# Optimal plane curves of degree $q-1$ over a finite field

Walteir de Paula Ferreira  and  Pietro Speziali

(Date: November 2022)

###### Abstract.

Let $q\geq 5$ be a prime power. In this note, we prove that if a plane curve
$\mathcal{X}$ of degree $q-1$ defined over $\mathbb{F}_{q}$ without
$\mathbb{F}_{q}$-linear components attains the Sziklai upper bound
$(d-1)q+1=(q-1)^{2}$ for the number of its $\mathbb{F}_{q}$-rational points,
then $\mathcal{X}$ is projectively equivalent over $\mathbb{F}_{q}$ to the
curve $\mathcal{C}_{(\alpha,\beta,\gamma)}:\alpha X^{q-1}+\beta Y^{q-1}+\gamma
Z^{q-1}=0$ for some $\alpha,\beta,\gamma\in\mathbb{F}_{q}^{*}$ such that
$\alpha+\beta+\gamma=0$. This completes the classification of curves that are
extremal with respect to the Sziklai bound. Also, since the Sziklai bound is
equal to the Stöhr-Voloch’s bound for plane curves of degree $q-1$, our main
result classifies the $\mathbb{F}_{q}$-Frobenius extremal classical plane
curves of degree $q-1$.

Walteir de Paula Ferreira is with the Instituto de Matemática, Estatística e
Computação Científica, Universidade Estadual de Campinas, Campinas, SP
13083-859, Brazil.
e-mail<EMAIL_ADDRESS>
Pietro Speziali is with the Instituto de Matemática, Estatística e Computação
Científica, Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil.
e-mail<EMAIL_ADDRESS>
Keywords: Algebraic curves; Rational points; Sziklai bound; optimal curves;
arcs

Mathematics Subject Classification (2010): 114H37, 14H05

##  1\. Introduction

Let $\mathcal{X}$ be a (projective, geometrically irreducible, algebraic)
curve defined over a finite field $\mathbb{F}_{q}$ where $q=p^{h}$ for some
prime $p$ and some positive integer $h$. It is a classical problem to count
the number $\text{N}_{q}(\mathcal{X})$ of $\mathbb{F}_{q}$-rational points of
$\mathcal{X}$. However, since this problem is rather hard to solve, it is
often desirable to find good upper bounds for $\text{N}_{q}(\mathcal{X})$
depending on some invariants of the curve $\mathcal{X}$. For instance, the
famous Hasse-Weil upper bound states that $\text{N}_{q}(\mathcal{X})\leq
1+q+2g\sqrt{q}$ where $g$ is the genus of $\mathcal{X}$. Note that the same
bound holds for any curve defined over $\mathbb{F}_{q}$ and genus $g$. Once we
have a bound, it is a natural question to see whether such a bound is sharp or
not, and then, it is also natural to try and classify the optimal curves, that
is, the curves attaining said bound. In the context of the Hasse-Weil bound,
such optimal curves do exist and are called maximal curves. Maximal curves may
exist when $q=n^{2}$ is a square, and it is known that the genus of an
$\mathbb{F}_{q}$-maximal curve is upper bounded by $(n-1)(n-2)/2$. Up to
birational equivalence, there is exactly one $\mathbb{F}_{q}$-maximal curve of
genus $(n-1)(n-2)/2$: the Hermitian curve $\mathcal{H}_{n}$ given by the
homogeneous equation

$\mathcal{H}_{n}:Y^{n}Z+YZ^{n}=X^{n+1}.$

It is a classical and yet unsolved problem to find the spectrum of the genera
of $\mathbb{F}_{q}$-maximal curves; see [1].

Now, let $\mathcal{X}$ be a plane curve of degree $d\geq 2$ without
$\mathbb{F}_{q}$-linear components. In [19], Sziklai conjectured the following
result: $\text{N}_{q}(\mathcal{X})\leq(d-1)q+1$. The unique exception to
Sziklai’s conjecture was found in [9, Section 3] and it is given by the curve
over $\mathbb{F}_{4}$ with homogeneous equation

$X^{4}+Y^{4}+Z^{4}+X^{2}Y^{2}+Y^{2}Z^{2}+Z^{2}X^{2}+X^{2}YZ+XY^{2}Z+XYZ^{2}=0.$

The Sziklai bound was later proved by Homma and Kim in a sequence of three
papers [9, 13, 14]. We are interested in curves attaining the Sziklai bound.
Let $\mathcal{X}$ be a nonsingular plane curve that is optimal with respect to
the Sziklai upper bound; then [13, Section 5] its degree $d$ must belong to
the set $\\{2,\sqrt{q}+1,q-1,q,q+1,q+2\\}$. This means that the spectrum of
the degrees of optimal Sziklai curves is pretty small, hence, it seems
feasible to classify, up to projective equivalence, the curves attaining the
Sziklai bound. So far, this is known in the cases $d=2,\sqrt{q}+1,q,q+1$ or
$q+2$; see [8, Section 5.1]), [7, Theorem 8]), [15, Main Theorem], [16,
Theorem 1.3], [20] and [10]. For the case $d=q-1$, a family of optimal curves
is given by the homogeneous equation

$C_{(\alpha,\beta,\gamma)}:\alpha X^{q-1}+\beta Y^{q-1}+\gamma Z^{q-1}=0$

with $\alpha,\beta,\gamma\in\mathbb{F}_{q}^{*}$ and $\alpha+\beta+\gamma=0$.
Recently, Homma [12] has stated the following question:

######  Question 1.1.

Are there curves of degree $q-1$ that attain the Sziklai’s upper bound such
that are not projectively equivalent over $\mathbb{F}_{q}$ to a curve of type
$\mathcal{C}_{(\alpha,\beta,\gamma)}$?

In the same paper, he gives a positive solution to this problem for $q=4$,
since in this case, the Hermitian cubic attains Sziklai’s bound but is not
projectively equivalent to any $\mathcal{C}_{(\alpha,\beta,\gamma)}$.

In this paper, we give a negative answer to Question 1.1 for $q\geq 5$, thus
completing the classification of optimal Sziklai curves; see Theorem 4.4.

This paper is organized as follows.

In Section 2, we give the necessary background, as well as, a brief survey of
the existing literature on the Sziklai bound and related topics.

In Section 3, we will give several technical results that are necessary to
prove our classification of curves of degree $q-1$ that are optimal with
respect to the Sziklai bound.

Section 4 is devoted to the proof of Theorem 4.4, which is the main result of
our paper. Here, we remark that while our technique applies to all $q\geq 8$,
the cases $q=5,7$ need to be dealt with by using two different approaches,
which are of independent interest. The former needs the knowledge of
L-polynomial of curves of genus 3 with small defect [17], the latter is based
on the classification on $(36,6)$-arcs in $\mathbb{P}^{2}(\mathbb{F}_{7})$
[3].

Finally, in Section 5, we give a brief discussion regarding topics that are
directly linked to (or are possible applications of) our results. More in
detail, we show as our main result is related to the
$\mathbb{F}_{q}$-Frobenius classical plane curves of degree $q-1$ attaining
the Stöhr-Voloch upper bound. Further, curves attaining the Sziklai upper
bound are related to nonsingular hypersurfaces with many
$\mathbb{F}_{q}$-rational points in even-dimensional projective spaces; see
[4, 21].

##  2\. Background and preliminary results

###  2.1. Plane Curves

Our notation and terminology are standard. For definitions and basic
properties of plane curves, see [6, Chapter 1-5]. In the projective plane
$\mathbb{P}^{2}(\mathbb{F}_{q})$, a plane curve $\mathcal{C}$ in
$\mathbb{P}^{2}(\mathbb{F}_{q})$ of homogeneous equation $F(X,Y,Z)=0$, where
$F\in\mathbb{F}_{q}[X,Y,Z]$, is denoted by $\mathcal{C}=\textbf{{v}}(F)$ and
consists of all points $(x:y:z)\in\mathbb{P}^{2}(\mathbb{F}_{q})$ such that
$F(x,y,z)=0$. We regard $\mathcal{C}$ as a curve over a fixed algebraic
closure $K:=\overline{\mathbb{F}}_{q}$. By doing so, the points in
$(x:y:z)\in\mathbb{P}^{2}(\mathbb{F}_{q})$ such that $F(x,y,z)=0$ are called
$\mathbb{F}_{q}$-rational points (or simply, rational points) of $\mathcal{C}$
and
$\mathcal{C}(\mathbb{F}_{q}):=\mathbb{P}^{2}(\mathbb{F}_{q})\cap\mathcal{C}$.
A plane curve $\mathcal{C}=\textbf{{v}}(F)$ in
$\mathbb{P}^{2}:=\mathbb{P}^{2}(K)$, where $F\in K[X,Y,Z]$, is said to be
defined over $\mathbb{F}_{q}$ if there is $\lambda\in K^{*}$ such that
$\lambda\cdot F\in\mathbb{F}_{q}[X,Y,Z]$. A point $P=(x:y:z)$ of a plane curve
$\mathcal{C}$ in $\mathbb{P}^{2}$ is singular if

$\dfrac{\partial F}{\partial X}(x,y,z)=\dfrac{\partial F}{\partial
Y}(x,y,z)=\dfrac{\partial F}{\partial Z}(x,y,z)=0.$

Otherwise, $P$ is a nonsingular (or a simple) point. The degree of
$\mathcal{C}$, denoted by $\text{deg}(\mathcal{C})$, is $\text{deg}(F)$. A
component of $\mathcal{C}$ is a curve $\mathcal{X}$ of homogeneous equation
$G=0$ such that $G|F$. A curve $\mathcal{C}$ is said to be absolutely
irreducible if $F$ is irreducible over $K$. A projectivity
$\varphi_{A}:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ is given as follows:

$\varphi_{A}(x:y:z)=A\cdot\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)$

where $A\in\text{GL}_{3}(K)$. Two curves $\mathcal{C}$ and $\mathcal{X}$ are
said projectively equivalent over $\mathbb{F}_{q}$, denoted by
$\mathcal{C}\simeq_{\text{proj}}\mathcal{X}$, if there is a projectivity
$\varphi_{A}:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ with
$A\in\text{GL}_{3}(\mathbb{F}_{q})$ such that
$\varphi_{A}(\mathcal{C})=\mathcal{X}$. The $q$-Frobenius map
$\Psi_{q}:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ is defined by
$\Psi_{q}(x:y:z):=(x^{q}:y^{q}:z^{q})$. The dual projective space
$\check{\mathbb{P}}^{2}=\check{\mathbb{P}}^{2}(K)$ is the space of all line in
$\mathbb{P}^{2}$ and $\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$ we mean the set
of lines defined over $\mathbb{F}_{q}$ of $\check{\mathbb{P}}^{2}.$ For a
point $P\in\mathbb{P}^{2}(\mathbb{F}_{q}),$ define
$\check{P}(\mathbb{F}_{q}):=\\{l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})\ |\
P\in l\\}.$

The proof of the next result can be found in [6, section 4.5].

######  Theorem 2.1 (Nother’s “AF + BG” Theorem).

Let $\mathcal{F}=\textbf{{v}}(F)$ and $\mathcal{G}=\textbf{{v}}(G)$ be two
plane curves defined over $\mathbb{F}_{q}$ with no common components. If
$\mathcal{F}\cap\mathcal{G}=\\{P_{1},...,P_{s}\\}$ and the multiplicity of
$\mathcal{F}$ and $\mathcal{G}$ at each point $P_{i}$ is equal to $1$, then
for all plane curve $\mathcal{X}=\textbf{{v}}(H)$ defined over
$\mathbb{F}_{q}$ with $\mathcal{F}\cap\mathcal{G}\subseteq\mathcal{X}$ there
are $A,B\in\mathbb{F}_{q}[X,Y,Z]$ such that $H=AF+BG$.

Note that Theorem 2.1 gives a method to find all plane curves passing through
a given set of points of $\mathbb{P}^{2}(\mathbb{F}_{q})$.

###  2.2. Arcs of $\mathbb{P}^{2}(\mathbb{F}_{q})$ and Codes

The following brief account of the theory of plane arcs and their relationship
to linear codes is based on [2, Chapter 10 and 17], to which we refer the
reader for further details. In the projective plane
$\mathbb{P}^{2}(\mathbb{F}_{q})$, a $(k,n)$-arc $\mathcal{K}$ is a set of $k$
points such that each line contains at most $n$ point of $\mathcal{K}$ and
there is a line that contains exactly $n$ points of $\mathcal{K}$. A
$(k,2)$-arc is simply called an arc. Let
$\mathcal{K}\subseteq\mathbb{P}^{2}(\mathbb{F}_{q})$ be a $(k,n)$-arc, for
$0\leq i\leq q+1$, define
$\mathcal{A}_{i}(\mathcal{K}):=\\{l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})\
|\ \\#(l\cap\mathcal{K})=i\\}$,
$a_{i}(\mathcal{K}):=\\#\mathcal{A}_{i}(\mathcal{K})$ and
$k_{0}(\mathcal{K}):={\rm min}\\{i\ |\ a_{i}\neq 0\\}$. When there is no
possibility of confusion we will denote it simply by $\mathcal{A}_{i}$,
$a_{i}$ and $k_{0}$.

A linear subspace $C$ of $\mathbb{F}_{q}^{n}$ of dimension $k$ is called an
$[n,k]_{q}$-code. The elements of a linear code $C$ are called codewords. The
weight of a codeword $x=(x_{1},...,x_{n})\in C\subseteq\mathbb{F}_{q}^{n}$ is
the number of nonzero coordinates in $x$, denoted by $wt(x)$. The minimum
distance of $\mathcal{C}$ is ${\rm min}\\{wt(x)|x\in\mathcal{C},x\neq 0\\}$.
If the minimum distance of $\mathcal{C}$ is $d$, then we write that
$\mathcal{C}$ is an $[n,k,d]_{q}$-code. A generator matrix $G$ of an
$[n,k,d]_{q}$-code $\mathcal{C}$ is a matrix with $k$ rows and $n$ columns
whose rows form a basis of $\mathcal{C}$. The code $\mathcal{C}$ is recovered
from $G$ by taking all linear combinations of rows. If $\mathcal{C}$ contains
$c_{i}$ codewords of weight $i$, for $i=1,...,n$, then the weight enumerator
is defined by

$W_{\mathcal{C}}(z):=c_{0}+c_{1}z+c_{2}z^{2}+\cdots+c_{n}z^{n}\in\mathbb{Z}[z].$

Let $\mathcal{C}$ be a linear $[n,3,d]_{q}$-code described by a generator
matrix $G$. We assume that there is no $0$ column in $G$. We can then consider
the columns of $G$ as generators of points in
$\mathbb{P}^{2}(\mathbb{F}_{q})$. A linear $[n,3,d]_{q}$-code $\mathcal{C}$ is
called projective if there is a generator matrix whose columns generate
different points in $\mathbb{P}^{2}(\mathbb{F}_{q})$. For a projective
$[n,3,d]_{q}$-code $\mathcal{C}$ with a generator matrix $G$, the $n$ points
in $\mathbb{P}^{2}(\mathbb{F}_{q})$ corresponding to columns of $G$ form an
$(n,n-d)$-arc in $\mathbb{P}^{2}(\mathbb{F}_{q})$. For each $i$ in
$0,\ldots,n-d$, the number $a_{i}$ of lines in $\mathcal{A}_{i}$ is related to
the coefficients $c_{i}$ of the weight enumerator as follows:
$(q-1)\cdot(a_{0},\cdots,a_{n-d})=(c_{n},\cdots,c_{d})$.

