},3_{3},6_{3}$ |  $1_{3},3_{9},6_{1}$ |  4
$y^{4}+x^{3}+ay^{2}+1$ |  $C_{6}$, $(0;2,3^{2},6)$ |  $1_{1},3_{1},6_{4}$ |  $3_{4},6_{4}$ |  1
$1+y^{4}+x^{4}+a(x^{2}+y^{2})+bx^{2}y^{2}$ |  $D_{4}$, $(0;2^{5})$ |  $4_{5},8_{1}$ |  $1_{4},2_{4},4_{4},8_{1}$ |  4
$xy^{3}+x^{3}+1$ |  $C_{9}$, $(0;3,9^{2})$ |  $1_{1},9_{3}$ |  $9_{4}$ |  1
$y^{4}+ay^{2}+x^{4}+1$ |  $(C_{4}~{}\times~{}C_{2})\rtimes C_{2}{\cong(16,13)}$, $(0;2^{3},4)$ |  $4_{1},8_{3}$ |  $2_{6},8_{3}$ |  0
$1+y^{4}+x^{4}+a(y^{2}+x^{2}+y^{2}x^{2})$ |  $S_{4}$, $(0;2^{3},3)$ |  $4_{1},12_{2}$ |  $1_{2}$, $3_{2}$, $4_{1}$, $6_{2}$, $12_{1}$ |  2
$x^{4}+y^{4}+x$ |  $((C_{4}\times C_{2})\rtimes{C_{2})\rtimes C_{3}}$ ${\cong(48,33)}$, $(0;2,3,12)$ |  $4_{1},24_{1}$ |  $6_{2},24_{1}$ |  0
$y^{4}+x^{4}+1$ |  $(C_{4}^{2}~{}\rtimes~{}C_{3})\rtimes C_{2}{\cong(96,64)}$, $(0;2,3,8)$ |  $12_{1},16_{1}$ |  $4_{2},12_{1},16_{1}$ |  0
$xy^{3}+x^{3}+y$ |  $\operatorname{PSL}_{3}(\mathbb{F}_{2})$, $(0;2,3,7)$ |  $28_{1}$ |  $1_{1},7_{2},21_{1}$ |  1
$y^{2}-(x^{8}+ax^{6}+bx^{4}+cx^{2}+1)$ |  $C_{2}$, $(1;2^{4})$ |  $1_{4}$, $2_{12}$ |  $1_{12}$, $2_{12}$ |  $16$
$y^{2}-x(x^{2}-1)(x^{4}+ax^{2}+b)$ |  $C_{2}$ |  $1_{4}$, $2_{12}$ |  $1_{4}$, $2_{16}$ |  $8$
$y^{2}-(x^{4}+ax^{2}+1)(x^{4}+bx^{2}+1)$ |  $V_{4}$, $(0;2^{6})$ |  $2_{6}$, $4_{4}$ |  $1_{8}$, $2_{6}$, $4_{4}$ |  $8$
$y^{2}-(x^{4}-1)(x^{4}+ax^{2}+1)$ |  $V_{4}$ |  $1_{2}$, $2_{3}$, $4_{5}$ |  $1_{2}$, $2_{7}$, $4_{5}$ |  $4$
$y^{2}-x(x^{6}+ax^{3}+1)$ |  $S_{3}$, $(0;2^{4},3)$ |  $1_{1}$, $3_{3}$, $6_{3}$ |  $1_{3}$, $3_{9}$, $6_{1}$ |  $4$
$y^{2}-(x^{8}+ax^{4}+1)$ |  $D_{4}$, $(0;2^{2},4^{2})$ |  $4_{5}$, $8_{1}$ |  $1_{4}$, $2_{4}$, $4_{4}$, $8_{1}$ |  $4$
$y^{2}-(x^{7}-1)$ |  $C_{7}$, $(0;7^{3})$ |  $7_{4}$ |  $1_{1}$, $7_{5}$ |  $1$
$y^{2}-x(x^{6}-1)$ |  $D_{6}$ |  $1_{1}$, $3_{1}$, $6_{2}$, $12_{1}$ |  $1_{1}$, $2_{1}$, $3_{1}$, $6_{5}$ |  $2$
$y^{2}-(x^{8}-1)$ |  $D_{8}$ |  $4_{1}$, $8_{3}$ |  $1_{2}$, $2_{1}$, $4_{2}$, $8_{3}$ |  $2$
$y^{2}-(x^{8}+14x^{4}+1)$ |  $S_{4}$, $(0;3,4^{2})$ |  $4_{1},12_{2}$ |  $1_{2}$, $3_{2}$, $4_{1}$, $6_{2}$, $12_{1}$ |  2
