Enumerating hyperelliptic curves
over finite fields in quasilinear time

There are many circumstances in which one may want to enumerate all hyperelliptic curves of a given genus over a given finite field. One may wish to determine whether a hyperelliptic curve with certain special properties exists — for instance, with a certain number of points [14, p. 393], or a certain zeta function, or a certain $a$-number [7, §4], or some other property of interest — and explicit enumeration allows for a direct search. Or perhaps one may wish to gather data about all hyperelliptic curves of a given genus over a given finite field — for example, in order to compute the distribution of the number of points on such curves, as in [1], or to determine experimental results [18] that can inspire future theorems [11].

In this paper we present, for every fixed genus $g>1$, an algorithm to calculate a list of all hyperelliptic curves of genus $g$ over a given finite field of odd characteristic.¹¹1Hyperelliptic curves in characteristic $2$ are different from those in other characteristics in some basic ways, and, as we discuss later, there already exists an efficient algorithm for enumerating them.

From [3, Proposition 7.1] and from the fact that a generic hyperelliptic curve has automorphism group of order $2$, we see that there are roughly $2q^{2g-1}$ hyperelliptic curves of genus $g$ over ${\mathbf{F}}_{q}$, so our algorithm runs in quasilinear time.

We also present an apparently new explicit enumeration of all monic irreducible homogeneous bivariate quartics over a finite field $k$ of odd characteristic, up to the natural action of $\operatorname{PGL}_{2}(k)$ (see Section 2), which may be of independent interest.

We prove this in Section 3. Also, here and elsewhere, when we say we have a “complete set of unique representatives” for an action of a group on a set, we mean that we have a set of orbit representatives that contains exactly one member from each orbit.

Briefly, there are two main ideas behind the algorithms that give us Theorem 1.1. The first is that there is an easy way to tell whether two irreducible homogeneous polynomials in ${\mathbf{F}}_{q}[x,y]$ lie in the same orbit under the action of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ — see Theorem 4.2. We use this fact to reduce the problem of getting a list of all hyperelliptic curves without duplicates in quasilinear time to the problem of getting a list of hyperelliptic curves with a bounded number of duplicates in quasilinear time. The second is that if one understands the cosets of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ in $\operatorname{PGL}_{2}({\mathbf{F}}_{q^{2}})$, and if one has a list of orbit representatives for $\operatorname{PGL}_{2}({\mathbf{F}}_{q^{2}})$ acting on irreducible homogenous polynomials of degree $n$ in ${\mathbf{F}}_{q^{2}}[x,y]$, then one can get a list of orbit representatives for $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ acting on irreducible homogenous polynomials of degree $2n$ in ${\mathbf{F}}_{q}[x,y]$ — see Section 7.4. The point of this observation is that the number of $\operatorname{PGL}_{2}({\mathbf{F}}_{q^{2}})$ orbits of irreducible degree-$n$ polynomials over ${\mathbf{F}}_{q^{2}}$ is on the order of $q^{2n-6}$, while the number of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ orbits of degree-$2n$ polynomials over ${\mathbf{F}}_{q}$ is roughly $q^{2n-3}$, so the former is easier to compute than the latter.

The algorithm for enumerating hyperelliptic curves that we present is designed for simplicity of argument, rather than for efficiency of computation. In Section 8 we describe modifications that will make the algorithm more efficient, and in Section 9 we present even more details and timings for the case of genus $2$ and genus $3$.

An obvious issue with our algorithm, as we present it here, is that it requires $O(q^{2g-1})$ space. This is because our initial computations often produce some duplicate entries, and we eliminate these duplicates by collecting all the output, computing some invariants, and then discarding entries whose invariants have already been seen. In a followup paper [9], we show how it is possible to modify the techniques presented here in order to get a quasilinear time algorithm for computing genus-$g$ hyperelliptic curves over ${\mathbf{F}}_{q}$ that only requires $O(\log q)$ space. Our implementation of the genus-$2$ case of the algorithm from the present paper uses a basic version of the ideas from [9], and it would not be hard to modify the genus-$2$ code we provide in [10] so that it requires only $O(q)$ space.

