On the size of the Jacobians of curves over finite fields

Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field Fq of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the Fq -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 → ∞.