\section{Introduction}

In computational arithmetic geometry, it is an important task to develop an efficient group law for the Jacobian variety of algebraic curves defined over a finite field. One of the most important and recent application of such efficient arithmetic comes from cryptography \cite{Koblitz,Koblitz2,Cantor}. In \cite{FOR}, the authors introduced an efficient algorithm to perform arithmetic in the Jacobian of smooth plane quartics. The  presented algorithm depends on the existence of a rational line $l$ intersecting the quartic in rational points only. Moreover, the more special $l$ is (for instance, tangent, tangent at a flex,\ldots), the better is the complexity of the algorithm. Motivated by the above efficiency argument, we prove here the following theorems.

\begin{theorem}\label{conditionetoile1}
Let $C$ be a smooth plane  quartic over the finite field $\FF_q$ with $q$ elements. If $q \geq 127,$ then there
exists a line $l$ which intersects $C$ at rational points only.
\end{theorem}


\begin{theorem}\label{conditionetoile2}
Let $C$ be a smooth plane quartic over $\FF_q$. If $q \geq 66^2+1,$ then there
exists a tangent to $C$ which intersects $C$ at rational points only.
\end{theorem}

In  \cite{FOR}, the authors gave heuristic arguments and computational evidences that the probability for a plane smooth quartic over a finite field to have a rational flex is about $0.63$. In this article, we present a `proof' which depends on a conjectural analogue over finite fields of a result of \cite{harris-enu} on the Galois group $G$ of the $24$ flexes of a general quartic. Harris proved that, over $\CC$, the group $G$ is as big as possible, namely the symmetric group $S_{24}$. 
Unfortunately, Harris's proof uses monodromy arguments which cannot be adapted so easily in positive characteristic. Worse, we surprisingly found that Harris's result is not valid over fields of characteristic $3$ and we prove that for this field the Galois group is $S_8$. This is a consequence of the peculiar fact that, in characteristic $3$, a smooth plane quartic $C$ has generically only $8$ flexes (with multiplicity $3$), which belong to a conic. We suspect that this is the only exceptional case.\\

The methods used for these three problems are various and can be  generalized or adapted to other questions. This was our principal motivation to write down our approaches in the case of quartics. It illustrates also the very unusual behavior of special points and lines in small characteristics. \\

Coming back to our initial motivation, it appears nowadays unlikely (due to recent progress in index calculus \cite{DiTh}) that smooth plane quartics may be used for building discrete logarithm cryptosystems. 
 However, it is interesting to mention that the complexity analysis of the index calculus attack of \cite{DiTh} uses an asymptotic bound for the number of  lines intersecting a smooth plane quartic in four distinct rational points, in the spirit of Theorem \ref{conditionetoile1}.\\

 The paper is organized as follows: Section \ref{wfact} gives a brief overview on the possible geometric intersections of a line and a smooth plane quartic. 
 
  In Section \ref{tcase}, we give a proof of Theorem \ref{conditionetoile1} using Chebotarev density theorem for covers of curves. In Section \ref{ttcase}, we prove Theorem \ref{conditionetoile2} using the tangential correspondence and its associated curve $X_C \subset C \times C$. When $\echr \FF_q \ne 2$, the crucial point is to prove that $X_C$ is geometrically irreducible in order to apply Hasse-Weil bound for (possibly singular) geometrically irreducible curves.
 For the characteristic $2$ case, $X_C$ is not always absolutely irreducible. However, if it is not, we prove it is reducible (over $\FF_q$) and we can apply Hasse-Weil bound on each component.
Finally, in Section \ref{open}, we also address the question of the probability of existence of a rational flex and show that in characteristic $3$, the flexes are on a conic.\\

\noindent {\bf Conventions and notation.} In the following, we denote by
$(x:y:z)$  the  coordinates in $\mathbb{P}^2$, and by $(x,y)$ the coordinates in $\mathbb{A}^2$. Let $p$ be a prime or $0$ and $n \geq 1$ an integer. We use the letter $K$ for an arbitrary field of characteristic $p$ and let $k=\FF_q$ be a finite field with $p \ne 0$ and $q=p^n$ elements. When $C$ is a smooth geometrically irreducible projective curve, we denote by $\kappa_C$ its canonical divisor. Operators such as $\Hom$, $\End$ or $\Aut$ applied to varieties
over a field $K$ will always refer to $K$-rational homomorphisms and endomorphisms.\\

\noindent {\bf Acknowledgments.} We would like to thank Noam Elkies and Pierre D\`ebes for their suggestions and references in Section \ref{flexcar3}. We are also very grateful to Felipe Voloch for showing us how to prove Theorem \ref{conditionetoile2} in characteristic $2$.
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