Table 7. Orbit decomposition, all non-hyperelliptic genus-4 curves stratified by whether the corresponding quadric is singular, and separately some hyperelliptic genus-4 curve with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$ |  Odd |  Even |  I
---|---|---|---|---
$y^{3}+y(ax^{4}+bx^{2}+c)+(dx^{6}+ex^{4}+fx^{2}+g)$ |  $C_{2}$, $(1;2^{6})$ |  $1_{24}$, $2_{48}$ |  $1_{40}$, $2_{48}$ |  $64$
$y^{3}+y(ax^{4}+bx^{2}+c)+dx(x^{4}+ex^{2}+f)$ |  $C_{2}$, $(2;2^{2})$ |  $2_{60}$ |  $1_{16}$, $2_{60}$ |  $16$
$y^{3}+y[a(x^{4}+1)+bx^{2}]+x[c(x^{4}+1)+dx^{2}]$ |  $V_{4}$, $(1;2^{3})$ |  $2_{24}$, $4_{18}$ |  $1_{16}$, $2_{24}$, $4_{18}$ |  $16$
$y^{3}+y[a(x^{4}+1)+bx^{2}]+x(x^{4}-1)$ |  $V_{4}$, $(1;2^{3})$ |  $4_{30}$ |  $1_{4}$, $2_{18}$, $4_{24}$ |  $4$
$y^{3}+ayx^{2}+x(x^{4}+1)$ |  $D_{4}$, $(0;2^{4},4)$ |  $4_{12}$, $8_{9}$ |  $1_{4}$, $2_{6}$, $4_{18}$, $8_{6}$ |  $4$
$y^{3}+y(x^{4}+a)+(bx^{4}+c)$ |  $C_{4}$, $(0;2,4^{4})$ |  $1_{4}$, $2_{10}$, $4_{24}$ |  $1_{4}$, $2_{18}$, $4_{24}$ |  $8$
$y^{3}+ayx^{2}+(x^{6}+bx^{3}+1)$ |  $S_{3}$, $(0;2^{6})$ |  $1_{6}$, $3_{18}$, $6_{10}$ |  $1_{10}$, $3_{30}$, $6_{6}$ |  $16$
$y^{3}+ayx^{2}+(x^{6}+1)$ |  $D_{6}$, $(0;2^{5})$ |  $2_{3}$, $6_{13}$, $12_{3}$ |  $1_{4}$, $2_{3}$, $3_{12}$, $6_{9}$, $12_{3}$ |  $4$
$y^{3}+y(ax^{3}+b)+(x^{6}+cx^{3}+d)$ |  $C_{3}$, $(1;3^{3})$ |  $1_{3}$, $3_{39}$ |  $1_{1}$, $3_{45}$ |  $4$
$y^{3}+ay(x^{3}+1)+(x^{6}+20x^{3}-8)$ |  $A_{4}$, $(0;2,3^{3})$ |  $4_{3}$, $12_{9}$ |  $1_{1}$, $3_{1}$, $6_{6}$, $12_{8}$ |  $1$
$y^{3}+ay+(x^{6}+b)$ |  $C_{6}$, $(0;2,6^{3})$ |  $1_{3}$, $3_{7}$, $6_{16}$ |  $1_{1}$, $3_{13}$, $6_{16}$ |  $4$
$y^{3}+y+x^{6}$ |  $C_{12}$, $(0;4,6,12)$ |  $1_{1}$, $2_{1}$, $3_{1}$, $6_{3}$, $12_{8}$ |  $1_{1}$, $3_{1}$, $6_{6}$, $12_{8}$ |  $2$
$y^{3}+ay+(x^{5}+b)$ |  $C_{5}$, $(0;5^{4})$ |  $5_{24}$ |  $1_{1}$, $5_{27}$ |  $1$
$y^{3}+y+x^{5}$ |  $C_{10}$, $(0;5,10^{2})$ |  $10_{12}$ |  $1_{1}$, $5_{3}$, $10_{12}$ |  $1$
$y^{3}-(x^{6}+ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f)$ |  $C_{3}$, $(0;3^{6})$ |  $3_{40}$ |  $1_{1}$, $3_{45}$ |  $1$
$y^{3}-(x^{6}+ax^{4}+bx^{2}+1)$ |  $C_{6}$, $(0;2^{2},3^{3})$ |  $3_{8}$, $6_{16}$ |  $1_{3}$, $3_{13}$, $6_{16}$ |  $1$