###  2.3. Stöhr-Voloch Theory

In [18], Stöhr and Voloch gave a geometric method to obtain upper bounds for
the number of rational points of a curve of $\mathbb{P}^{n}$. Here, we give
the necessary background on a particular case of the Stöhr-Voloch theorem that
we will need in Section 4.

Let $\mathcal{C}$ a plane curve defined over $\mathbb{F}_{q}$ and
$0=\epsilon_{0}<\epsilon_{1}=1<\epsilon_{2}$ the order sequence of
$\mathcal{C}$. Let $P\in\mathcal{C}$. If $\mathcal{C}$ is nonsingular, then
the hermitian $P$-invariants are $j_{0}(P)=0,j_{1}(P)=1$ and
$j_{2}(P)=\text{I}(P,\mathbb{T}_{P}(\mathcal{C})\cap\mathcal{C})$ where
$\mathbb{T}_{P}(\mathcal{C})$ is the tangent line to $\mathcal{C}$ at $P$ and
$\text{I}(P,\mathbb{T}_{P}(\mathcal{C})\cap\mathcal{C})$ is the intersection
multiplicity of $\mathbb{T}_{P}(\mathcal{C})$ and $\mathcal{C}$ at $P$. Since
$\mathcal{C}$ is defined over $\mathbb{F}_{q}$, then there is a smallest
integer $\nu\in\\{1,\epsilon_{2}\\}$ such that

$\text{det}\left(\begin{array}[]{ccc}x_{0}^{q}&x_{1}^{q}&x_{2}^{q}\\\
x_{0}&x_{1}&x_{2}\\\
D_{\zeta}^{(\nu)}x_{0}&D_{\zeta}^{(\nu)}x_{1}&D_{\zeta}^{(\nu)}x_{2}\\\
\end{array}\right)\not\equiv 0$

where $D_{\zeta}^{(k)}$ is the $k$-th Hasse derivative with respect to a
separating variable $\zeta$ of $K(\mathcal{C})|K$, and $x_{0},x_{1},x_{2}$ are
the coordinate functions on $\mathcal{C}\subseteq\mathbb{P}^{2}$. The number
$\nu$ is called the $q$-Frobenius order of $\mathcal{C}$, and such a curve is
called $q$-Frobenius classical if $\nu=1$. The following is a slight rewording
of [18, Theorem 0.1].

######  Theorem 2.2.

Let $\mathcal{C}$ be an irreducible plane curve of degree $d$ defined over a
finite field $\mathbb{F}_{q}$. If $\mathcal{C}$ is $q$-Frobenius classical,
then

${\rm N}_{q}(\mathcal{C})\leq\frac{1}{2}d(d+q-1).$

A refined version of this theorem can be obtained if one can gather sufficient
information on the number and the weight of $\mathbb{F}_{q}$-rational
inflection points.

######  Theorem 2.3.

Let $\mathcal{C}\subseteq\mathbb{P}^{2}$ be an irreducible nonsingular
algebraic curve of genus $g$ and degree $d$ defined over $\mathbb{F}_{q}$. If
$\nu$ is the $q$-Frobenius order of $\mathcal{C}$, then

$2\cdot{\rm
N}_{q}(\mathcal{C})\leq\nu(2g-2)+(q+2)d-\sum_{P\in\mathcal{C}}A(P)$

where $A(P)=j_{2}(P)-\nu-1$ if $P\in\mathcal{C}(\mathbb{F}_{q})$ and $A(P)=0$
otherwise.

###  2.4. Sziklai’s upper bound and optimal curves

Let $C_{d}(\mathbb{F}_{q})$ be the set of plane curves of degree $d\geq 2$
defined over $\mathbb{F}_{q}$ without $\mathbb{F}_{q}$-linear components. For
$\mathcal{C}\in C_{d}(\mathbb{F}_{q})$, let
$\text{N}_{q}(\mathcal{C}):=|\mathcal{C}(\mathbb{F}_{q})|$. In [19], Sziklai
conjectured the bound $\text{N}_{q}(\mathcal{C})\leq(d-1)q+1$. Actually,
Sziklai’s conjecture fails for curves of degree $4$ over $\mathbb{F}_{4}$, as
the plane curve with equation

(1) $\displaystyle
X^{4}+Y^{4}+Z^{4}+X^{2}Y^{2}+Y^{2}Z^{2}+Z^{2}X^{2}+X^{2}YZ+XY^{2}Z+XYZ^{2}$
$\displaystyle=0$

has $14$ points over $\mathbb{F}_{4}$ (see [9, section 3]), while Sziklai’s
bound is equal to $13$. Later on, in a sequence of three papers [9, 13, 14],
Homma and Kim proved the modified Sziklai’s Conjecture:

######  Theorem 2.4 (Sziklai’s upper bound).

If $\mathcal{C}\in\mathcal{C}_{d}(\mathbb{F}_{q}),$ then

(2) $\displaystyle{\rm N}_{q}(\mathcal{C})$ $\displaystyle\leq(d-1)q+1,$

except for the curve over $\mathbb{F}_{4}$ which is projectively equivalent to
curve defined by (1).

######  Remark 2.5.

Let $\mathcal{C}\in C_{q}(\mathbb{F}_{q})$ with $(d,q)\neq(4,4)$. If
$N_{q}(\mathcal{C})=(q-1)q+1$, then $\mathcal{C}$ is absolutely irreducible
and any rational point of $\mathcal{C}$ is nonsingular, see [13, section 2].

In [13, section 5], Homma and Kim observe that the possible degrees $d$ of a
nonsingular curve with $(d-1)q+1$ rational points are
$q+2,q+1,q,q-1,\sqrt{q}+1$ and $2$. Also, for each degree $d$ in the list,
there is a nonsingular curve of degree $d$ attaining the bound. For $d\neq
q-1$, the complete classification of such optimal curves is known; we
summarize these results in the following Theorem.

######  Theorem 2.6.

Let $\mathcal{C}\in\text{C}_{d}(\mathbb{F}_{q})$ a nonsingular curve with
${\rm N}_{q}(\mathcal{C})=(d-1)q+1$.

  * (i)

If $d=2$, then $\mathcal{C}\simeq_{{\rm proj}}\textbf{{v}}(X^{2}+YZ)$ over
$\mathbb{F}_{q}$ ([8, Section 5.1]).

  * (ii)

If $d=\sqrt{q}+1$, then $\mathcal{C}\simeq_{{\rm
proj}}\textbf{{v}}(X^{\sqrt{q}+1}+Y^{\sqrt{q}+1}+Z^{\sqrt{q}+1})$ over
$\mathbb{F}_{q}$ when $q>4$ is a square ([7]).

  * (iii)

If $d=q+2$, then $\mathcal{C}$ is projectively equivalent over
$\mathbb{F}_{q}$ to the curve of type
$\textbf{{v}}(Y(Y^{q}Z-YZ^{q})+Z(Z^{q}X-ZX^{q})+(aX+bY+cZ)(X^{q}Y-XY^{q}))$
where $t^{3}-(ct^{q}+bt+a)$ is irreducible over $\mathbb{F}_{q}$ ([20] and
[10]).

  * (iv)

If $d=q+1$, then $\mathcal{C}$ is projectively equivalent over
$\mathbb{F}_{q}$ to the curve

$\mathcal{C}_{q+1}:=\textbf{{v}}(X^{q+1}-X^{2}Z^{q-1}+Y^{q}Z-YZ^{q})$

when $q\geq 5$ or $q=2$. If $q=4$, then $\mathcal{C}$ is projectively
equivalent over $\mathbb{F}_{4}$ to either $\mathcal{C}_{5}$ or the curve

$\textbf{{v}}(\mu G(X,Y,Z)+XYZ(\mu^{2}(X^{2}+Y^{2}+Z^{2})+XY+YZ+ZX))$

where $G(X,Y,Z)=X^{4}Y+XY^{4}+Y^{4}Z+YZ^{4}+Z^{4}X+ZX^{4}$ and
$\mu^{2}+\mu+1=0$. Moreover, those two curves are not projectively equivalent
to each other over $\mathbb{F}_{4}$. If $q=3$, then $\mathcal{C}$ is
projectively equivalent over $\mathbb{F}_{4}$ either to $\mathcal{C}_{4}$ or
to the curve

$\textbf{{v}}(X^{3}Y-XY^{3}+Y^{3}Z-YZ^{3}+Z^{3}X-ZX^{3}+XYZ(X+Y-Z)).$

Moreover, those two curves are not projectively equivalent to each other over
$\mathbb{F}_{3}$ ([16]) .

  * (v)

If $d=q$, then $\mathcal{C}\simeq_{{\rm
proj}}\textbf{{v}}(X^{q}-XZ^{q-1}+Y^{q-1}Z-Z^{q})$ over $\mathbb{F}_{q}$
([15]).

For $d=q-1$, as was mentioned by Sziklai in [19], the curve

$\mathcal{C}_{(\alpha,\beta,\gamma)}:=\textbf{{v}}(\alpha X^{q-1}+\beta
Y^{q-1}+\gamma Z^{q-1})$

with $\alpha,\beta,\gamma\in\mathbb{F}_{q}^{*}$ and $\alpha+\beta+\gamma=0$
has $(q-1)^{2}$ rational points. This curve is nonsingular and the set of its
$\mathbb{F}_{q}$-rational points is

$\mathcal{C}_{(\alpha,\beta,\gamma)}(\mathbb{F}_{q})=\mathbb{P}^{2}(\mathbb{F}_{q})\backslash\\{X=0\\}\cup\\{Y=0\\}\cup\\{Z=0\\}.$

Recently, Homma [12], has studied the number of projective equivalence classes
over $\mathbb{F}_{q}$ in this family of curves. More precisely, he proves the
following Theorem.

######  Theorem 2.7.

[12, Theorem 1.3] The number $\nu_{q}$ of projective equivalence classes over
$\mathbb{F}_{q}$ in the family of curves
$\\{\mathcal{C}_{(\alpha,\beta,\gamma)}\ |\
\alpha,\beta,\gamma\in\mathbb{F}_{q}^{*},\ \alpha+\beta+\gamma=0\\}$ is as
follows:

  * (i)

Suppose that the characteristic of $\mathbb{F}_{q}$ is neither $2$ or $3$.

    * (1)

If $q\equiv 2\mod 3$, then $\nu_{q}=(q+1)/6$.

    * (2)

If $q\equiv 1\mod 3$, then $\nu_{q}=(q+5)/6$.

  * (ii)

Suppose that $q$ is a power of $3$. Then $\nu_{q}=(q+3)/6$.

  * (iii)

Suppose that $q$ is a power of $2$:

    * (1)

If $q\equiv 2\mod 3$, then $\nu_{q}=(q-2)/6$.

    * (2)

If $q\equiv 1\mod 3$, then $\nu_{q}=(q+2)/6$.

In the same paper, the curves of degree $3$ over $\mathbb{F}_{4}$ are
classified.

######  Theorem 2.8.

[12, Theorem 3.1] Let $\mathcal{C}$ be a nonsingular plane curve of degree $3$
over $\mathbb{F}_{4}$. If $\text{N}_{4}(\mathcal{C})=9$, then $\mathcal{C}$ is
either

  * (i)

the Hermitian cubic $\mathcal{H}_{3}$ given by
$\textbf{{v}}(X^{3}+Y^{3}+Z^{3})$ or

  * (ii)

projectively equivalent to the curve $\mathcal{C}_{\alpha}$ given by
$\textbf{{v}}(X^{3}+\alpha Y^{3}+\alpha^{2}Z^{3})$ where
$\mathbb{F}_{4}=\\{0,1,\alpha,\alpha^{2}\\}$.

######  Remark 2.9.

It can be shown that the Hermitian cubic $\mathcal{H}_{3}$ and the curve
$\mathcal{C}_{\alpha}$ are birationally equivalent over $\mathbb{F}_{4}$.
Also, they are projectively equivalent over $\mathbb{F}_{2^{6}}$ (see, [12,
section 4]).

##  3\. Preliminary results

In this section, we give several technical results that are necessary to prove
our main result Theorem 4.4. First of all, not that, by Theorem 2.8, we may
assume $q\geq 5$. We begin by proving the following Proposition.

######  Proposition 3.1.

[12, Proposition 2.1] Let $\mathcal{C}$ be a (possibly reducible) plane curve
over $\mathbb{F}_{q}$ of degree $q-1$. Then
$\mathcal{C}\in\\{\mathcal{C}_{(\alpha,\beta,\gamma)}\ |\
\alpha,\beta,\gamma\in\mathbb{F}_{q}^{*},\ \alpha+\beta+\gamma=0\\}$ if and
only if

$\mathcal{C}(\mathbb{F}_{q})=\mathbb{P}^{2}(\mathbb{F}_{q})\backslash(\textbf{{v}}(X)\cup\textbf{{v}}(Y)\cup\textbf{{v}}(Z)).$

Fix a curve $\mathcal{X}\in C_{q-1}(\mathbb{F}_{q})$ with
$\text{N}_{q}(\mathcal{X})=(q-1)^{2}$. Let
$Z(\mathcal{X}):=\mathbb{P}^{2}(\mathbb{F}_{q})\backslash\mathcal{X}(\mathbb{F}_{q})$.
By Proposition 3.1, if
$Z(\mathcal{X})=(\textbf{{v}}(X)\cup\textbf{{v}}(Y)\cup\textbf{{v}}(Z))(\mathbb{F}_{q})$
then $\mathcal{X}=\mathcal{C}_{(\alpha,\beta,\gamma)}$ for some
$\alpha,\beta,\gamma\in\mathbb{F}_{q}^{\times}$ such that
$\alpha+\beta+\gamma=0$. Since the general projective linear group ${\rm
PGL}(2,q)$ $3$-transitively on the set of lines of
$\mathbb{P}^{2}(\mathbb{F}_{q})$ and $\\#Z(\mathcal{X})=3q$, if there are
three lines $l_{1},l_{2},l_{3}\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$ such
that $Z(\mathcal{X})=(l_{1}\cup l_{2}\cup l_{3})(\mathbb{F}_{q})$, then
$l_{1},l_{2},l_{3}$ are not concurrent and we can choose coordinates $X,Y,Z$
of $\mathbb{P}^{2}$ such that $l_{1}=\textbf{{v}}(X),l_{2}=\textbf{{v}}(Y)$ e
$l_{3}=\textbf{{v}}(Z)$. This means that, in order to prove our main result,
it is enough to show the existence of such three lines. We start by proving
the following lemma.

