On the size of the Jacobians of curves over finite fields

Abstract

Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field $$ \mathbb{F}_q $$ of q elements, we obtain the asymptotic formula q g+o(g) for the size of set of the $$ \mathbb{F}_q $$ -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.