$y^{3}-x(x^{4}+ax^{2}+1)$ |  ${C_{6}\times C_{2}}$, $(0;2^{2},3,6)$ |  $6_{8}$, $12_{6}$ |  $1_{1}$, $3_{5}$, $6_{8}$, $12_{6}$ |  $1$
$y^{3}-(x^{6}+ax^{3}+1)$ |  ${C_{3}\times S_{3}}$, $(0;2^{2},3^{2})$ |  $3_{2}$, $6_{1}$, $9_{6}$, $18_{3}$ |  $1_{1}$, $3_{3}$, $9_{10}$, $18_{2}$ |  $1$
$y^{3}-(x^{5}+1)$ |  $C_{15}$, $(0;3,5,15)$ |  $15_{8}$ |  $1_{1}$, $15_{9}$ |  $1$
$y^{3}-(x^{6}+1)$ |  ${C_{6}\times S_{3}}$, $(0;2,6^{2})$ |  $6_{2}$, $18_{4}$, $36_{1}$ |  $1_{1}$, $3_{1}$, $6_{1}$, $9_{4}$, $18_{3}$, $36_{1}$ |  $1$
$y^{3}-x(x^{4}+1)$ |  ${C_{3}\times S_{4}}$, $(0;2,3,12)$ |  $12_{2}$, $24_{1}$, $36_{2}$ |  $1_{1}$, $3_{1}$, $12_{2}$, $18_{2}$, $36_{2}$ |  $1$
$y^{4}(x+1)+y^{3}(x^{2}+ax+1)+y^{2}[b(x^{3}+1)+cx(x+1)]+y[dx(x^{2}+1)+ex^{2}]+fx^{2}(x+1)$ |  $C_{2}$, $(1;2^{6})$ |  $1_{24}$, $2_{48}$ |  $1_{40}$, $2_{48}$ |  $64$
$y^{6}+y^{4}(x^{2}+ax+1)+y^{2}x(dx^{2}+bx+e)+cx^{3}$ |  $C_{2}$, $(2;2^{2})$ |  $2_{60}$ |  $1_{16}$, $2_{60}$ |  $16$
$y^{6}+y^{4}(x^{2}+ax+1)+y^{2}x[d(x^{2}+1)+bx]+cx^{3}$ |  $V_{4}$, $(1;2^{3})$ |  $2_{24}$, $4_{18}$ |  $1_{16}$, $2_{24}$, $4_{18}$ |  $16$
$y^{6}+y^{2}[cx(xy^{2}+1)+b(x^{3}+y^{2})+ax(y^{2}+x)]+x^{3}$ |  $V_{4}$, $(1;2^{3})$ |  $4_{30}$ |  $1_{4}$, $2_{18}$, $4_{24}$ |  $4$
$by^{2}(y^{2}-x)(x^{2}-1)-y^{6}-axy^{2}(y^{2}+x^{2})-x^{3}$ |  $D_{4}$, $(0;2^{4},4)$ |  $4_{12}$, $8_{9}$ |  $1_{4}$, $2_{6}$, $4_{18}$, $8_{6}$ |  $4$
$y^{6}+ay^{3}(x^{3}+1)+bxy^{4}+cx^{2}y^{2}+x^{3}$ |  $S_{3}$, $(0;2^{6})$ |  $1_{6}$, $3_{18}$, $6_{10}$ |  $1_{10}$, $3_{30}$, $6_{6}$ |  $16$
$y^{6}+ay^{3}(x^{3}+1)+bxy^{4}+bx^{2}y^{2}+x^{3}$ |  $D_{6}$, $(0;2^{5})$ |  $2_{3}$, $6_{13}$, $12_{3}$ |  $1_{4}$, $2_{3}$, $3_{12}$, $6_{9}$, $12_{3}$ |  $4$
$by^{2}(y^{2}-x)(x^{2}-1)-(y^{2}+x)^{3}$ |  $S_{4}$, $(0;2^{3},4)$ |  $4_{6}$, $12_{6}$, $24_{1}$ |  $1_{4}$, $4_{6}$, $6_{6}$, $12_{6}$ |  4
$x^{3}y^{3}+y^{6}+(a+b+1)y^{2}(x^{3}-y^{3})+(ab+a+b)y(x^{3}-y^{3})+ab(x^{3}-y^{3})$ |  $S_{3}$, $(0;2^{2},3^{3})$ |  $6_{20}$ |  $1_{1}$, $3_{15}$, $6_{15}$ |  $1$
$y^{4}(a+y^{2})+x^{3}(1+ay^{2})$ |  $D_{6}$, $(0;2^{2},3,6)$ |  $12_{10}$ |  $1_{1}$, $3_{3}$, $6_{13}$, $12_{4}$ |  $1$