######  Lemma 3.2.

The set $\mathcal{X}(\mathbb{F}_{q})\subseteq\mathbb{P}^{2}(\mathbb{F}_{q})$
is a $((q-1)^{2},q-1)$-arc.

###### Proof.

Since $\text{deg}(\mathcal{X})=q-1$, then
$\\#(l\cap\mathcal{X}(\mathbb{F}_{q}))\leq q-1$ for every line
$l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$. Let
$t:=\max\\{\\#(l\cap\mathcal{X}(\mathbb{F}_{q}))\ |\
l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})\\}\leq q-1.$ If
$P\in\mathcal{X}(\mathbb{F}_{q})$ then each line in
$\check{P}(\mathbb{F}_{q})$ contains at most $t$ points of
$\mathcal{X}(\mathbb{F}_{q})$. Since $\\#\check{P}(\mathbb{F}_{q})=q+1$ then
$1+(q+1)(t-1)\geq(q-1)^{2}$. Hence,

$q-1\geq t\geq\dfrac{q(q-2)}{q+1}+1=q-2+\dfrac{3}{q+1}>q-2.$

This implies that $t=q-1$. Therefore, $\mathcal{X}(\mathbb{F}_{q})$ is a
$((q-1)^{2},q-1)$-arc in $\mathbb{P}^{2}(\mathbb{F}_{q})$. ∎

For $0\leq i\leq q+1$, recall the definition of

$\mathcal{A}_{i}=\\{l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})\ |\
\\#(l\cap\mathcal{X}(\mathbb{F}_{q}))=i\\}\text{ and
}a_{i}=\\#\mathcal{A}_{i}.$

Since $\text{deg}(\mathcal{X})=q-1$, then $a_{q}=a_{q+1}=0$. A line
$l\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$ is called an $i$-line if
$l\in\mathcal{A}_{i}$. A point $P\in\mathbb{P}(\mathbb{F}_{q})$ is said to be
of type $i_{1}^{r_{1}}...i_{t}^{r_{t}}$ ($i_{1}>\cdots>i_{t}$ and
$r_{1},...,r_{t}\geq 0$) if the number of $i_{j}$-lines through $P$ is $r_{j}$
for $j=1,...,t$. Also, as $\mathcal{X}(\mathbb{F}_{q})$ is a
$((q-1)^{2},q-1)$-arc, we may use the following result:

######  Lemma 3.3.

[8, Lemma 12.1.1] With the same notation as above, we have the following
equalities.

  * (i)

$\displaystyle\sum_{i=0}^{q-1}a_{i}=q^{2}+q+1.$

  * (ii)

$\displaystyle\sum_{i=1}^{q-1}ia_{i}=(q+1)(q-1)^{2}.$

  * (iii)

$\displaystyle\sum_{i=2}^{q-1}i(i-1)a_{i}=q(q-2)(q-1)^{2}.$

######  Lemma 3.4.

Let $P\in\mathbb{P}^{2}(\mathbb{F}_{q})$ be a point of type
$i_{1}^{r_{1}}...i_{t}^{r_{t}}$. Then $r_{1}+\cdots+r_{t}=q+1$. Moreover,

  * (i)

If $P\in\mathcal{X}$, then $i_{j}\geq 1$ for all $j=1,...,t$ and $1+\sum
r_{j}(i_{j}-1)=(q-1)^{2}$.

  * (ii)

If $P\notin\mathcal{X}$ then $\sum r_{j}i_{j}=(q-1)^{2}$.

###### Proof.

Since the $\mathbb{F}_{q}$-lines through $P$ cover the whole plane
$\mathbb{P}^{2}(\mathbb{F}_{q})$ and $\text{N}_{q}(\mathcal{X})=(q-1)^{2}$,
the proof is straightforward. ∎

######  Corollary 3.5.

Let $i$ and $j$ be (not necessarily distinct) non-negative integers. Suppose
that there are different $\mathbb{F}_{q}$-lines $l_{1},l_{2}$ with
$l_{1}\in\mathcal{A}_{i}$ and $l_{2}\in\mathcal{A}_{j}$. If $P=l_{1}\cap
l_{2}\in\mathcal{X}(\mathbb{F}_{q})$, then $i+j\geq q$.

###### Proof.

Suppose that $P$ is of type $i_{1}^{r_{1}}...i_{t}^{r_{t}}$. By Lemma 3.4 item
(i), we have

$\displaystyle(q-1)^{2}$ $\displaystyle=1+\sum r_{j}(i_{j}-1)$
$\displaystyle\leq 1+(i-1)+(j-1)+(q-1)(q-2)$ $\displaystyle=i+j-1+q^{2}-3q+2$
$\displaystyle=i+j-q+(q-1)^{2}.$

Therefore, $i+j\geq q$. ∎

######  Definition 1.

For $i=0,\ldots,q-1$ we define
$\psi_{i}:\mathbb{P}^{2}(\mathbb{F}_{q})\rightarrow\\{0,1,...,q+1\\}$ as

$\psi_{i}(P):=\\#(\check{P}(\mathbb{F}_{q})\cap\mathcal{A}_{i}).$

######  Lemma 3.6.

If $P\in\mathcal{X}(\mathbb{F}_{q})$, then $\psi_{q-1}(P)\geq 3$. In
particular, $a_{q-1}\geq 3(q-1)$. Also, if $\psi_{q-1}(P)=3$, then $P$ is of
type $(q-1)^{3}(q-2)^{q-2}$.

###### Proof.

Let $r_{P}=\psi_{q-1}(P)$. By Lemma 3.4 item (i), if $P$ is of type
$i_{1}^{r_{1}}...i_{t}^{r_{t}}$ then

$\displaystyle(q-1)^{2}$ $\displaystyle=1+\sum r_{j}(i_{j}-1)$
$\displaystyle\leq 1+r_{P}(q-2)+(q+1-r_{P})(q-3)$
$\displaystyle=1+qr_{P}-2r_{P}+q^{2}-3q+q-3-qr_{P}+3r_{P}$
$\displaystyle=r_{P}+(q-1)^{2}-3,$

hence, $r_{P}\geq 3$. We get $3(q-1)^{2}$ lines in $\mathcal{A}_{q-1}$.
However, each line was counted at most $(q-1)$ times. This implies
$a_{q-1}\geq 3(q-1)$. If $\psi_{q-1}(P)=3$, let $s_{P}=\psi_{q-2}(P)$, then

$\displaystyle(q-1)^{2}$ $\displaystyle=1+\sum r_{j}(i_{j}-1)$
$\displaystyle\leq 1+3(q-2)+s_{P}(q-3)+(q-2-s_{P})(q-4)$
$\displaystyle=3q-5+qs_{P}-3s_{P}+q^{2}-4q-2q+8-qs_{P}+4s_{P}$
$\displaystyle=s_{P}+(q-1)^{2}+2-q,$

hence, $s_{P}\geq q-2$. This means that the other lines in
$\check{P}(\mathbb{F}_{q})$ are in $\mathcal{A}_{q-2}$. Therefore, $P$ is of
type $(q-1)^{3}(q-2)^{q-2}$. ∎

Loosely speaking, in order to prove our result, we need that there exists a
point $Q_{0}\in Z(\mathcal{X})$ such that $\psi_{q-1}(Q_{0})$ is _big enough_.
In order to do so, we give the following proposition, which is inspired by
[13, Proposition 3.1 and Proposition 3.2].

######  Proposition 3.7.

Fix a point $Q_{0}\in Z(\mathcal{X})$ and
$l_{\infty}\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})\backslash\check{Q}_{0}(\mathbb{F}_{q})$.
Suppose there are lines $l_{1},...,l_{t}\in\check{Q}_{0}(\mathbb{F}_{q})\
(2\leq t\leq q-1)$ such that $l_{i}(\mathbb{F}_{q})\backslash(\\{Q_{0}\\}\cup
l_{\infty})\subseteq\mathcal{X}(\mathbb{F}_{q}).$ For a line
$l\in\check{Q}_{0}(\mathbb{F}_{q})$ other than these $t$ lines, if
$\\#(l\backslash l_{\infty})\cap\mathcal{X}(\mathbb{F}_{q})\geq q-t$, then
$l(\mathbb{F}_{q})\backslash(\\{Q_{0}\\}\cup
l_{\infty}(\mathbb{F}_{q}))\subseteq\mathcal{X}(\mathbb{F}_{q}).$

###### Proof.

Choose coordinates $X,Y,Z$ of $\mathbb{P}^{2}$ such that
$l_{1}=\textbf{{v}}(X),l_{2}=\textbf{{v}}(Y)$ and
$l_{\infty}=\textbf{{v}}(Z)$, whence $Q_{0}=(0:0:1)$. Let
$G_{0}=Z^{q-1}-X^{q-1}-Y^{q-1},G_{1}=XY\in\mathbb{F}_{q}[X,Y,Z]$. Note that
$\textbf{{v}}(G_{0})$ and $\textbf{{v}}(G_{1})$ are plane curves with no
common components. A direct computation shows that
$\\#(\textbf{{v}}(G_{0})\cap\textbf{{v}}(G_{1}))=2(q-1)$. Also, since
$l_{i}(\mathbb{F}_{q})\backslash(\\{Q_{0}\\}\cup
l_{\infty})\subseteq\mathcal{X}(\mathbb{F}_{q})$ for $i=1,2$ then
$\textbf{{v}}(G_{0})\cap\textbf{{v}}(G_{1})\subseteq\mathcal{X}(\mathbb{F}_{q})$.
Let $F$ be a homogeneous equation for $\mathcal{X}$ over $\mathbb{F}_{q}$, by
Nother’s “AF + BG” Theorem 2.1, we can write

$F(X,Y,Z)=a_{00}(Z^{q-1}-X^{q-1}-Y^{q-1})+XY(g_{q-3}(X,Y)+g_{q-4}(X,Y)Z+\cdots+g_{0}Z^{q-3})$

where $g_{\nu}\in\mathbb{F}_{q}[X,Y,Z]$ is homogeneous of degree $\nu$ and
$a_{00}\in\mathbb{F}_{q}^{*}$. In general, any line
$L\in\check{Q}_{0}(\mathbb{F}_{q})\backslash(l_{1}\cup l_{2})$ is defined by
an equation of the form $Y-\mu X=0$ for some $\mu\in\mathbb{F}_{q}^{*}.$ So

$L(\mathbb{F}_{q})\backslash(Q_{0}\cup
l_{\infty}(\mathbb{F}_{q}))=\\{(1:\mu:\beta)\ |\
\beta\in\mathbb{F}_{q}^{*}\\}.$

Since $a_{00}(\beta^{q-1}-1-\mu^{q-1})=-a_{00}$ when
$\beta,\mu\in\mathbb{F}_{q}^{\times}$, then

(3) $F(1,\mu,\beta)=(\mu(g_{q-3}(1,\mu)-a_{00})+\mu
q_{q-4}(1,\mu)\beta+\cdots+\mu g_{0}\beta^{q-3}.$

In particular, if $l_{2+\mu}=\textbf{{v}}(Y-a_{\mu}X)$ with $\mu=1,...,t-2$,
we must have $a_{\mu}\neq 0$. Let

$B=\left(\begin{array}[]{ccccc}&&\vdots&&\\\
1&\beta&\beta^{2}&\cdots&\beta^{q-3}\\\
&&\vdots&&\end{array}\right)_{\beta\in\mathbb{F}_{q}^{*}\backslash\\{1\\}}.$

Since $l_{2+\mu}(\mathbb{F}_{q})\backslash(\\{Q_{0}\\}\cup
l_{\infty})\subseteq\mathcal{X}(\mathbb{F}_{q})$, by equation (3), then

$B\cdot\left(\begin{array}[]{c}a_{\mu}g_{q-3}(1,a_{\mu})-a_{00}\\\
a_{\mu}g_{q-4}(1,a_{\mu})\\\ \vdots\\\
a_{\mu}g_{0}\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ \vdots\\\
0\end{array}\right).$

Since $B$ is a Vandermonde matrix, we have $\text{det}(B)\neq 0$. This implies
that

$g_{q-4}(1,a_{\mu})=\cdots=g_{0}(1,a_{\mu})=0.$

If $\nu<t-2$, since $g_{\nu}(1,y)$ has $t-2$ roots $\\{a_{1},...,a_{t-2}\\}$
but its degree is less than $t-2$, then $g_{\nu}(1,y)\equiv 0$ as a polynomial
in $y$. So

$F(1,y,z)=a_{00}(z^{q-1}-y^{q-1})+(yg_{q-3}(1,y)-a_{00})+yq_{q-4}(1,y)z+\cdots+yg_{t-2}(1,y)z^{q-t-1}.$

Let $l=\textbf{{v}}(Y-\mu X)$, where $\mu\in\mathbb{F}_{q}^{*}$, and
$\\{(1,\mu,\beta_{i})\ |\ 1\leq i\leq q-t\\}$ a set of chosen points of
$(l\backslash l_{\infty})(\mathbb{F}_{q})\cap\mathcal{X}.$ So $\mu\neq 0$ and
$\beta_{i}\neq 0$ for $i=1,...,q-t$. Hence,

$\left(\begin{array}[]{ccccc}&&\vdots&&\\\
1&\beta_{i}&\beta_{i}^{2}&\cdots&\beta_{i}^{q-t-1}\\\
&&\vdots&&\end{array}\right)_{i=1,...,q-t}\left(\begin{array}[]{c}\mu
g_{q-3}(1,v)-a_{00}\\\ \mu g_{q-4}(1,v)\\\ \vdots\\\ \mu
g_{t-2}(1,v)\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ \vdots\\\
0\end{array}\right).$

This implies that $\mu g_{q-3}(1,\mu)-a_{00}=\mu g_{q-4}(1,\mu)=\mu
g_{t-2}(1,\mu)=0$. So $F(1,\mu,\beta)=0$ for any $\beta\in\mathbb{F}_{q}^{*}$.
Therefore, $l(\mathbb{F}_{q})\backslash(\\{Q_{0}\\}\cup
l_{\infty}(\mathbb{F}_{q}))\subseteq\mathcal{X}(\mathbb{F}_{q})$.

