On the linear bounds on genera of pointless hyperelliptic curves

Ivan Pogildiakov (1)

1 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

An irreducible smooth projective curve over $\mathbb{F}_q$ is called \textit{pointless} if it has no $\mathbb{F}_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field $\mathbb{F}_q$. Using some explicit constructions of hyperelliptic curves, we establish two new bounds that depend linearly on the number $q$. In the case of odd characteristic this improves upon a result of R. Becker and D. Glass. We also provide a similar new bound when $q$ is even.

Mots clés: Curves over finite fields; Rational points; Pointless curves

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Ivan Pogildiakov. On the linear bounds on genera of pointless hyperelliptic curves. 2017. ⟨hal-01494793⟩