$x^{3}y^{3}+y^{6}+ax^{3}+y^{3}$ |  ${S_{3}\times S_{3}}$, $(0;2^{3},3)$ |  $6_{2}$, $18_{6}$ |  $1_{1}$, $3_{6}$, $9_{9}$, $36_{1}$ |  $1$
$x^{3}y^{3}+y^{6}-x^{3}+y^{3}$ |  $(S_{3}\times S_{3})\rtimes C_{2}{\cong(72,40)}$, $(0;2,4,6)$ |  $12_{1}$, $36_{3}$ |  $1_{1}$, $6_{3}$, $9_{3}$, $18_{3}$, $36_{1}$ |  $1$
$x^{2}y^{3}+y^{4}+a^{5}x^{3}+xy$ |  $D_{5}$, $(0;2^{2},5^{5})$ |  $10_{12}$ |  $1_{1}$, $5_{15}$, $10_{6}$ |  $1$
$xy-x^{3}+y^{4}+x^{2}y^{3}$ |  $S_{5}$, $(0;2,4,5)$ |  $20_{3}$, $60_{1}$ |  $1_{1}$, $5_{3}$, $10_{3}$, $30_{3}$ |  $1$
$y^{2}-(x^{9}-1)$ |  $C_{9}$, $(0;9^{3})$ |  $3_{1}$, $9_{13}$ |  $1_{1}$, $9_{15}$ |  $1$
$y^{2}-x(x^{4}-1)(x^{4}+2i\sqrt{3}x^{2}+1)$ |  $A_{4}$ |  $4_{3}$, $6_{4}$, $12_{7}$ |  $4_{1}$, $6_{4}$, $12_{9}$ |  0
$y^{2}-x(x^{8}-1)$ |  $D_{8}$ |  $8_{5}$, $16_{5}$ |  $2_{2}$, $4_{1}$, $8_{10}$, $16_{3}$ |  $0$
$y^{2}-(x^{10}-1)$ |  $D_{10}$ |  $10_{4}$, $20_{4}$ |  $1_{1}$, $5_{3}$, $10_{8}$, $20_{2}$ |  $1$
Table 8. Orbit decomposition, three non-hyperelliptic genus-5 curves, and separately all hyperelliptic genus-5 curves with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$y^{4}-4(x^{4}-ax^{2}+1)y^{2}+b^{2}x^{4}$ |  $C_{2}^{4}$ |  $4_{40}$, $8_{30}$, $16_{6}$ |  $1_{32}$, $4_{40}$, $8_{30}$, $16_{6}$ |  $32$
$y^{3}-x^{2}(x^{5}-1)$ |  $C_{3}\times D_{5}$, $(0;2,6,15)$ |  $1_{1}$, $15_{5}$, $30_{14}$ |  $3_{1}$, $15_{15}$, $30_{10}$ |  $1$
$4x^{8}+36x^{4}y^{4}+81y^{8}+8x^{6}+30x^{2}y^{4}+5x^{4}+14y^{4}+2x^{2}+1$ |  $(({(C_{4}\times C_{2})}\rtimes{C_{4})\rtimes C_{3})}\rtimes C_{2}$ $\cong(192,181)$, $(0;2,3,8)$ |  $16_{1}$, $24_{2}$, $48_{1}$, $96_{4}$ |  $1_{1}$, $3_{1}$, $4_{1}$, $6_{2}$, $12_{1}$, $16_{1}$, $24_{4}$, $48_{4}$, $96_{2}$ |  $1$
$y^{2}-(x^{11}-1)$ |  $C_{11}$, $(0;11^{3})$ |  $1_{1}$, $11_{45}$ |  $11_{48}$ |  $1$
$y^{2}-x(x^{10}-1)$ |  $D_{10}$ |  $1_{1}$, $5_{3}$, $10_{12}$, $20_{18}$ |  $1_{1}$, $2_{1}$, $5_{3}$, $10_{27}$, $20_{12}$ |  $2$
$y^{2}-(x^{12}-1)$ |  $D_{12}$ |  $1_{1}$, $3_{1}$, $6_{2}$, $12_{12}$, $24_{14}$ |  $1_{1}$, $2_{1}$, $3_{1}$, $6_{5}$, $12_{25}$, $24_{8}$ |  $2$