We start by considering various reductions, special cases, and lemmas. In Section 2 we show that enumerating hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic can be reduced to enumerating Galois-stable sets of $2g+2$ elements of ${\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ up to the natural action of $\operatorname{PGL}_{2}(k)$. The rest of the paper is therefore concerned mostly with the latter problem, which we solve for all finite fields, not just those of odd characteristic. In Section 3 we prove Theorem 1.2, as well as similar results for quartics with one or two irreducible quadratic factors and generalizations to characteristic $2$. In Section 4 we introduce an invariant for $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ orbits of monic irreducible homogeneous bivariate polynomials of degree $n\geq 4$ over ${\mathbf{F}}_{q}$, and we use it to provide a very straightforward algorithm for giving a complete set of unique representatives for these orbits in time $\mathchoice{\accentset{\displaystyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\textstyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptstyle\text{\smash{\raisebox{-4.2194pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptscriptstyle\text{\smash{\raisebox{-3.01385pt}{$\widetildesym$}}}}{O}}(q^{n-2})$. In Sections 5 and 6 we give an explicit enumeration of a complete set of unique representatives for the cosets of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ in $\operatorname{PGL}_{2}({\mathbf{F}}_{q^{p}})$ for primes $p$. The case $p=2$ is the key result needed for the most difficult case of our algorithm to enumerate hyperelliptic curves. In Section 7 we use the results of the earlier sections to present a collection of algorithms that, together, give a complete set of unique representatives for the $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ orbits of Galois stable sets of ${2g+2}$ elements of ${\mathbf{P}}^{1}(\overline{{\mathbf{F}}}_{q})$. We close with Sections 8 and 9, described above.

For many of our algorithms we require an easily computable total ordering of the elements of ${\mathbf{F}}_{q}$ or of polynomials in ${\mathbf{F}}_{q}[x]$ or in ${\mathbf{F}}_{q}[x,y]$. We will always denote such an ordering by “$<$” and we leave the reader to choose their favorite one. Also, if the proof of a proposition is clear, we indicate that the proof will be skipped by including an end-of-proof mark in the statement of the result.

We begin by setting some general notation. Given a finite field $k$, let $R$ be the graded polynomial ring $k[x,y]$, with the grading given by the degree. For each $n$ let $R_{n}$ be the set of homogeneous polynomials in $R$ of degree $n$ and let $R_{\textup{hom}}$ be the union of the $R_{n}$. We say that $f\in R_{\textup{hom}}$ is monic if $f(x,1)$ is monic as a univariate polynomial, and we say that $f$ is separable if in $\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}[x,y]$ it can be written as a constant times a product of distinct monic linear factors. We say that $\alpha\in\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}$ is a root of $f$ if $f(\alpha,1)=0$, and that ${[\alpha\,{:}\,\beta]}\in{\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ is a zero of $f$ if $f(\alpha,\beta)=0$.

We also define a left action of $\operatorname{PGL}_{2}(k)$ on $R_{\textup{hom}}/k^{\times}$ as follows. If $\Gamma$ is an element of $\operatorname{PGL}_{2}(k)$ represented by a matrix $M\colonequals\bigl{[}\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr{]}$ and if $f$ lies in $R_{\textup{hom}}$, we define $\Gamma(f\bmod k^{\times})$ to be the class in $R_{\textup{hom}}/k^{\times}$ of $f(dx-by,-cx+ay)$.

Every class of $R_{\textup{hom}}/k^{\times}$ contains a unique monic element, and we define an action of $\operatorname{PGL}_{2}(k)$ on the monic elements of $R_{\textup{hom}}$ by writing $\Gamma(f)=g$ when $g$ is the monic element of $\Gamma(f\bmod k^{\times})$. If $\Gamma$ is represented by a matrix $M\colonequals\bigl{[}\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr{]}$ and if $\Gamma(f)=g$, then $f(dx-by,-cx+ay)=eg(x,y)$ for some $e\in k^{\times}$, and if we choose a different matrix to represent $\Gamma$, then the constant $e$ will be multiplied by an $n$th power. When $n$ is even, the class of $e$ in $k^{\times}/k^{\times 2}$ therefore depends only on $\Gamma$, and we denote this square class by $s_{\Gamma,f}$.

Given a separable polynomial $f\in R_{\textup{hom}}$, we let $\operatorname{Zeros}(f)$ denote the set of zeros of $f$ in ${\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$, so that $\operatorname{Zeros}(f)$ consists of the roots of $f$ under the usual inclusion $\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}\subset{\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ given by $\alpha\mapsto{[\alpha\,{:}\,1]}$, together with $\infty\colonequals{[1\,{:}\,0]}\in{\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ if $f$ is divisible by $y$. For every integer $n\geq 0$ we let $\operatorname{Sym}^{n}(k)$ denote the set of all Galois-stable sets of $n$ distinct points in ${\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$, so that the natural action of $\operatorname{PGL}_{2}(k)$ on ${\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ leads to an action of $\operatorname{PGL}_{2}(k)$ on $\operatorname{Sym}^{n}(k)$. We see that $\operatorname{Zeros}$ gives us a bijection between the set of monic separable polynomials in $R_{n}$ and the set $\operatorname{Sym}^{n}(k)$, and we check that $\operatorname{Zeros}(\Gamma(f))=\Gamma(\operatorname{Zeros}(f))$ for all $f\in R_{\textup{hom}}$.

