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Article Dans Une Revue Acta Arithmetica Année : 2010

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.

Dates et versions

hal-01079449 , version 1 (02-11-2014)

Identifiants

Citer

Stéphane Ballet, Christophe Ritzenthaler, Robert Rolland. On the existence of dimension zero divisors in algebraic function fields defined over F_q.. Acta Arithmetica, 2010, 143 (4), pp.377-392. ⟨10.4064/aa143-4-4⟩. ⟨hal-01079449⟩
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