$y^{2}-(x^{12}-33x^{8}-33x^{4}+1)$ |  $S_{4}$ |  $1_{1}$, $3_{1}$, $6_{2}$, $12_{8}$, $24_{16}$ |  $3_{2}$, $4_{1}$, $6_{3}$, $8_{1}$, $12_{13}$, $24_{14}$ |  $1$
$y^{2}-x(x^{10}+11x^{5}-1)$ |  $A_{5}$, $(0;3^{2},5)$ |  $1_{1}$, $15_{1}$, $30_{6}$, $60_{5}$ |  $6_{3}$, $10_{3}$, $15_{4}$, $30_{12}$, $60_{1}$ |  $1$
Table 9. Orbit decomposition, some non-hyperelliptic genus-6 curves, and separately all hyperelliptic genus-6 curves with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$L_{5}(x,y)+L_{3}(x,y)+L_{1}(x,y)$ |  $C_{2}$ |  $1_{96}$, $2_{960}$ |  $1_{160}$, $2_{960}$ |  $256$
$x^{5}+ax^{2}y^{3}+bx^{3}y+y^{4}+cxy^{2}+dx^{2}+ey$ |  $C_{3}$ |  $1_{6}$, $3_{670}$ |  $1_{10}$, $3_{690}$ |  $16$
$L_{5}(x,y)+L_{1}(x,y)$ |  $C_{4}$ |  $1_{16}$, $2_{40}$, $4_{480}$ |  $2_{80}$, $4_{480}$ |  $16$
$x^{5}+axy^{4}+bx^{2}y^{2}+cx^{3}+dy^{2}+ex$ |  $C_{4}$ |  $1_{8}$, $2_{44}$, $4_{480}$ |  $1_{8}$, $2_{76}$, $4_{480}$ |  $16$
$1+L_{5}(x,y)$ |  $C_{5}$ |  $1_{1}$, $5_{403}$ |  $5_{416}$ |  $1$
$x^{5}+ax^{2}y^{3}+bx^{3}y+y^{4}+cxy^{2}+dx^{2}+y$ |  $S_{3}$ |  $1_{6}$, $3_{90}$, $6_{290}$ |  $1_{10}$, $3_{150}$, $6_{270}$ |  $16$
$x^{5}+y^{4}+ax^{3}+x$ |  $C_{8}$ |  $1_{4}$, $2_{6}$, $4_{20}$, $8_{240}$ |  $4_{40}$, $8_{240}$ |  $4$
$x^{5}+y^{5}+axy^{3}+bx^{2}y+1$ |  $D_{5}$ |  $1_{6}$, $5_{90}$, $10_{156}$ |  $1_{10}$, $5_{150}$, $10_{132}$ |  $16$
$x^{5}+y^{5}+ax^{3}+x$ |  $C_{10}$ |  $1_{1}$, $5_{19}$, $10_{192}$ |  $5_{32}$, $10_{192}$ |  $1$
$x^{5}+y^{4}+x$ |  $C_{16}$ |  $1_{2}$, $2_{1}$, $4_{3}$, $8_{10}$, $16_{120}$ |  $8_{20}$, $16_{120}$ |  $2$
$x^{5}+y^{5}+x$ |  $C_{20}$ |  $1_{1}$, $5_{3}$, $10_{8}$, $20_{96}$ |  $10_{16}$, $20_{96}$ |  $1$
$x^{5}+y^{4}+y$ |  ${C_{5}\times S_{3}}$, $(0;2,10,15)$ |  $1_{1}$, $5_{1}$, $15_{18}$, $30_{58}$ |  $5_{2}$, $15_{30}$, $30_{54}$ |  $1$
$x^{4}y+y^{4}+x$ |  ${C_{13}\rtimes C_{3}}$ ${\cong(39,1)}$, $(0;3^{2},13)$ |  $1_{1}$, $13_{5}$, $39_{50}$ |  $13_{10}$, $39_{50}$ |  $1$
$y^{3}-(x^{4}-1)^{2}(x^{4}+1)$ |  $C_{3}\rtimes D_{8}\cong(48,15)$, $(0;2,6,8)$ |  $24_{4}$, $48_{40}$ |  $1_{1}$, $3_{1}$, $6_{2}$, $12_{8}$, $24_{34}$, $48_{24}$ |  $1$
