On the existence of dimension zero divisors in algebraic function fields defined over F_q.
Résumé
Let $\mathbf{F}/\mathbb{F}_q$ be an algebraic function field of genus $g$ defined over a finite field $\mathbb{F}_q$. We obtain new results on the existence, the number and the density of dimension zero divisors of degree $g-k$ in $\mathbf{F}/\mathbb{F}_q$ where $k$ is an integer $\geq 1$. In particular, for $q=2,3$ we prove that there always exists a dimension zero divisor of degree $\gamma-1$ where $\gamma$ is the $q$-rank of $\mathbf{F}/\mathbb{F}_q$ and in particular a non-special divisor of degree $g-1$ when the Jacobian of $\mathbf{F}/\mathbb{F}_q$ is ordinary. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree $g-k$ for a hyperelliptic field $\mathbf{F}/\mathbb{F}_q$ in terms of its Zeta function.