Our goal in this paper is to produce an algorithm to enumerate hyperelliptic curves of a given genus $g$ over finite fields $k$ of odd characteristic. As a first step, we reduce this problem to the problem of computing representatives for all $\operatorname{PGL}_{2}(k)$ orbits of $\operatorname{Sym}^{n}(k)$, where $n=2g+2$.

Let $k$ be a finite field of odd characteristic and let $C$ be a hyperelliptic curve over $k$, that is, a curve of genus $g>1$ with a degree-$2$ map to a curve of genus $0$. Every genus-$0$ curve over a finite field is isomorphic to ${\mathbf{P}}^{1}$, and since our $k$ has odd characteristic $C$ can be written in the form $z^{2}=\tilde{f}$, where $\tilde{f}\in k[x]$ is a separable polynomial of degree $2g+1$ or $2g+2$. We can rewrite this as a model $z^{2}=f(x,y)$ in weighted projective space by homogenizing $\tilde{f}$ into a polynomial $f\in R_{2g+2}$; here we give the coordinates $x$ and $y$ weight $1$ and the coordinate $z$ weight $g+1$. Then the map ${[x\,{:}\,y\,{:}\,z]}\mapsto{[x\,{:}\,y]}$ gives the double cover $C\to{\mathbf{P}}^{1}$, and the points of ${\mathbf{P}}^{1}$ that ramify in this map are exactly the elements of $\operatorname{Zeros}(f)$. If $C_{1}$ and $C_{2}$ are two hyperelliptic curves, given by equations $z^{2}=f_{1}$ and $z^{2}=f_{2}$, then every isomorphism from $C_{1}$ to $C_{2}$ can be written in the form

where $ad-bc$ and $e$ are nonzero, and where

see [17, Corollary 7.4.33]. Then $e^{2}f_{1}(dx-by,-cx+ay)=(ad-bc)^{2g+2}f_{2}(x,y)$, so we see that $f_{2}\bmod k^{\times}=\Gamma(f_{1}\bmod k^{\times})$, where $\Gamma\in\operatorname{PGL}_{2}(k)$ is the element represented by the matrix $M\colonequals\bigl{[}\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr{]}$. Thus, if $C_{1}$ and $C_{2}$ are isomorphic, the element $\Gamma$ of $\operatorname{PGL}_{2}(k)$ takes the ramification points of $C_{1}\to{\mathbf{P}}^{1}$ to the ramification points of $C_{2}\to{\mathbf{P}}^{1}$. Conversely, if $\bigl{[}\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr{]}\in\operatorname{PGL}_{2}(k)$ takes the ramification points of $C_{1}\to{\mathbf{P}}^{1}$ to the ramification points of $C_{2}\to{\mathbf{P}}^{1}$, then there is an $e\in\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}$, with $e^{2}\in k$, that makes (1) hold. If $e$ lies in $k$, we have an isomorphism between $C_{1}$ and $C_{2}$; if $e$ does not lie in $k$, we have an isomorphism between $C_{1}$ and the quadratic twist of $C_{2}$, that is, the curve $y^{2}=\nu f_{2}$, where $\nu$ is a nonsquare in $k$. (In general, a twist of a curve $C$ over a finite field $k$ is another curve over $k$ that becomes isomorphic to $C$ when the base field is extended to an algebraic closure of $k$. Twists of $C$ correspond to elements of the cohomology set $H^{1}(\operatorname{Gal}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}/k),\operatorname{Aut}_{\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}}C)$ — see [21, §III.1.3] — and by “the quadratic twist” of a hyperelliptic curve we mean the twist corresponding to the cocycle that sends the Frobenius element of $\operatorname{Gal}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}/k)$ to the hyperelliptic involution. Note that sometimes the quadratic twist may in fact be the trivial twist.)

Thus, we have a map from the set of isomorphism classes of genus-$g$ hyperelliptic curves over $k$ to the set of $\operatorname{PGL}_{2}(k)$ orbits of $\operatorname{Sym}^{n}(k)$, where $n=2g+2$. This map is clearly surjective, and the $\operatorname{PGL}_{2}(k)$ orbit of an element $A$ of $\operatorname{Sym}^{n}(k)$ has at most two preimages in the set of isomorphism classes of hyperelliptic curves: the isomorphism classes of $y^{2}=f$ and of $y^{2}=\nu f$, where $f$ is the unique monic polynomial with $\operatorname{Zeros}(f)=A$ and where $\nu$ is a nonsquare in $k$. (We say “at most two” preimages because, as we noted above, these two curves may be isomorphic to one another.)