$y^{3}-(x^{4}-2i\sqrt{3}x^{2}+1)(x^{4}+2i\sqrt{3}x^{2}+1)^{2}$ |  $(V_{4}\rtimes C_{9})\rtimes{C_{2}\cong}(72,15)$, $(0;2,4,9)$ |  $18_{4}$, $36_{6}$, $72_{24}$ |  $1_{1}$, $9_{7}$, $18_{4}$, $36_{30}$, $72_{12}$ |  $1$
$(x^{6}+y^{6}+1)+(x^{2}+y^{2}+1)(x^{4}+y^{4}+1)-12x^{2}y^{2}$ |  $S_{5}$, $(0;2,4,6)$ |  $6_{2}$, $12_{2}$, $20_{3}$, $30_{2}$, $60_{17}$, $120_{7}$ |  $1_{2}$, $2_{1}$, $10_{2}$, $12_{3}$, $15_{6}$, $20_{2}$, $30_{9}$, $60_{21}$, $120_{3}$ |  $2$
$x^{5}+y^{5}+1$ |  ${C_{5}^{2}\rtimes S_{3}}$ ${\cong(150,5)}$, $(0;2,3,10)$ |  $1_{1}$, $15_{1}$, $25_{5}$, $75_{13}$, $150_{6}$ |  $15_{2}$, $25_{10}$, $75_{20}$, $150_{2}$ |  $1$
$y^{2}-(x^{13}-1)$ |  $C_{13}$, $(0;13^{3})$ |  $1_{1}$, $13_{155}$ |  $13_{160}$ |  $1$
$y^{2}-x(x^{12}-1)$ |  $D_{12}$ |  $2_{1}$, $4_{1}$, $6_{1}$, $12_{17}$, $24_{75}$ |  $2_{1}$, $4_{2}$, $6_{3}$, $12_{43}$, $24_{64}$ |  $0$
$y^{2}-x(x^{4}-1)(x^{8}+14x^{4}+1)$ |  $S_{4}$ |  $6_{4}$, $8_{3}$, $12_{6}$, $24_{79}$ |  $4_{4}$, $6_{4}$, $8_{3}$, $12_{30}$, $24_{69}$ |  $0$
$y^{2}-(x^{14}-1)$ |  $D_{14}$ |  $14_{16}$, $28_{64}$ |  $1_{1}$, $7_{7}$, $14_{41}$, $28_{52}$ |  $1$
Table 10. Orbit decomposition, some non-hyperelliptic genus-7 curves, and separately some hyperelliptic genus-7 curves including all with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$(x^{3}+y^{3})^{2}-x^{2}y^{2}-1$ |  $D_{6}$ |  $1_{4}$, $2_{12}$, $3_{12}$, $6_{232}$, $12_{556}$ |  $1_{12}$, $2_{12}$, $3_{36}$, $6_{336}$, $12_{508}$ |  $16$
$x^{6}+y^{6}-x^{3}-y^{3}$ |  $C_{3}\times S_{3}$ |  $1_{1}$, $3_{1}$, $6_{7}$, $9_{20}$, $18_{439}$ |  $3_{4}$, $6_{6}$, $9_{60}$, $18_{426}$ |  $1$
$x^{6}+y^{4}-1$ |  ${C_{12}\times C_{2}}$, $(0;4,6,12)$ |  $1_{1}$, $3_{1}$, $6_{6}$, $12_{42}$, $24_{316}$ |  $1_{1}$, $2_{1}$, $3_{1}$, $6_{11}$, $12_{66}$, $24_{308}$ |  $2$
$(x^{4}+y^{4})^{2}-x^{3}y^{3}-x^{2}y^{2}$ |  ${C_{8}\rtimes V_{4}}$ ${\cong(32,43)}$, $(0;2^{3},8)$ |  $8_{16}$, $16_{84}$, $32_{208}$ |  $2_{4}$, $4_{22}$, $8_{42}$, $16_{129}$, $32_{180}$ |  $0$
$x^{7}+y^{7}-x^{2}y^{2}$ |  ${C_{3}\times D_{7}}$ |  $1_{1}$, $21_{21}$, $42_{183}$ |  $3_{1}$, $21_{63}$, $42_{165}$ |  $1$
$y^{21}-x(x+1)^{13}(x-1)^{7}$ |  ${C_{3}\times D_{7}}$, $(0;2,6,21)$ |  $1_{1}$, $21_{21}$, $42_{183}$ |  $3_{1}$, $21_{63}$, $42_{165}$ |  $1$