∎

######  Corollary 3.8.

Suppose there is a point $Q\in Z(\mathcal{X})$ such that $r=\psi_{q-1}(Q)\geq
2$. If these $r$ lines are $l_{1},...,l_{r}$ then
$\\{(l_{i}(\mathbb{F}_{q})\backslash\\{Q\\})\cap Z(\mathcal{X})\ |\
i=1,...,r\\}$ is contained in a line.

###### Proof.

Let $Q_{i}=(l_{i}(\mathbb{F}_{q})\backslash\\{Q\\})\cap Z(\mathcal{X})$ with
$i=1,...,r$. Since $r\geq 2$, we can consider the line
$l_{\infty}:=\overline{Q_{1}Q_{2}}$. Since $l_{i}\in\mathcal{A}_{q-1}$, then
$\\#(l_{i}\backslash
l_{\infty})\cap\mathcal{X}(\mathbb{F}_{q})\geq(q+1)-3=q-2$ for $i=3,...,r$. By
Proposition 3.7, we have
$l_{i}\backslash(l_{\infty}\cup\\{Q\\})\subseteq\mathcal{X}(\mathbb{F}_{q})$.
Therefore,

$\\{(l_{i}(\mathbb{F}_{q})\backslash\\{Q\\})\cap Z(\mathcal{X})\ |\
i=1,...,r\\}\subseteq l_{\infty}.$

∎

We now prove some interesting Corollaries to Proposition 3.7. They can be
thought of as partial negative answers to Question 1 when assuming stronger
conditions on the structure of $Z(\mathcal{X})$.

######  Corollary 3.9.

If there are two lines
$l_{\infty}^{1},l_{\infty}^{2}\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$ such
that $(l_{\infty}^{1}\cup l_{\infty}^{2})(\mathbb{F}_{q})\subseteq
Z(\mathcal{X})$, then $a_{0}=3$.

###### Proof.

Let $Z^{*}(\mathcal{X}):=Z(\mathcal{X})\backslash(l_{\infty}^{1}\cup
l_{\infty}^{2})$ and $Q\in l_{\infty}^{1}\cap l_{\infty}^{2}$. Since
$\\#Z(\mathcal{X})=3q$, we have $\\#Z^{*}(\mathcal{X})=q-1$. Since
$\\#\check{Q}(\mathbb{F}_{q})\backslash\\{l_{\infty}^{1},l_{\infty}^{2}\\}=q-1$
and $\text{deg}(\mathcal{X})=q-1$, then each line
$l\in\check{Q}(\mathbb{F}_{q})\backslash\\{l_{\infty}^{1},l_{\infty}^{2}\\}$
contains exactly one point of $Z^{*}(\mathcal{X})$. By Corollary 3.8,
$Z^{*}(\mathcal{X})$ is contained in a line. Therefore, $a_{0}=3$. ∎

######  Corollary 3.10.

Let $q\geq 8$. If $Q\in Z(\mathcal{X})$ is a point such that
$r=\psi_{q-1}(Q)\geq 4$, then $r=q-1$. In particular, $a_{0}=3$.

###### Proof.

Since $r\geq 4$, by Lemma 3.7 and Corollary 3.8, the lines in
$\check{Q}(\mathbb{F}_{q})\backslash\mathcal{A}_{q-1}$ have at most $q-r$
points of $\mathcal{X}(\mathbb{F}_{q})$. Then

$\displaystyle q^{2}-2q+1=(q-1)^{2}$ $\displaystyle\leq r(q-1)+(q+1-r)(q-r)$
$\displaystyle=qr-r+q^{2}+q-qr-qr-r+r^{2}$ $\displaystyle=q^{2}+q-qr+r(r-2).$

Hence $q(r-3)\leq r(r-2)-1$. Since $q\geq 8$, we must have $r\geq 7$. So

$q\leq\dfrac{r(r-2)-1}{r-3}=\dfrac{(r+1)(r-3)+2}{r-3}=r+1+\dfrac{2}{r-3}.$

Since $r\leq(q-1)$, this implies that $r=q-1$. Therefore, $Q$ is of type
$(q-1)^{q-1}0^{2}$ and, by Corollary 3.9, $a_{0}=3$. ∎

######  Corollary 3.11.

Let $q\geq 7$. If there is a line
$l_{1}^{\infty}\in\check{\mathbb{P}}^{2}(\mathbb{F}_{q})$ such that
$l_{1}^{\infty}(\mathbb{F}_{q})\subseteq Z(\mathcal{X})$, then $a_{0}=3$.

###### Proof.

By Lemma 3.6, $a_{q-1}\geq 3(q-1)$. Since $q\geq 7$, if $\psi_{q-1}(Q)\leq 2$
for every point $Q\in l_{1}^{\infty}$ then $a_{q-1}\leq 2(q+1)<3(q-1)$, a
contradiction. Hence, there is a point $Q\in l_{1}^{\infty}$ such that
$r=\psi_{q-1}(Q)\geq 3$. Also, by Lemma 3.7 and Corollary 3.8, the other lines
have at most $q-r$ points of $\mathcal{X}(\mathbb{F}_{q})$. This implies that

$\displaystyle q^{2}-2q+1=(q-1)^{2}$ $\displaystyle\leq 0+r(q-1)+(q-r)(q-r)$
$\displaystyle=qr-r+q^{2}-2qr+r^{2}$ $\displaystyle=q^{2}-qr+r(r-1).$

Hence $q(r-2)\leq r(r-1)-1$. Since $q\geq 7$, we must have $r\geq 6$. So

$q\leq\dfrac{r(r-1)-1}{r-2}=\dfrac{(r+1)(r-2)+1}{r-2}=r+1+\dfrac{1}{r-2}.$

Since $r\leq q-1$, this implies that $r=q-1$. Therefore, $Q$ is of type
$(q-1)^{q-1}0^{2}$. Then the result follows from Corollary 3.9. ∎

Let $k_{0}={\rm min}\\{i\ |\ a_{i}\neq 0\\}$. By the previous Corollary, in
order to prove our main result for $q\geq 7$, it is enough to show that
$k_{0}=0$. We start by giving an upper bound for $k_{0}$.

######  Lemma 3.12.

We have that $k_{0}\leq q-4$.

###### Proof.

Suppose that $k_{0}\geq q-3$. Hence, by Lemma 3.3, we have

  * (1)

$a_{q-3}+a_{q-2}+a_{q-1}=q^{2}+q+1$;

  * (2)

$(q-3)a_{q-3}+(q-2)a_{q-2}+(q-1)a_{q-1}=(q+1)(q-1)^{2}$;

  * (3)

$(q-3)(q-4)a_{q-3}+(q-2)(q-3)a_{q-2}+(q-1)(q-2)a_{q-1}=q(q-2)(q-1)^{2}$.

A direct computation shows that the system above implies that
$2a_{q-3}=3(q^{2}-3q+2)$, $a_{q-2}=-2(q^{2}-5q+4)$ and
$2a_{q-1}=3(q^{2}-3q+4)$. Since $q\geq 5$, then

$a_{q-2}=-2(q^{2}-5q+4)<0,$

a contradiction. Therefore, $k_{0}\leq q-4$. ∎

We now prove a lower bound for $k_{0}$ whenever $k_{0}\neq 0$. We start with
the following lemma:

######  Lemma 3.13.

$\sum_{i=k_{0}}^{q-1}(i-k_{0})(i-q+2)a_{i}=3(q-1)^{2}-3k_{0}$. In particular,

$(q-k_{0}-1)a_{q-1}\geq 3(q-1)^{2}-3k_{0}.$

###### Proof.

First, note that

$\displaystyle(i-k_{0})(i-q+2)$ $\displaystyle=i(i-1)+i(3-q)+k_{0}(q-2-i)$
$\displaystyle=i(i-1)+i(3-q)-ik_{0}+k_{0}(q-2)$
$\displaystyle=i(i-1)+i(3-q-k_{0})+k_{0}(q-2).$

By Lemma 3.3, we have

$\displaystyle\displaystyle\sum_{i=k_{0}}^{q-1}(i-k_{0})(i-q+2)a_{i}$
$\displaystyle=\displaystyle\sum_{i=k_{0}}^{q-1}i(i-1)a_{i}+(3-q-k_{0})\displaystyle\sum_{i=k_{0}}^{q-1}ia_{i}+k_{0}(q-2)\displaystyle\sum_{i=k_{0}}^{q-1}a_{i}$
$\displaystyle=q(q-2)(q-1)^{2}+(3-q-k_{0})(q+1)(q-1)^{2}+$ $\displaystyle\ \ \
\ \ +k_{0}(q-2)(q^{2}+q+1)$ $\displaystyle=3(q-1)^{2}-3k_{0}.$

Moreover, if $i=q-1$ then $(i-k_{0})(i-q+2)=(q-k_{0}-1)$ and if $k_{0}\leq
i\leq q-2$ then $(i-k_{0})(i-q+2)\leq 0$. Also, by Lemma 3.12, $k_{0}\leq
q-4$. Hence, $3(q-1)^{3}-3k_{0}\geq 0$. Therefore, $(q-k_{0}-1)a_{q-1}\geq
3(q-1)^{2}-3k_{0}$. ∎

######  Proposition 3.14.

Let $q\geq 7$. If $k_{0}\neq 0$, then $k_{0}\geq 2$.

###### Proof.

Suppose that $k_{0}=1$. Let $l_{1}\in\mathcal{A}_{1}$. Also, consider
$P_{0}=l_{1}\cap\mathcal{X}(\mathbb{F}_{q})$ and

$l_{1}\cap Z(\mathcal{X})=\\{Q_{1},...,Q_{q}\\}.$

By Corollary 3.5, $P_{0}$ is of type $(q-1)^{q}1^{1}$. If
$r_{i}:=\psi_{q-1}(Q_{i})$ for $i=1,...,q$ then

$r_{i}(q-1)+1\leq\text{N}_{q}(\mathcal{X})=(q-1)^{2}$

Hence, $r_{i}\leq q-2$. If $r_{i}\geq 3$, by Lemma 3.7 and Corollary 3.8, the
lines in $\check{Q}_{i}(\mathbb{F}_{q})\backslash\mathcal{A}_{q-1}$ have at
most $q-r_{i}$ points in $\mathcal{X}(\mathbb{F}_{q})$. On the other hand, we
have

$\displaystyle q^{2}-2q+1=(q-1)^{2}$ $\displaystyle\leq
1+r_{i}(q-1)+(q-r_{i})(q-r_{i})$
$\displaystyle=1+r_{i}q-r_{i}+q^{2}-2qr_{i}+r_{i}^{2}$
$\displaystyle=q^{2}+1-qr_{i}+r_{i}(r_{i}-1),$

hence, $q(r_{i}-2)\leq(r_{i}-1)r_{i}.$ Since $q\geq 7$, we must have
$r_{i}\geq 6$. So

$q\leq\dfrac{(r_{i}-1)r_{i}}{(r_{i}-2)}=\dfrac{(r_{i}+1)(r_{i}-2)+2}{r_{i}-2}=r_{i}+1+\dfrac{2}{r_{i}-2}.$

This implies that $r_{i}\geq q-1$, a contradiction. Therefore $r_{i}\leq 2$
for $i=1,...,q$, hence, $a_{q-1}\leq q+2q=3q$. By Lemma 3.13,
$(q-2)a_{q-1}\geq 3(q-1)^{2}-3=3q(q-2).$ So $a_{q-1}=3q$. By Lemma 3.13, we
have

$\sum_{i=1}^{q-1}(i-1)(i-q+2)a_{i}=3q(q-2).$

Since $(q-2)a_{q-1}=3q(q-2)$, this implies that $a_{2}=\cdots=a_{q-3}=0$. By
Lemma 3.3, we have

  * (1)

$a_{1}+a_{q-2}+3q=q^{2}+q+1$,

  * (2)

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

A direct computation shows that the system above implies that
$(q-3)a_{1}=3(q-1)$ and $(q-3)a_{q-2}=q(q^{2}-5q+4)$. Note that

$a_{1}=\dfrac{3(q-1)}{q-3}=3+\dfrac{6}{q-3}$

Since $q\geq 7$ then $(q-3)\nmid 6$, but $a_{1}$ is a integer, a
contradiction. Therefore, $k_{0}\geq 2$. ∎

##  4\. Characterization of optimal Sziklai curves of degree $q-1$

In this section, we provide the characterization of optimal Sziklai curves of
degree $q-1$. First, we deal with the cases $q=5,7$, as they need some _ad
hoc_ techniques.

######  Proposition 4.1.

If $\mathcal{X}\in C_{4}(\mathbb{F}_{5})$ and $\text{N}_{5}(\mathcal{X})=16$,
then there exist $\alpha,\gamma,\beta\in\mathbb{F}_{5}^{*}$ with
$\alpha+\beta+\gamma=0$ such that $\mathcal{X}\simeq_{{\rm
proj}}\mathcal{C}_{(\alpha,\beta,\gamma)}$ over $\mathbb{F}_{5}$.