Whether an element $A$ of $\operatorname{Sym}^{n}(k)$ has one or two preimages is easy to determine: Let $f\in R_{n}$ be the unique monic polynomial with $\operatorname{Zeros}(f)=A$. Compute all elements $\Gamma$ of $\operatorname{PGL}_{2}(k)$ that take the set $A$ to itself; at worst this takes time ${O(n(n-1)(n-2))}$, and since $n$ is fixed in our context, this is $O(1)$ operations. For each such $\Gamma$ compute the element $s_{\Gamma,f}$ of $k^{\times}/k^{\times 2}$. If any of these elements is nontrivial, then the curve $y^{2}=f$ is isomorphic to its twist $y^{2}=\nu f$. (Compare to [20, Lemma 1.2].)

This shows that if we can compute a complete set of unique representatives for the orbits of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ acting on $\operatorname{Sym}^{2g+2}({\mathbf{F}}_{q})$ in time $\mathchoice{\accentset{\displaystyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\textstyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptstyle\text{\smash{\raisebox{-4.2194pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptscriptstyle\text{\smash{\raisebox{-3.01385pt}{$\widetildesym$}}}}{O}}(q^{2g-1})$, we can also compute a complete set of unique representatives for the hyperelliptic curves of genus $g$ over ${\mathbf{F}}_{q}$ in time $\mathchoice{\accentset{\displaystyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\textstyle\text{\smash{\raisebox{-6.02773pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptstyle\text{\smash{\raisebox{-4.2194pt}{$\widetildesym$}}}}{O}}{\accentset{\scriptscriptstyle\text{\smash{\raisebox{-3.01385pt}{$\widetildesym$}}}}{O}}(q^{2g-1})$. Thus, for the rest of this paper we focus on enumerating the $\operatorname{PGL}_{2}(k)$ orbits of $\operatorname{Sym}^{n}(k)$, for $n=2g+2$. In particular, for our application to enumerating hyperelliptic curves we only need to consider even $n$ that are at least $6$. The algorithms that we present for the latter problem work for all finite fields, not just those of odd characteristic.

In this section we prove Theorem 1.2. We also prove similar results that give complete sets of unique representatives for the action of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ on monic homogeneous quartics that have one or two irreducible quadratic factors, and we state generalizations to finite fields of characteristic $2$. We begin with an elementary lemma.

For products of two distinct irreducible quadratics, we have a similar result.

We also have a similar result for quartics with exactly one irreducible quadratic factor, which works in all characteristics.

Theorem 3.6 applies to all finite fields, while Theorems 1.2 and 3.4 require the characteristic to be odd. The following theorem generalizes the latter two results to characteristic $2$. The proof is analogous to those of the earlier theorems, but is made much simpler by the fact that in this case every involution in $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ is conjugate to $\bigl{[}\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\bigr{]}$. We leave the details to the reader.

In this section we define an easily computable invariant²²2Classically, an invariant on the set $R_{n}\subset k[x,y]$ of homogeneous bivariate polynomials of degree $n$ is a function $R_{n}\to k$ that is constant on $\operatorname{PGL}_{2}(k)$ orbits. We use the term more generally here, and simply mean a function from $R_{n}$ to any set that is constant on $\operatorname{PGL}_{2}(k)$ orbits. for monic irreducible homogeneous bivariate polynomials of arbitrary degree $n\geq 4$ over a finite field ${\mathbf{F}}_{q}$ under the action of $\operatorname{PGL}_{2}({\mathbf{F}}_{q})$ mentioned in Section 2. The invariant is based on the classical cross ratio, which is the function that assigns to an ordered quadruple $(P_{1},P_{2},P_{3},P_{4})$ of distinct elements of ${\mathbf{P}}^{1}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ the element $\alpha\in\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}}$ for which ${\Gamma(P_{4})={[\alpha\,{:}\,1]}}$, where $\Gamma$ is the unique element of $\operatorname{PGL}_{2}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ that sends $P_{1}$ to $\infty$, $P_{2}$ to $0$, and $P_{3}$ to $1$. It follows that the cross ratio is constant on the orbits of the diagonal action of $\operatorname{PGL}_{2}(\mathchoice{k\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm k}}$}}{k\hbox to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm k}}$}})$ on such quadruples, and takes distinct values on distinct orbits. (Compare to [5, Definition III.3.7] and the propositions following it.)