On the number of points on Abelian and Jacobian varieties over finite fields (Q5891161)

Let \(A\) be an abelian variety of dimension \(g\) defined over \(\mathbb{F}_q\). In this paper the authors prove the inequality \((q+1-\lfloor2q^{1/2}\rfloor)^g\leq|(A(\mathbb{F}_q)|\leq(q+1+\lfloor2q^{1/2}\rfloor)^g\), which gives an improvement of Weil's inequality. When \(A=J_C\) is the Jacobian variety of a smooth projective curve \(C\) over \(\mathbb{F}_q\), they offer several improvements of the lower bound proved in [\textit{G. Lachaud} and \textit{M. Martin-Deschamps}, Acta Arith. 56, No. 4, 329--340 (1990; Zbl 0727.14019)]. For example they show that \(|J_C(\mathbb{F}_q)|\geq(1-2/q)(q+1+(N-(q+1))/g)^g\). Furthermore they give exact values for the maximum and the minimum number of rational points on Jacobian varieties of dimension 2.