###### Proof.

By Lemma 3.12, we have $k_{0}\leq 1$. Suppose that $k_{0}=1$. Let
$l_{1}\in\mathcal{A}_{1}$ with
$l_{1}\cap\mathcal{X}(\mathbb{F}_{q})=\\{P_{0}\\}$. By Corollary 3.5, $P_{0}$
is of type $4^{5}1^{1}$. In this case, $l_{1}$ is the tangent line to
$\mathcal{X}$ at $P_{0}$. Since $\text{deg}(\mathcal{X})=4$, then the divisor

$l_{1}\cdot\mathcal{X}:=\sum\text{I}(P,l_{1}\cap\mathcal{X})P=2P_{0}+P_{1}+P_{2}$

for some points $P_{1},P_{2}\in l_{1}\cap\mathcal{X}$, where
$\text{I}(P,l_{1}\cap\mathcal{X})$ is the intersection multiplicity of $l_{1}$
and $\mathcal{X}$ at $P$. Since the divisor $l_{1}\cdot\mathcal{X}$ is defined
over $\mathbb{F}_{q}$, applying the $5$-Frobenius map $\Psi_{5}$, we have
$P_{1}+P_{2}+2P_{0}=\Psi_{5}(P_{1})+\Psi_{5}(P_{2})+2P_{0}$, which implies
that $P_{1},P_{2}\in\mathbb{P}^{2}(\mathbb{F}_{25})$. If $P_{1}=P_{2}$ then
$P_{1}=P_{2}=P_{0}$ and $P_{0}$ is a inflexion point. On the other hand, by
Stohr-Voloch’s Theorem 2.3, we have

$32=2\text{N}_{q}(\mathcal{X})\leq 32-\sum_{P\in\mathcal{X}}A(P).$

This implies that $0=A(P_{0})=j_{2}(P_{0})-2$, hence,
$\text{I}(P_{0},l_{1}\cap\mathcal{X})=j_{2}(P_{0})=2,$ a contradiction. So
$P_{1}\neq P_{2}$ and
$P_{1},P_{2}\in\mathbb{P}^{2}(\mathbb{F}_{25})\backslash\mathbb{P}^{2}(\mathbb{F}_{5})$.
By Lemma 3.3, we have

  * (i)

$a_{1}+a_{2}+a_{3}+a_{4}=31;$

  * (ii)

$a_{1}+2a_{2}+3a_{3}+4a_{4}=96;$

  * (iii)

$2a_{2}+6a_{3}+12a_{4}=240.$

A direct computation shows that the system above implies that
$a_{1}=21-a_{4}$, $a_{2}=3(a_{4}-15)$ and $a_{3}=55-3a_{4}$. Since
$a_{1},a_{2},a_{3}\geq 0$, then $15\leq a_{4}\leq 18$. Hence $a_{1}\geq 3$.
Follows from the above that $a_{1}\geq 3$ implies that
$\text{N}_{25}(\mathcal{X})\geq 16+3*2=22$. On the other hand, let
$L(t)=\prod_{i=1}^{6}(1-\omega_{i}t)$ the $L$-polynomial of $\mathcal{X}$ into
linear factors in some finite extension of $\mathbb{Q}$. By [6, Theorem 9.10],
we have

$\text{N}_{q^{n}}(\mathcal{X})=q^{n}+1-\sum_{i=1}^{2g}\omega_{i}^{n}.$

Since $16=\text{N}_{5}(\mathcal{X})=5+1+3\cdot 4-2=5+3\cdot 4-1$. As mentioned
in [17, fact 3.3], the curve $\mathcal{X}$ has zeta function of type
$[4,4,2]$, this means that

$\omega_{1}+\overline{\omega_{1}}=-4,\
\omega_{2}+\overline{\omega_{2}}=-4\text{ and
}\omega_{3}+\overline{\omega_{3}}=-2.$

Hence,
$\sum_{i=1}^{6}\omega_{i}^{2}=36-2(|\omega_{1}|^{2}+|\omega_{2}|^{2}+|\omega_{3}|^{2}).$
By [6, Theorem 9.16], $|\omega_{i}|^{2}=5$. This implies that

$22\leq\text{N}_{25}(\mathcal{X})=26-\sum_{i=1}^{6}\omega_{i}^{2}=26-(36-30)=20,$

a contradiction. Then $k_{0}=0$.

Suppose that $a_{1}\neq 0$. Let $l_{0}\in\mathcal{A}_{0}$ and
$l_{1}\in\mathcal{A}_{1}$. By Corollary 3.5, we have

$Q=l_{0}\cap l_{1}\in Z(\mathcal{X}).$

If $r_{Q}=\psi_{4}(Q)$, by Lemma 3.4, we have $16\leq 0+1+4r+3(4-r)=13+r,$
hence, $r\geq 3$. This means that $Q$ is of type $4^{3}3^{1}1^{1}0^{1}$. On
the other hand, since $\psi_{4}(Q)\geq 3$, by Proposition 3.7, we must have
$\psi_{3}(Q)=0$, a contradiction. So $a_{1}=0$. By Lemma 3.3, we have

  * (1)

$a_{0}+a_{2}+a_{3}+a_{4}=31;$

  * (2)

$2a_{2}+3a_{3}+4a_{4}=96;$

  * (3)

$2a_{2}+6a_{3}+12a_{4}=240.$

A direct computation shows that the system above implies that
$3a_{0}=21-a_{4}$, $a_{2}=2(a_{4}-12)$ and $3a_{3}=144-8a_{4}$. If $a_{0}=1$,
then $a_{2}=12$ and $a_{3}=0$. Let $l_{2}\in\mathcal{A}_{2}$ and $P_{0}\in
l_{2}\cap\mathcal{X}(\mathbb{F}_{5})$. If $r=\psi_{4}(P_{0})$, by Lemma 3.4,
we have $16\leq 2+3r+2(5-r)=12+r.$ Since $2+3r\leq 16$, this implies that
$r=4$. Hence, $\psi_{2}(P_{0})=1$, $\psi_{3}(P_{0})=1$ and
$\psi_{4}(P_{0})=4$, a contradiction. Therefore, $a_{0}\geq 2$ and the result
follows Corollary 3.9. ∎

######  Proposition 4.2.

If $\mathcal{X}\in C_{6}(\mathbb{F}_{7})$ and $\text{N}_{7}(\mathcal{X})=36$,
then there exist $\alpha,\gamma,\beta\in\mathbb{F}_{7}^{*}$ with
$\alpha+\beta+\gamma=0$ such that $\mathcal{X}\simeq_{{\rm
proj}}\mathcal{C}_{(\alpha,\beta,\gamma)}$ over $\mathbb{F}_{7}$.

###### Proof.

First, recall the definition of $k_{0}={\rm min}\\{i\ |\ a_{i}\neq 0\\}$.
Also, by Proposition 3.14 and Corollary 3.11, it is enough to prove that
$k_{0}\leq 1$.

By Lemma 3.2, we have that
$\mathcal{X}(\mathbb{F}_{7})\subseteq\mathbb{P}^{2}(\mathbb{F}_{7})$ is a
$(36,6)$-arc. Up to projective equivalence, there are exactly 194
$(36,6)$-arcs in $\mathbb{P}^{2}(\mathbb{F}_{7})$, see [3, Remark 1]. The full
list can be found online at http://mars39.lomo.jp/opu/36˙3˙30.txt, where the
points of such arcs are arranged as a generator matrix for a
$[36,3,30]_{7}$-code together with the weight enumerator. Recall that, by the
relation between projective $[n,3,d]_{q}$-codes and $(n,n-d)$-arcs in
$\mathbb{P}^{2}(\mathbb{F}_{q})$, we have

$6\cdot(a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})=(c_{36},c_{35},c_{34},c_{33},c_{32},c_{31},c_{30}).$

By Lemma 3.12, $k_{0}\leq 3$. Hence, from our initial observation, it is
enough to prove that the condition $k_{0}\in\\{2,3\\}$ leads to a
contradiction, equivalently, the exactly six $(36,6)$-arcs of the
$[36,3,30]_{7}$-codes such that the weight of a codeword is at most $34$ is
not projectively equivalent to $\mathcal{X}(\mathbb{F}_{7})$.

Suppose that $k_{0}=3$. Since there is only one $[36,3,30]_{7}$-code such that
the maximal weight of a codeword is exactly 33, then there is only one
$(36,6)$-arc with $k_{0}=3$, up to projective equivalence. So we can choose
coordinates $X,Y,Z$ of $\mathbb{P}^{2}$ such that

$\displaystyle\mathcal{X}(\mathbb{F}_{7})=\\{$
$\displaystyle(1:1:3),(1:1:5),(1:2:3),(1:2:5),(1:2:6),(1:3:3),$
$\displaystyle(1:3:4),(1:3:5),(1:3:6),(1:3:0),(1:4:2),(1:4:4),$
$\displaystyle(1:4:5),(1:4:6),(1:4:0),(1:5:2),(1:5:3),(1:5:4),$
$\displaystyle(1:5:6),(1:5:0),(1:6:1),(1:6:3),(1:6:4),(1:6:5),$
$\displaystyle(1:6:0),(1:0:1),(1:0:2),(1:0:4),(1:0:6),(1:0:0),$
$\displaystyle(0:1:3),(0:1:4),(0:1:5),(0:1:6),(0:1:0),(0:0:1)\\}.$

Let

$G=YZ(Z-3Y)(Z-4Y)(Z-5Y)(Z-6Y)\in\mathbb{F}_{7}[X,Y,Z]$
$H=X(X+Y-Z)(2X+Y-Z)(X+2Y-2Z)\in\mathbb{F}_{7}[X,Y,Z]$

A direct computation shows that $\\#(\textbf{{v}}(G)\cap\textbf{{v}}(H))=24$
and $\textbf{{v}}(G)\cap\textbf{{v}}(H)\subseteq\mathcal{X}(\mathbb{F}_{7})$.
Let $F\in\mathbb{F}_{7}[X,Y,Z]$ be a homogeneous equation for $\mathcal{X}$
over $\mathbb{F}_{7}$, by Noether’s “AF + BG” Theorem 2.1, we have

$F=G+(\alpha_{1}X^{2}+\alpha_{2}Y^{2}+\alpha_{3}Z^{2}+\alpha_{4}XY+\alpha_{5}XZ+\alpha_{6}YZ)\cdot
H$

for some $\alpha_{1},...,\alpha_{6}\in\mathbb{F}_{7}$. Since
$0=F(1,0,0)=2\alpha_{1}$,

$\displaystyle 0=F(1,5,4)$
$\displaystyle=G(1,5,4)+(5^{2}\alpha_{2}+4^{2}\alpha_{3}+5\alpha_{4}+4\alpha_{5}4+20\alpha_{6})H(1,5,4)$
$\displaystyle=4(4\alpha_{2}+2\alpha_{3}+5\alpha_{4}+4\alpha_{5}+6\alpha_{6}),$
$\displaystyle 0=F(1,4,0)$
$\displaystyle=G(1,4,0)+(4^{2}\alpha_{2}+4\alpha_{4})H(1,4,0)$
$\displaystyle=4(2\alpha_{2}+4\alpha_{4}),$ $\displaystyle 0=F(1,0,6)$
$\displaystyle=G(1,0,6)+(6^{2}\alpha_{3}+6\alpha_{5})H(1,0,6)$
$\displaystyle=4(\alpha_{3}+6\alpha_{5}),$ $\displaystyle 0=F(1,2,5)$
$\displaystyle=G(1,2,5)+(2^{2}\alpha_{2}+5^{2}\alpha_{3}+2\alpha_{4}+5\alpha_{5}+10\alpha_{6})H(1,2,5)$
$\displaystyle=4(4\alpha_{2}+4\alpha_{3}+2\alpha_{4}+5\alpha_{5}+3\alpha_{6}),$
$\displaystyle 0=F(1,6,4)$
$\displaystyle=G(1,6,4)+(6^{2}\alpha_{2}+4^{2}\alpha_{3}+6\alpha_{4}+4\alpha_{5}+24\alpha_{6}YZ)H(1,6,4)$
$\displaystyle=4(\alpha_{2}+2\alpha_{3}+6\alpha_{4}+4\alpha_{5}+3\alpha_{6}YZ),$

It is easy to check that this implies that $\alpha_{1}=\cdots=\alpha_{6}=0$.
Therefore, $F=G$ and $\mathcal{X}$ has $\mathbb{F}_{7}$-linear components, a
contradiction.

Suppose that $k_{0}=2$. Since there are five $[36,3,30]_{7}$-codes such that
the maximal weight of a codeword is exactly 34, then there are five
$(36,6)$-arc with $k_{0}=2$, up to projective equivalence. For instance, we
can choose coordinates $X,Y,Z$ of $\mathbb{P}^{2}$ such that

$\displaystyle\mathcal{X}(\mathbb{F}_{7})=\\{$
$\displaystyle(1:1:4),(1:1:5),(1:1:6),(1:1:0),(1:2:2),(1:2:4),$
$\displaystyle(1:2:6),(1:2:0),(1:3:2),(1:3:3),(1:3:4),(1:3:5),$
$\displaystyle(1:4:2),(1:4:3),(1:4:5),(1:4:6),(1:5:3),(1:5:5),$
$\displaystyle(1:5:6),(1:5:0),(1:6:1),(1:6:2),(1:6:4),(1:6:6),$
$\displaystyle(1:6:0),(1:0:2),(1:0:3),(1:0:4),(1:0:5),(1:0:0),$
$\displaystyle(0:1:3),(0:1:4),(0:1:5),(0:1:6),(0:1:0),(0:0:1)\\}.$

Let $Q_{0}=(1:5:2)\in Z(\mathcal{X})$. A direct computation shows that

$\check{Q_{0}}(\mathbb{F}_{7})\cap\mathcal{A}_{5}=\\{x+3y+6z,x+4y\\}\text{ and
}\check{Q_{0}}(\mathbb{F}_{7})\cap\mathcal{A}_{6}=\\{y+z,4x+3y+z,x+3z\\}.$

Since $\psi_{6}(Q_{0})=3$, by Corollary 3.8 and Proposition 3.7, we must have
$\psi_{5}(Q_{0})=0$, a contradiction.

The other four $(36,6)$-arc with $k_{0}=2$ can be dealt with by using a
similar argument. In each of these cases, we have a contradiction. Therefore,
none of these $(36,6)$-arcs can be equal to $\mathcal{X}(\mathbb{F}_{7})$, and
our assertion follows. ∎

######  Remark 4.3.

As a byproduct of Proposition 4.2, we have that, of the 194 nonequivalent
$(36,6)$-arcs in $\mathbb{P}^{2}(\mathbb{F}_{7})$, only one is obtained as the
set of rational points of an irreducible plane curve of degree $6$.

We are now in a position to prove our main result.

######  Theorem 4.4.

Let $\mathcal{X}\in C_{q-1}(\mathbb{F}_{q})$. If
$\text{N}_{q}(\mathcal{X})=(q-1)^{2}$ and $q\geq 5$, then there exist
$\alpha,\gamma,\beta\in\mathbb{F}_{q}^{*}$ with $\alpha+\beta+\gamma=0$ such
that $\mathcal{X}\simeq_{{\rm proj}}\mathcal{C}_{(\alpha,\beta,\gamma)}$ over
$\mathbb{F}_{q}$.

###### Proof.

Let $k_{0}={\rm min}\\{i\ |\ a_{i}\neq 0\\}.$ If $q\geq 8$, then by
Proposition 3.14 and Corollary 3.10, it is enough to prove that either
$k_{0}\leq 1$ or there is a point $Q\in Z(\mathcal{X})$ such that
$\psi_{q-1}(Q)\geq 4$.