$x^{9}+y^{9}-x^{6}$ |  ${C_{3}\times D_{9}}$, $(0;2,6,9)$ |  $1_{1}$, $18_{3}$, $27_{21}$, $54_{139}$ |  $3_{1}$, $18_{4}$, $27_{63}$, $54_{120}$ |  $1$
$x^{9}+y^{9}-x^{3}y^{3}$ |  ${C_{3}\times D_{9}}$, $(0;2,6,9)$ |  $1_{1}$, $18_{3}$, $27_{21}$, $54_{139}$ |  $3_{1}$, $18_{4}$, $27_{63}$, $54_{120}$ |  $1$
$y^{8}-(x^{2}-1)(x^{2}+1)^{3}$ |  ${(C_{16}\rtimes C_{2})}\rtimes C_{2}{\cong(64,41)}$, $(0;2,4,16)$ |  $16_{8}$, $32_{42}$, $64_{104}$ |  $4_{2}$, $8_{9}$, $16_{15}$, $32_{64}$, $64_{92}$ |  $0$
$y^{16}-x(x-1)^{9}(x+1)^{6}$ |  ${(C_{8}\rtimes C_{4})}\rtimes C_{2}{\cong(64,41)}$, $(0;2,4,16)$ |  $16_{8}$, $32_{42}$, $64_{104}$ |  $4_{2}$, $8_{9}$, $16_{15}$, $32_{64}$, $64_{92}$ |  $0$
$y^{9}-6y^{6}+3(9x^{4}-5)y^{3}-8$ |  $((C_{4}\times S_{3})\rtimes{C_{2})\rtimes C_{3}}$ ${\cong(144,127)}$, $(0;2,3,12)$ |  $4_{1}$, $12_{1}$, $24_{2}$, $36_{2}$, $72_{41}$, $144_{35}$ |  $6_{2}$, $12_{2}$, $18_{6}$, $24_{2}$, $36_{28}$, $72_{28}$, $144_{35}$ |  $0$
$28x^{4}y^{4}+2x^{7}+2y^{7}+35x^{3}y^{3}+21x^{2}y^{2}+7xy+1$ |  $\operatorname{PSL}_{2}(\mathbb{F}_{8})$, $(0;2,3,7)$ |  $28_{1}$, $36_{1}$, $252_{14}$, $504_{9}$ |  $28_{3}$, $36_{3}$, $126_{16}$, $252_{18}$, $504_{3}$ |  $0$
$y^{2}-x(x^{6}-1)(x^{8}-2)$ |  $C_{2}$ |  $1_{64}$, $2_{4032}$ |  $1_{64}$, $2_{4096}$ |  $128$
$y^{2}-x(x^{7}-1)(x^{7}-2)$ |  $D_{7}$ |  $1_{1}$, $7_{63}$, $14_{549}$ |  $1_{3}$, $7_{189}$, $14_{495}$ |  $4$
$y^{2}-(x^{15}-1)$ |  $C_{15}$, $(0;15^{3})$ |  $3_{1}$, $5_{2}$, $15_{541}$ |  $1_{1}$, $5_{1}$, $15_{550}$ |  $1$
$y^{2}-(x^{8}-1)(x^{8}-2)$ |  $D_{8}$ |  $8_{72}$, $16_{472}$ |  $1_{4}$, $2_{4}$, $4_{25}$, $8_{170}$, $16_{424}$ |  $4$
$y^{2}-x(x^{14}-1)$ |  $D_{14}$ |  $1_{1}$, $7_{7}$, $14_{57}$, $28_{260}$ |  $1_{1}$, $2_{1}$, $7_{7}$, $14_{120}$, $28_{233}$ |  $2$
$y^{2}-(x^{16}-1)$ |  $D_{16}$ |  $8_{4}$, $16_{62}$, $32_{222}$ |  $1_{2}$, $2_{1}$, $4_{3}$, $8_{14}$, $16_{112}$, $32_{198}$ |  $2$
$y^{2}-(x^{16}+1)$ |  $D_{16}$ |  $8_{4}$, $16_{62}$, $32_{222}$ |  $1_{2}$, $2_{1}$, $4_{3}$, $8_{14}$, $16_{112}$, $32_{198}$ |  $2$
