Two-torsion in the Jacobian of hyperelliptic curves over finite fields

Abstract

We determine the exact dimension of the $$ {\bf{F}}_2 $$ -vector space of $$ {\bf{F}}_q $$ -rational 2-torsion points in the Jacobian of a hyperelliptic curve over $$ {\bf{F}}_q $$ (q odd) in terms of the degrees of the rational factors of its discriminant, and relate this to genus theory for the corresponding function field. As a corollary, we prove the existence of a point of order > 2 in the Jacobian of certain real hyperelliptic curves.