By way of contradiction, assume that $k_{0}\geq 2$ and
$r_{Q}:=\\#\check{Q}(\mathbb{F}_{q})\cap\mathcal{A}_{q-1}\leq 3$ for every
point $Q\in Z(\mathcal{X})$. Since $\\#Z(\mathcal{X})=3q$, the latter
hypothesis implies that $2a_{q-1}\leq 9q$.

Now, assume that $q\geq 11$. If for every point
$P\in\mathcal{X}(\mathbb{F}_{q})$ we have $\psi_{q-1}(P)\geq 5$, then
$a_{q-1}\geq 5(q-1)>9q/2$, a contradiction. Let
$P_{0}\in\mathcal{X}(\mathbb{F}_{q})$ such that $r_{0}=\psi_{q-1}(P)\leq 4$.
By Lemma 3.6, we have that $r_{0}\in\\{3,4\\}$. We distinguish two cases,
namely $r_{0}=3$ or $r_{0}=4$.

If $r_{0}=3$, then by Lemma 3.6, $P_{0}$ is of type $(q-1)^{3}(q-2)^{q-2}$.
Let $l\in\check{P}(\mathbb{F}_{q})\cap\mathcal{A}_{q-2}$ and $Q\in l\cap
Z(\mathcal{X})$. If $r_{Q}:=\psi_{q-1}(Q)=3$, by Lemma 3.7 and Corollary 3.8,
then $l\in\mathcal{A}_{q-1}$, a contradiction. So there are at least $3(q-2)$
points in $Z(\mathcal{X})$ such that $r_{Q}\leq 2$. Therefore, $2a_{q-1}\leq
2(3(q-2))+3*6$. So $a_{q-1}\leq 3(q-2)+9=3(q+1)$. By Lemma 3.12, $k_{0}\leq
q-4$ and, by Lemma 3.13, we have

$\dfrac{3(q-1)^{2}-3k_{0}}{q-k_{0}-1}\leq a_{q-1}\leq 3(q+1).$

This implies that $3q(k_{0}-2)+6=3(q-1)^{2}-3k_{0}-(q-k_{0}-1)(3q+3)\leq 0$.
Therefore, $k_{0}\leq 2-2/q<2,$ a contradiction.

Now, suppose that $r_{0}=4$. Let $s=\psi_{q-2}(P)$. We have

$\displaystyle(q-1)^{2}$ $\displaystyle\leq 1+4(q-2)+s(q-3)+(q-3-s)(q-4)$
$\displaystyle=4q-7+sq-3s+q^{2}-3q-sq-4q+12+4s$
$\displaystyle=(q-1)^{2}+4+s-q.$

This implies that $s\geq q-4$, hence $P$ is of type
$(q-1)^{4}(q-2)^{q-4}(q-5)^{1}.$ As in the previous case, there are at least
$3(q-4)$ points in $Z(\mathcal{X})$ such that the image by $\psi_{q-1}$ is
less than or equal to $2$. Therefore, $2a_{q-1}\leq 2(3(q-4))+3\cdot 12$,
hence, $a_{q-1}\leq 3(q-4)+18=3(q+2)$. Since $q\geq 11$, there is point
$P\in\mathcal{X}(\mathbb{F}_{q})$ such that $\psi_{q-1}(P)=3$, otherwise,
$a_{q-1}\geq 4(q-1)>3(q+2).$ Again, this implies that $k_{0}<2$, a
contradiction.

We are then left with the cases $q=5,7,8,9$. As the cases $q=5,7$ have already
been dealt with in Propositions 4.1 and 4.2 respectively, we only need to
consider the cases $q=8,9$. We proceed with a careful case-by-case analysis.

For $q=9$, by Lemma 3.13, we have

(4) $\displaystyle\dfrac{3(64-k_{0})}{8-k_{0}}\leq a_{8}\leq 40.$

By Lemma 3.12, we have $k_{0}\leq 5$, and then
$37k_{0}-128=3(64-k_{0})-40(8-k_{0})\leq 0$. This in turn implies
$k_{0}\in\\{2,3\\}$.

Now, we prove that the condition $k_{0}=2$ leads to a contradiction for $q=9$.
If $k_{0}=2$, by the inequality (4), $a_{8}\geq 31$. Let
$l_{2}\in\mathcal{A}_{2}$ with
$l_{2}\cap\mathcal{X}(\mathbb{F}_{9})=\\{P_{0},P_{1}\\}$ and $l_{2}\cap
Z(\mathcal{X})=\\{P_{2},...,P_{9}\\}.$ Let $r_{j}:=\psi_{8}(P_{j})$ for
$j=0,1,2,...,9$. By Lemma 3.4, if $j\in\\{0,1\\}$, then $64\leq
2+7r_{j}+6(9-r_{j})=56+r_{j}.$ Since $2+7r_{j}\leq 64$, this implies that
$r_{j}=8$ for $j=0,1$. If $r_{j}=3$ for some $j\in\\{2,...,9\\}$, follows from
Lemma 3.4 and Proposition 3.7 that the other lines in
$\check{Q}_{j}(\mathbb{F}_{q})$ contain at most $6$ points of
$\mathcal{X}(\mathbb{F}_{9})$, hence, $64\leq 2+3\cdot 8+6\cdot 6=62,$ a
contradiction. Then $r_{j}\leq 2$ when $j=2,...,9$. This implies that
$a_{8}\leq 2\cdot 8+8\cdot 2=32$. If $r_{P}=\psi_{8}(P)\geq 4$ for every point
$P\in\mathcal{X}(\mathbb{F}_{q})\backslash\\{P_{1},P_{2}\\}$, then

$a_{8}\geq\dfrac{8\cdot 2+4((q-1)^{2}-2)}{q-1}=33,$

a contradiction. Then there is a point $P\in\mathcal{X}(\mathbb{F}_{q})$ such
that $r_{P}=3$. Again, this implies that $a_{8}\leq 3(q+1)=30$, a
contradiction.

We now prove that the condition $k_{0}=3$ leads to a contradiction for $q=9$.
If $k_{0}=3$, by inequality (4), $a_{8}\geq 37$. Let $l_{3}\in\mathcal{A}_{3}$
with $l_{3}\cap\mathcal{X}(\mathbb{F}_{9})=\\{P_{0},P_{1},P_{2}\\}$ and
$l_{3}\cap Z(\mathcal{X})=\\{P_{3},...,P_{9}\\}.$ Let $r_{j}:=\psi_{8}(P_{j})$
where $i=0,1,2,...,9$. By Lemma 3.4 if $j\in\\{0,1,2\\}$, then $64\leq
3+7r_{j}+6(9-r_{j})=57+r_{j}.$ Since $3+7r_{j}\leq 64$, this implies that
$r_{j}\in\\{7,8\\}$ for $j=0,1,2$. If $r_{j}=3$ for some $j\in\\{3,...,9\\}$,
it follows from Lemma 3.4 and Proposition 3.7 that the other lines in
$\check{Q}_{j}(\mathbb{F}_{q})$ contain at most $6$ points of
$\mathcal{X}(\mathbb{F}_{9})$, hence, $64\leq 3+3\cdot 8+6\cdot 6=63,$ a
contradiction. Then $r_{j}\leq 2$ when $j=3,...,9$. This implies that
$a_{8}\leq 3\cdot 8+7\cdot 2=38$. If $r_{P}=\psi_{8}(P)\geq 5$ for every point
$P\in\mathcal{X}(\mathbb{F}_{q})$, then $a_{8}\geq 5(q-1)=40,$ a
contradiction. There is then a point $P\in\mathcal{X}(\mathbb{F}_{q})$ such
that $\psi_{8}(P)\leq 4$. Again, this implies that $a_{8}\leq 3(q+2)=33$, a
contradiction.

For $q=8$, by Lemma 3.13, we have

(5) $\displaystyle\dfrac{3(49-k_{0})}{7-k_{0}}\leq a_{7}\leq 36$

By Lemma 3.12, we know that $k_{0}\leq 4$. Then
$3(11k_{0}-35)=3(49-k_{0})-36(7-k_{0})\leq 0.$ Hence, $k_{0}\in\\{2,3\\}$.

We are then left with the case $q=8$. As in the previous case, we prove that
the condition $k_{0}=2$ leads to a contradiction. If $k_{0}=2$, by the
inequality (5), $a_{7}\geq 29.$ Let $l_{2}\in\mathcal{A}_{2}$ with
$l_{2}\cap\mathcal{X}(\mathbb{F}_{8})=\\{P_{0},P_{1}\\}$ and $l_{2}\cap
Z(\mathcal{X})=\\{P_{2},...,P_{8}\\}.$ Let $r_{j}:=\psi_{7}(P_{j})$ where
$j=0,1,2,...,8$. By Lemma 3.4 if $j\in\\{0,1\\}$, then $49\leq
2+6r_{j}+5(8-r_{j})=42+r_{j}.$ Since $2+6r_{j}\leq 49$, then $r_{j}=7$ for
$i=0,1$. If $r_{j}=3$ for some $j=2,...,8$, by Lemma 3.4 and Proposition 3.7,
the other lines in $\check{Q}_{j}(\mathbb{F}_{q})$ contain at most $5$ points
of $\mathcal{X}(\mathbb{F}_{8})$. We thus obtain that $49\leq 2+3\cdot
7+5\cdot 5=48,$ a contradiction. Hence, $r_{j}\leq 2$ when $j=2,...,8$.
Therefore $a_{7}\leq 2\cdot 7+7\cdot 2=28$, a contradiction.

Finally, we prove that the condition $k_{0}=3$ leads to a contradiction for
$q=8$. If $k_{0}=3$, by the inequality (5), $a_{7}\geq 35$. Let
$l_{3}\in\mathcal{A}_{3}$ with
$l_{3}\cap\mathcal{X}(\mathbb{F}_{8})=\\{P_{0},P_{1},P_{2}\\}$ and $l_{2}\cap
Z(\mathcal{X})=\\{P_{3},...,P_{8}\\}$. Let $r_{j}:=\psi_{7}(P_{j})$ where
$i=0,1,2,...,8$. By Lemma 3.4, if $j\in\\{0,1,2\\}$ then $49\leq
3+6r_{j}+5(8-r_{j})=43+r_{j}.$ Since $3+6r_{j}\leq(q-1)^{2}=49$, then
$r_{j}\in\\{6,7\\}$ for $i=0,1,2$. If $r_{j}=3$ for some $j=2,...,8$, by Lemma
3.4 and Proposition 3.7 the other lines in $\check{Q}_{j}(\mathbb{F}_{q})$
contain at most $5$ points of $\mathcal{X}(\mathbb{F}_{8})$, then $49\leq
3+3\cdot 7+5\cdot 5=49.$ This means that the other $5$ lines in
$\check{Q}_{j}$ are in $\mathcal{A}_{5}$. By Lemma 3.13, we have
$-2a_{4}-2a_{5}+4a_{7}=138$. This implies that $4a_{7}\geq 138+2a_{5}\geq
138+10=148$, hence $a_{7}\geq 37$, a contradiction. So $r_{j}\leq 2$ when
$j=2,\ldots,8$. Therefore, $a_{7}\leq 3\cdot 7+6\cdot 2=33$, a contradiction.
The proof of our theorem is then completed.

∎

##  5\. Final remarks

###  5.1. On $\mathbb{F}_{q}$-Frobenius classical curves with many points

Let $\mathcal{C}\subseteq\mathbb{P}^{2}$ be an irreducible nonsingular
algebraic curve of degree $d$ defined over $\mathbb{F}_{q}$. If $\mathcal{C}$
is $q$-Frobenius classical, by Theorem 2.2, we have

(6) $\text{N}_{q}(\mathcal{C})\leq\dfrac{d(d+q-1)}{2}.$

Note that if $d=q-1$ then $d(d+q-1)/2=(q-1)^{2}$. This means that the Stöhr-
Voloch upper bound for a nonsingular $q$-Frobenius classical plane curve of
degree $q-1$ is equal to Sziklai’s upper bound. Also, by the proof of our main
result, it is inferred that the curves attaining the Sziklai bound are
$q$-Frobenius classical and have no $\mathbb{F}_{q}$-rational point of
inflection. In other words, Theorem 4.4 classifies the
$\mathbb{F}_{q}$-Frobenius classical curves of degree $q-1$ attaining the
Stöhr-Voloch upper bound (6) up to projective equivalence.

###  5.2. On hypersurfaces with many rational points

In [11], Homma and Kim gave an upper bound for the number
$\text{N}_{q}(\mathcal{X})$ of $\mathbb{F}_{q}$-rational points of a
nonsingular hypersurface $\mathcal{X}$ defined over $\mathbb{F}_{q}$ in an
odd-dimensional projective space $\mathbb{P}^{n}$, namely

(7) $\text{N}_{q}(\mathcal{X})\leq\theta_{q}\left(m\right)((d-1)q^{m}+1),$

where $m=(n-1)/2$ and
$\theta_{q}(m):=|\mathbb{P}^{m}(\mathbb{F}_{q})|=q^{m}+\cdots+q+1$. In [11,
Theorem 1.1], they also characterized, up to projective equivalence, the
hypersurfaces attaining the bound (7).

In the same paper, they also conjectured the following for the even-
dimensional case: if $\mathcal{X}\subseteq\mathbb{P}^{n}$ is a nonsingular
hypersurface defined over $\mathbb{F}_{q}$ of degree $d$ with $n$ even, then

(8)
$\text{N}_{q}(\mathcal{X})\leq\Theta_{n}^{d,q}:=\theta_{q}\left(m-1\right)((d-1)q^{m}+1)$

where $m=n/2$.

This conjecture was then proved by Datta in the case $n=4$ [4], and by Tironi
[21] in the general case. Here, surprisingly, a link with curves that are
optimal with respect to the Sziklai bound appears when considering
hypersurfaces attaining (8). In fact, let $\mathcal{X}$ be a hypersurface in
$\mathbb{P}^{4}$ attaining the bound (8); then by [4, Theorem 4.8] there
exists a point $P\in\mathcal{X}(\mathbb{F}_{q})$ such that
$\mathcal{X}\cap\mathbb{T}_{P}(\mathcal{X})$ is a cone, with center at $P$,
over a plane curve $\mathcal{C}$ of degree $d$ defined over $\mathbb{F}_{q}$
without $\mathbb{F}_{q}$-linear components and $\text{N}_{q}(\mathcal{C})$
attains the Sziklai bound. Also, by [21, Theorem 1], this curve must be
nonsingular. For $n\geq 6$, we have an analogous result; by [21, Theorem 2]
the bound (8) is attained by a nonsingular hypersurface
$\mathcal{X}\subseteq\mathbb{P}^{n}$ defined over $\mathbb{F}_{q}$ of degree
$d$ with $n\geq 6$ even only if there exists a point
$P\in\mathcal{X}(\mathbb{F}_{q})$ such that
$\mathcal{X}\cap\mathbb{T}_{P}(\mathcal{X})$ is a cone, with center at $P$,
over a nonsingular hypersurface $\mathcal{Y}\subseteq\mathbb{P}^{n-2}$ of
degree $d$ defined over $\mathbb{F}_{q}$ without $\mathbb{F}_{q}$-linear
components and $\text{N}_{q}(\mathcal{Y})=\Theta_{n-2}^{d,q}$. Therefore, the
extremal hypersurfaces in even dimension can be characterized inductively
starting with the ones in $\mathbb{P}^{4}$, which in turn can be constructed
from optimal Sziklai curves. This gives a possible application of Theorem 4.4.