Table 11. Orbit decomposition, all hyperelliptic curves of genus 8 with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$y^{2}-(x^{17}-1)$ |  $C_{17}$, $(0;17^{3})$ |  $17_{1920}$ |  $1_{1}$, $17_{1935}$ |  $1$
$y^{2}-x(x^{4}-1)(x^{12}-33x^{8}-33x^{4}+1)$ |  $S_{4}$ |  $4_{2}$, $8_{2}$, $12_{94}$, $24_{1312}$ |  $4_{2}$, $6_{8}$, $8_{4}$, $12_{90}$, $24_{1322}$ |  $0$
$y^{2}-x(x^{16}-1)$ |  $D_{16}$ |  $16_{72}$, $32_{984}$ |  $2_{2}$, $4_{1}$, $8_{7}$, $16_{188}$, $32_{932}$ |  $0$
$y^{2}-(x^{18}-1)$ |  $D_{18}$ |  $6_{1}$, $18_{63}$, $36_{875}$ |  $1_{1}$, $3_{1}$, $6_{1}$, $9_{14}$, $18_{182}$, $36_{819}$ |  $1$
Table 12. Orbit decomposition, two non-hyperelliptic curves of genus 9, and all hyperelliptic curves of genus 9 with many automorphisms $f$ |  $\operatorname{\overline{\operatorname{Aut}}}$, $\bm{c}$ |  Odd |  Even |  I
---|---|---|---|---
$x^{5}y^{2}+y^{5}+x^{2}$ |  ${C_{19}\rtimes C_{3}}$ ${\cong(57,1)}$, $(0;3^{2},19)$ |  $1_{1}$, $19_{27}$, $57_{2286}$ |  $19_{36}$, $57_{2292}$ |  1
$y-x^{3}-x^{4}y^{3}+xy^{4}+3x^{2}y^{2}$ |  $S_{5}$, $(0;2,5,6)$ |  $6_{1}$, $10_{4}$, $20_{12}$, $30_{35}$, $60_{300}$, $120_{929}$ |  $1_{2}$, $5_{6}$, $6_{1}$, $10_{10}$, $15_{24}$, $20_{6}$, $30_{85}$, $60_{402}$, $120_{867}$ |  2
$y^{2}-(x^{19}-1)$ |  $C_{19}$, $(0;19^{3})$ |  $1_{1}$, $19_{6885}$ |  $19_{6912}$ |  1
$y^{2}-(x^{12}-33x^{8}-33x^{4}+1)(x^{8}+14x^{4}+1)$ |  $S_{4}$ |  $1_{2}$, $2_{1}$, $3_{2}$, $4_{2}$, $6_{13}$, $8_{11}$, $12_{172}$, $24_{5357}$ |  $3_{4}$, $4_{4}$, $6_{18}$, $8_{16}$, $12_{290}$, $24_{5316}$ |  2
$y^{2}-x(x^{18}-1)$ |  $D_{18}$ |  $1_{1}$, $3_{1}$, $6_{2}$, $9_{14}$, $12_{1}$, $18_{245}$, $36_{3507}$ |  $1_{1}$, $2_{1}$, $3_{1}$, $6_{5}$, $9_{14}$, $18_{497}$, $36_{3395}$ |  2
$y^{2}-(x^{20}-1)$ |  $D_{20}$ |  $1_{1}$, $5_{3}$, $10_{12}$, $20_{246}$, $40_{3144}$ |  $1_{1}$, $2_{1}$, $5_{3}$, $10_{27}$, $20_{504}$, $40_{3024}$ |  2
$y^{2}-(x^{20}-228x^{15}+494x^{10}+228x^{5}+1)$ |  $A_{5}$, $(0;3,5^{2})$ |  $1_{1}$, $5_{3}$, $10_{3}$, $15_{4}$, $20_{9}$, $30_{117}$, $60_{2117}$ |  $6_{3}$, $10_{12}$, $15_{16}$, $20_{12}$, $30_{345}$, $60_{2006}$ |  $1$

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