It should be noted, however, that it is still unknown whether all
hypersurfaces with a point $P$ such that is a cone, with center at $P$, over a
nonsingular hypersurface $\mathcal{Y}\subseteq\mathbb{P}^{n-2}$ of degree $d$
defined over $\mathbb{F}_{q}$ without $\mathbb{F}_{q}$-linear components and
$\text{N}_{q}(\mathcal{Y})=\Theta_{n-2}^{d,q}$ attains the bound (8).

## References

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# Supplementary Material

##  I Experimental Data

Figure S1 presents our experimental chamber in its entirety, with all of the
shim coils and the front MOT and mini-MOT coils removed for clarity. The
science region of our vacuum chamber is comprised of a double-sided octagon
glass cell (Precision Glassblowing) treated with nano-texture etching
(TelAztec). Our MOT beams are oriented such that the 2 radial beams are
orthogonal in the radial plane, while the axial beam is 30∘ off axis. Due to
Lithium’s low vapor pressure, we introduce atoms into the gas-phase by use of
an atomic beam source. A stainless steel oven containing 14g of isotopically
enriched 6Li is heated to 300∘ Celsius with band heaters. A nozzle comprised
of a triangular array of 21 high aspect ratio stainless steel microcapillary
tubes couples the lithium oven to the ultra-high vacuum chamber, thereby
forming an effusive beam of 6Li atoms that delivers a flux of $\sim$109
atoms/s into the science region of our chamber.

We utilize two stages of differential pumping by separating the three regions
of our vacuum chamber with two small stainless steel tubes, both of which have
internal diameters that are twice that necessitated by the aspect ratio of the
atomic beam. Atoms are decelerated via a Zeeman slower laser that is passed
through the far-end of the octagon glass cell via a sapphire viewport. Laser
light that addresses the D1 and D2 lines of Lithium are generated by two
external cavity diode lasers (Toptica DL pros). The D1 laser is locked to the
D1 central crossover transition via saturated absorption spectroscopy and the
D2 laser is offset phase-locked to the D1 laser. Repumper light for D2 MOT
loading and D1 cooling is generated with a free-space electro-optic modulator
(Qubig).

To circumvent considerable eddy currents after the main MOT coils are switched
off, a pair of smaller mini-MOT coils remain on for 50 ms, after which time
all residual magnetic fields are switched off and $\Lambda$-enhanced gray
molasses ($\Lambda$-GM) is performed on the D1 line of the bulk gas, reducing
the temperature of the atoms from 1 mK to 100 $\mu$K. Atoms are then
probabilistically loaded into a 1064 nm optical tweezer, which is generated
from an ultralow noise high power fiber laser (Azurlight Systems, 130 W). The
1064 nm light is coupled to the experiment via a polarization-maintaining
optical fiber, and is split into an arbitrary array of individual trap sites
using an acouso-optic deflector (IntraAction Corp) driven by an arbitrary
waveform generator (16bit 625MS/s M4i6622-x8). The tweezer light is then
passed through a dichroic filter and then through a high numerical aperture
objective (Special Optics, 0.55 NA), through which it is focused down to
approximately 1.0 $\mu$m at the center of the glass cell.

Figure S1: Full view of our vacuum chamber, which is oriented at 45∘ with
respect to the optics table. Additionally, the entire chamber is mounted on a
translational stage. These features allow for convenient optical access and
beam alignment at the atoms. The lithium-6 effusive oven, located at the back
of the chamber, is separated from the double-sided glass octagon cell by two
stages of differential pumping (marked by the three red Saes ion/getter
pumps). The optically accessible 4-way cross just after the effusive oven is
useful for characterizing and shaping the atomic beam. Shim coils, and the
front MOT and mini MOT coils are removed for clarity.  Figure S2:  Cartoon
schematic of atom imaging. 1064 nm tweezer light is filtered and split into an
array by an acousto-optic modulator. The light is expanded by a factor of ten
and focused down through an objective to the center of the glass cell.
Scattered photons from trapped atoms are collected back through the same
objective, bounced off of a dichroic mirror and imaged onto an EMCCD camera.
Additionally, a second back objective is used to image both the tweezer sites
and the MOT onto a CMOS camera.  Figure S3: Atom survival vs. number of
images. Inset shows early timescale. Each imaging cycle lasts 15 ms.

The survival plots for continuous $\Lambda$-GM (blue) and simply holding in
the tweezer (orange) are reproduced in Figure S3. As mentioned in the main
text, the survival decay curve of simply holding the atom in the tweezer took
the form of a sigmoid, rather than an exponential, function. We fit the
tweezer hold time decay curve to an error function of a normalized Gaussian
with zero offset. The inset of the plot shows early timescales, where the
goodness of the fit is apparent. The good agreement implies that that
probability density function (PDF) of the time required to heat the atom out
of the trap is, to a good approximation, a Gaussian function. By a simple
conversion with $\Gamma$, the PDF of the number of photons required to heat
the atom out of the trap is also a Gaussian function. We can see from Figure
S3 that holding the atom in the tweezer for the time equivalent of taking 300
$\Lambda$-GM images knocks the atom out of the trap with 50$\%$ likelihood.
This corresponds to a tweezer hold time of 4.5 s. Using an off-resonant
scattering rate of 10 Hz implies that a lithium atom in the tweezer, initially
at 100 $\mu$K, needs to only scatter 45 photons before 50 $\%$ likelihood of
heating out of the trap.

Figure S4:  Atom survival vs. trap depth modulation frequency. Lithium’s light
mass leads to relatively high radial trap frequencies of 600 and 700 kHz.
Figure S5:  These plots highlight the sensitivity of $\Lambda$-GM parameters
on repeated imaging survival. At $\Lambda$-GM conditions corresponding to the
lowest achieved temperature (measured via release and recapture (a)) the atom
survival under constant application of the cooling light is much worse than
that of the $\Lambda$-GM optimized for 100 $\mu$K (b). As mentioned in the
main text, the survival $1/e$ lifetime of the 70 $\mu$K data is nearly
identical to that of simply holding the atom in the tweezer. Figure S6: The
above two plots show the time-dependant effects of application of cooling
light vs. application of no light in the tweezer. In the first case (a) we see
that the temperature of the atom is the same after 15 ms and 1,000 ms of
applied $\Lambda$-GM light. In the second case (b), we see that the atom heats
rapidly by off-resonant scattering of the trap light.

##  II Tweezer cooling simulation

Figure S7: Lithium 6 D1 level structure and definition of frequencies,
detunings, and hyperfine splittings.

The tweezer cooling simulation is performed by solving the steady state master
equation for the D1 levels of lithium 6 in a harmonic oscillator. We simulate
up to 14 harmonic levels in 1D and obtain accurate predictions for temperature
and scattering rate. We also develop a new picture for cooling in gray
molasses (GM) and $\Lambda$-enhanced GM in and near the Lamb-Dicke regime,
which leads to a simple criteria for efficient cooling: the bright state AC
stark shift must be similar to the trapping frequency energy $\hbar\omega$. In
a future paper, we will show how this picture can also describe polarization
gradient (PG), Doppler cooling, and sideband cooling.

###  II.0.1 Master Equation Derivation

Before diving into the derivation, first we work out the dipole moment
operator and electric field operator in the preferred basis. The atom-dipole
notation follows the conventions of Ref. [1]. The dipole operator is expressed
in spherical vector form as

$\mathbf{d}=\sum_{q}d_{q}\hat{\mathbf{e}}_{q}^{*},$ (1)

with components $d_{\pm 1}=\mp\frac{1}{\sqrt{2}}(d_{x}\pm id_{y})$ and
$d_{0}=d_{z}$. The unit spherical vectors are similarly defined as
$\hat{\mathbf{e}}_{\pm 1}=\mp\frac{1}{\sqrt{2}}(\hat{\mathbf{e}}_{x}\pm
i\hat{\mathbf{e}}_{y})$ and $\hat{\mathbf{e}}_{0}=\hat{\mathbf{e}}_{z}$.
Spherical vectors possess properties such as $(d_{q})^{*}=(-1)^{q}d_{-q}$ and
dot product
$\mathbf{A}\cdot\mathbf{B}=\sum_{q}(-1)^{q}A_{q}B_{-q}=\sum_{q}A_{q}B_{q}^{*}$,
with orthogonality
$\hat{e}_{q}\cdot\hat{e}_{q^{\prime}}^{*}=\delta_{qq^{\prime}}$.

The identity operator in the atom basis is a sum of all ground and D1 excited
states:

$\hat{\mathcal{I}}=\sum_{F_{g}m_{g}}|F_{g}m_{g}\rangle\langle
F_{g}m_{g}|+\sum_{F_{e}m_{e}}|F_{e}m_{e}\rangle\langle F_{e}m_{e}|.$ (2)

Multiplying the dipole operator on both sides by the identity
$\hat{\mathcal{I}}d_{q}\hat{\mathcal{I}}$ and using the Wigner-Eckert theorem
and angular momentum identities yields an expression for the dipole momentum
operator in terms of a collective lowering operator $\Sigma_{q}$ [1]:

$d_{q}=\langle
J_{g}||d||J_{e}\rangle\left(\Sigma_{q}+(-1)^{q}\Sigma_{-q}^{\dagger}\right)=d_{q}^{(+)}+d_{q}^{(-)}.$
(3)

We further separate the collective lowering operator into a lowering operator
for both ground states:

$\Sigma_{q}=\sum_{F_{g}}\Sigma_{q,F_{g}},$ (4)

with lowering operator to a single hyperfine ground state as:

$\Sigma_{q,F_{g}}=\sum_{m_{g}F_{e}m_{e}}(-1)^{F_{e}+J_{g}+I+1}\sqrt{(2F_{e}+1)(2J_{g}+1)}\langle
F_{g}m_{g}|F_{e}m_{e};1q\rangle\begin{Bmatrix}J_{e}&J_{g}&1\\\
F_{g}&F_{e}&1\end{Bmatrix}|F_{g}m_{g}\rangle\langle F_{e}m_{e}|.$ (5)

This lowering operator lowers all excited states to the $F_{g}$ ground state
while changing $m$ by $q$.

The electric field operator written in complex form with a rotating $(+)$ and
counter rotating $(-)$ terms is

$\mathbf{E}(\mathbf{r},t)=\mathbf{E}^{(+)}(\mathbf{r},t)+\mathbf{E}^{(-)}(\mathbf{r},t)=\sum_{i}\left(\bm{\mathcal{E}}^{(+)}_{i}(\mathbf{r})e^{-i\omega_{i}t}+\text{c.c.}\right)$
(6)

Here $i$ denotes the frequency of the laser that addresses either the $F=3/2$
or $F=1/2$ transition. These frequencies are typically detuned relative to
each by the ground state hyperfine splitting in order to make a $\Lambda$
system. The complex spatial field for a particular frequency $i$ is expressed
in terms of a sum of plane waves

$\bm{\mathcal{E}}^{(+)}_{i}(\mathbf{r})=\sum_{j}\bm{\epsilon}_{ij}\frac{E_{ij}}{2}e^{i\mathbf{k}_{ij}\cdot\mathbf{r}}$
(7)

The index $j$ denotes the beam index for one of the frequencies,
$\mathbf{k}_{ij}$ is the wave-vector for frequency $i$ and beam $j$,
$\bm{\epsilon}_{ij}$ is the polarization vector, and $E_{ij}$ is the complex
electric field amplitude. The intensity is related to the amplitude by
$I=\frac{c\epsilon_{0}}{2}|E|^{2}$.

The Hamiltonian for the interaction between the atom and the electric field of
the laser in the dipole approximation and rotating wave approximation (but not
in the rotating frame) is

$\displaystyle H_{\text{AF}}$
$\displaystyle=-\mathbf{d}^{(+)}\cdot\mathbf{E}^{(-)}-\mathbf{d}^{(-)}\cdot\mathbf{E}^{(+)}$
$\displaystyle=-\sum_{q}(-1)^{q}\left(d_{q}^{(+)}E_{-q}^{(-)}+d_{q}^{(-)}E_{-q}^{(+)}\right)$
(8)

We define the position-dependent but time-independent Rabi frequencies

$\Omega_{q,i}(\mathbf{r})=-2\langle
J_{g}||\mathbf{d}||J_{e}\rangle\frac{1}{\hbar}\mathcal{E}^{(+)}_{q,i}(\mathbf{r})$
(9)

The atom-field interaction is then defined in terms of a sum over the two
laser frequencies and all three polarizations

$H_{AF}=\frac{1}{2}\sum_{q,i}\left(\Omega_{q,i}(\mathbf{r})\Sigma_{q}^{(+)}e^{-i\omega_{i}t}+h.c.\right)$
(10)

We now go to the rotating frame of the lasers to get rid of the time-
dependence. Because the cooling light is mostly addressing the $F=3/2$ state
and the repumper is mostly addressing the $F=1/2$ state, we put the two
hyperfine ground states in the rotating frame of their respective laser by
applying a unitary transformation

$U=e^{-i\omega_{3/2}t\sum_{m_{g}}|3/2,m_{g}\rangle\langle
3/2,m_{g}|}e^{-i\omega_{1/2}t\sum_{m_{g}}|1/2,m_{g}\rangle\langle 1/2,m_{g}|}$
(11)

The new Hamiltonian transforms to $H\rightarrow
UHU^{\dagger}+i\hbar(\frac{\delta}{\delta t}U)U^{\dagger}$ and the operators
transform to $\sigma\rightarrow U\sigma U^{\dagger}$. From now on, all
operators will be assumed to be in the rotating frame defined by this unitary
operator.

In this rotating frame, the non-interacting Hamiltonian is (defining
$F^{\prime}=3/2$ excited state as zero energy)

$\begin{split}H_{0}&=\sum_{m_{g}}\big{[}\delta_{1/2}|1/2,m_{g}\rangle\langle
1/2,m_{g}|+\delta_{3/2}|3/2,m_{g}\rangle\langle 3/2,m_{g}|\big{]}\\\
&+\sum_{m_{e}}\big{[}-\Delta^{\prime}_{\text{HF}}|1/2,m_{e}\rangle\langle
1/2,m_{e}|+0\times|3/2,m_{e}\rangle\langle 3/2,me|\big{]}\end{split}$ (12)

As defined in the level diagram, $\Delta^{\prime}_{HF}$ is the D1 excited
state hyperfine splitting and is positive(negative) for GM(PG) cooling.

The atom-field interaction has two terms. One is time-independent, where the
cooling light only addresses the $F=3/2$ state and repumper light only
addresses the $F=1/2$ state:

$H_{\text{AF}}=\frac{1}{2}\sum_{q}\left[\Omega_{q,3/2}(\mathbf{r})\Sigma^{\dagger}_{q,3/2}+\Omega_{q,1/2}(\mathbf{r})\Sigma^{\dagger}_{q,1/2}+h.c.\right]$
(13)

The other atom field interaction is the cross-coupling terms consisting of the
$F=3/2$ laser addressing the $F=1/2$ transition and $F=1/2$ laser addressing
the $F=3/2$ transition. These terms still have time-dependence in the rotating
frame:

$H_{\text{AF}}^{\text{cross}}=\sum_{q}\left[\Omega_{q,1/2}(\mathbf{r})e^{i(\omega_{3/2}-\omega_{1/2})t}\Sigma^{\dagger}_{q,3/2}+\Omega_{q,1/2}(\mathbf{r})e^{i(\omega_{1/2}-\omega_{3/2})t}\Sigma^{\dagger}_{q,1/2}+h.c.\right]$
(14)

This cross-coupling will be treated later as a perturbation, which is a good
approximation when $\delta\ll\Delta_{HF}$. Since the ground state hyperfine
splitting is several times larger than the detunings, they are off-resonant.

Lastly, the interaction with motional states is incorporated by first adding a
harmonic oscillator Hamiltonian.

$H_{\text{HO}}=\hbar\omega(a^{\dagger}a+1/2)$ (15)

where $\omega$ is the trapping frequency. Second, we promote the positions in
$\Omega_{q,i}(\mathbf{r})$ to operators. In our 1D simulation, we assume that
the incoming light is propagating in the $z$ direction as $E_{0}e^{ikz}$ with
real amplitude $E_{0}$. We also assume there is a retro-reflected beam with
the same magnitude $E_{0}e^{i\phi}e^{ikz}$ but with a phase determined by the
mirror position. We also assume that the polarization is in the $x$-$y$ plane.
We define recoil operators

$\hat{\mathcal{R}}=e^{ik\hat{z}},\quad\hat{\mathcal{R}}^{\dagger}=e^{-ik\hat{z}}$
(16)

which are momentum displacement operators corresponding to a recoil from
photon absorption or emission. We do not assume the Lamb-Dicke (LD) regime in
our simulations, but we find it useful to expand the recoil operator in the LD
regime to see what the dominant terms correspond to. The LD regime here would
correspond to the expansion

$\hat{\mathcal{R}}\approx 1+ik\hat{z}=1+i\eta_{LD}(\hat{a}+\hat{a}^{\dagger})$
(17)

where $\eta_{LD}=kz_{0}$ and $z_{0}=\sqrt{\hbar/2m\omega}$. The first term
corresponds to no motional state change, and the second term to either the
motional state gaining or loosing one quanta.

In terms of the recoil operators, the complex electric field can be expressed
as

$\Omega_{q,i}(\hat{z})=\Omega_{i}\left[\epsilon_{q,\rightarrow}\hat{\mathcal{R}}+e^{i\phi_{i}}\epsilon_{q,\leftarrow}\hat{\mathcal{R}}^{\dagger}\right]$
(18)

with scalar Rabi frequency

$\Omega_{i}=-\frac{2}{\hbar}\langle
J_{g}||\mathbf{d}||J_{e}\rangle\frac{E_{0,i}}{2}$ (19)

The arrows denote the direction of the beam.

The atom-field interaction that couples the ground and excited state and
motional states for a pair of counter-propagating beams is

$H_{AF}=\sum_{q,i}\frac{\Omega_{i}}{2}\left[(\epsilon_{q,\rightarrow}\hat{\mathcal{R}}+e^{i\phi_{i}}\epsilon_{q,\leftarrow}\hat{\mathcal{R}}^{\dagger})\Sigma_{q}^{\dagger}+h.c.\right]$
(20)

These equations can be used to simulate any combination of forward and
backward polarizations.

For the master equation, we use the six collapse operators corresponding to an
emission of a photon with polarization $q$ in either the forward or backward
direction:

$\hat{C}_{q}=\sqrt{\Gamma}\,{\Sigma}_{q},\hat{\mathcal{R}},\quad\sqrt{\Gamma}\,\Sigma_{q}\hat{\mathcal{R}}^{\dagger}$
(21)

Besides the cross-coupling terms which are described later, this concludes the
form of the master equation used in the simulation. The main approximations
are 1) 1D motional states 2) cross-coupling scattering and shifts are small 3)
photons can only be emitted in the forward and backward direction, and 4) only
a pair of counter-propagating beams.

Now we define a specific polarization for the two beams. The most conventional
polarizations configurations are lin$\perp$lin and $\sigma_{+}$-$\sigma_{-}$.
The lin$\perp$lin configuration corresponds to a polarization that changes
between circular and linear, and therefore has a strong vector shift. The
$\sigma_{+}$-$\sigma_{-}$ configuration is instead a linear rotating field and
therefore only has a tensor shift. Because the MOT is
$\sigma_{+}$-$\sigma_{-}$ and we use the same beams, the cooling also uses
$\sigma_{+}$-$\sigma_{-}$.

For $\sigma_{+}$-$\sigma_{-}$, the atom-field interaction Hamiltonian for the
counter-propagating beams is:

$H_{AF}^{\sigma_{+}-\sigma_{-}}=\sum_{i}\Omega_{i}\left[(\hat{\mathcal{R}}\Sigma_{1}^{\dagger}+e^{i\phi_{i}}\hat{\mathcal{R}}^{\dagger}\Sigma_{-1}^{\dagger})+h.c.\right]$
(22)

We assume that the atom is situated around a position $z=0$, where it
experiences linear polarization determined by the retro phases $\phi_{1}$ and
$\phi_{2}$ for the two different frequencies. Since the cooling and repumper
lights have different frequencies, their phase accumulation from the atom to
the mirror and back differs by $\phi_{1/2}-\phi_{3/2}\approx 0.3(2\pi)$. All
three retro mirrors are positioned about 10 inches away from the atoms. This
phase difference results in the linearly polarized light seen by the atom at
the two frequencies having different angles.

When the phase difference is zero, the two polarizations are linear and
aligned. However, in our experiment and simulation, this phase seems to be of
little importance, likely because the repumper has 10 times less power than
the cooling beam and does not contribute significantly to the AC Stark shifts.

Performing the LD expansion of this atom-field Hamiltonian gives (for a single
frequency)

$H_{AF}^{\sigma_{+}-\sigma_{-}}\approx\Omega_{i}\left[\left(\Sigma_{1}^{\dagger}+e^{i\phi_{1}}\Sigma_{-1}^{\dagger}\right)+ik\hat{z}\left(\Sigma_{1}^{\dagger}-e^{i\phi_{1}}\Sigma_{-1}^{\dagger}\right)\right]$
(23)

In the LD regime, the zeroth order term does not couple different motional
states $|n\rangle$, but does couple the levels within a single motional state
and results in the dressed states shown in the cartoons of Fig. 4 in the main
paper. The first order term results in coupling between different motional
states. Interestingly, it is this sign difference in the zeroth order and
first order term that allows the first order coupling term to mix bright and
dark states of different $|n\rangle$. Without that sign difference, the bright
and dark states of neighboring $n$ would not be coupled.

The steady-state solver is unable to handle time-dependent interactions. Given
that the cross-coupling is off-resonant, we assume that the scattering rate is
small compared to that of the main beams. To address this, we derive
additional effective cross-coupling collapse operators as follows:

$\hat{C}_{i}=\sqrt{\Gamma\frac{\Omega_{i}^{2}}{4\Delta_{i}^{2}}}\left[\Sigma_{1,i}^{\dagger}\hat{\mathcal{R}}+e^{i\phi_{i}}\Sigma_{-1,i}^{\dagger}\hat{\mathcal{R}}^{\dagger}\right]$
(24)

where $\Delta_{i}$ represents the detuning of the cross-coupling beams to the
$F^{\prime}=3/2$ excited state. These collapse operators are derived by first
moving to a rotating frame of the cross-coupling beams, then transitioning to
the dressed frame of the time-independent interaction, considering only first-
order terms in saturation, and finally returning to the rotating frame of the
main beams.

The cross-couplings introduce additional AC Stark shifts, which might affect
the $F=1/2$ and $F=3/2$ states. However, we find this impact to be minimal and
thus ignore it. The primary effect is the introduction of new collapse
operators that correspond to decay from the ground to excited state. These new
collapse operators serve two purposes: 1) they induce recoil heating at the
rate of off-resonant scattering of the cross-coupling beam, and 2) they cause
mixing between the dark and bright states.

###  II.0.2 Simulations

In the simulation, we simplify the trap by representing it as a one-
dimensional model, rather than its actual three-dimensional configuration.
Although the real experiment employs three beams with corresponding retro-
reflections, our simulations are limited to one dimension. The simulation
would more accurately model pulsing the three pairs of beams individually.
Using all three beams simultaneously might lead to interference and alter the
results. However, experimental evidence suggests that cooling remains robust
even with minor beam adjustments, indicating that interference is likely not a
significant factor. Future research will explore the differences between
simultaneous and pulse-based multi-axis cooling.

We use QuTip’s steady-state solver, employing an iterative approach with up to
14 harmonic states. This solver generates the steady-state density matrix,
from which we determine the motional state distribution and excited state
population. By fitting the motional state distribution to a Boltzmann
distribution $e^{-\beta\hbar\omega n}$, we can extract the temperature
$\beta=1/k_{B}T$. Fig. S8 shows the population distributions and Boltzmann
fits for several optimized simulations. The blue curve is when radial
temperature is minimized, and gives $T=0.8\,\hbar\omega/k_{B}$ = 27 $\mu$K, or
57 $\%$ ground state. The green curve is when radial is instead optimized to
minimize the high population. The red curve is the population when axial is
optimized for temperature, and give $T=3.7\hbar\omega_{\text{axial}}/k_{B}T=$
27 $\mu$K, or 7$\%$ ground state. In the 2d simulations of the main of the
main text corresponding to the experimental parameters, we get 41 $\mu$K for
radial and 82 $\mu$K for axial.

Figure S8:  Examples of distributions of population and fits to a Boltzmann
distribution to extract temperature. (blue) Radial population for simulation
optimized for radial temperature. (green) radial temperature optimized to
minimize high $n$. (red) axial optimized to minimize high $n$.

The radial trapping frequency was measured to be 700 kHz, corresponding to a
motional quanta of 33.6 $\mu$ K. The axial trapping frequency was not directly
observed, but is typically 5 -6 times smaller in similar experiments. From the
simulations, we can either extract average 1D energy
$\langle\hbar\omega(n+1/2)$ or fit the population to a Boltzmann distribution
to extract temperature. We find the population distributions typically follow
a Boltzmann distribution and that it converges for less harmonic states. That
temperature is also more accurate for comparing to the temperature extracted
from release and recapture. The temperature extract from both methods
typically did not differ much.

Importantly, we see that the axial and radial cooling have very different
conditions. Experimentally, we have optimized the combined temperature, but
the simulation suggests we could cool further by pulsing the radial and axial
cooling separately with different powers. Indeed in simulation we find the
temperature can be reduced by more than a factor of two by optimizing the
radial and axial separately.

The Lamb-Dicke (LD) regime approximation is not employed; however, the values
in the $\mathcal{R}$ matrix are discarded if they fall below $5\times 10^{-3}$
in order to speed up the simulations. The simulations were executed for all D1
lines for up to 10 harmonic states directly and 20 iteratively using the
BICGSTAB algorithm with an iLu preconditioner within QuTip using the Purdue
Negishi computing cluster.

###  II.0.3 D1 Saturation Intensity

In the main text we refer to the D1 saturation intensity. As there seems to be
confusion in literature, we here derive the D1 saturation intensity. The
reduced matrix element convention adopted is given by

$\tau^{-1}=\frac{\omega_{0}^{3}}{3\pi\epsilon_{0}\hbar
c^{3}}\frac{2J+1}{2J^{\prime}+1}|\langle
J||\mathbf{d}||J^{\prime}\rangle|^{2},$ (25)

with $\tau=27.102$ ns. The reduced matrix element is:

$\langle J=1/2||\mathbf{d}||J^{\prime}=1/2\rangle=1.989\times
10^{-29}\,\text{C}\cdot\text{m}.$ (26)

The saturation intensity of a transition is defined by

$\frac{I}{I_{\text{sat}}}=\frac{c\epsilon_{0}\Gamma^{2}\hbar^{2}}{4\,|\langle
F,m|\mathbf{d}|F^{\prime},m^{\prime}\rangle|^{2}}.$ (27)

By convention, the saturation intensity for D1 or D2 is cited as the smallest
saturation intensity for any two transitions with any polarization. The
largest dipole moment of any two transitions in D1 is for
$|F=1/2,m=1/2\rangle$ to $|F^{\prime}=3/2,m^{\prime}=3/2\rangle$, which yields
a saturation intensity of

$I_{\text{sat}}^{\text{D1}}=5.72\,\,\text{mW}/\text{cm}^{2}.$ (28)

We note the disagreement with Ref. [2], where the largest dipole moment was
not used. The D1 saturation intensity differs approximately by $\sqrt{2}$ from
the more commonly cited D2 saturation intensity,
$I_{\text{sat}}^{\text{D2}}=2.54\,\,\text{mW}/\text{cm}^{2}$.

## References

  * Steck [2023] D. A. Steck, Quantum and atom optics, available online at http://steck.us/teaching (revision 0.13.19, 6 april 2023) (2023).
  * [2] M. E. Gehm, Properties of 6Li, https://jet.physics.ncsu.edu/techdocs/pdf/PropertiesOfLi.pdf, accessed: 2023-04-28.