Bounds for the number of points on curves over finite fields

Abstract

Let X be a projective irreducible nonsingular algebraic curve defined over a finite field Fq. This paper presents a variation of the Stöhr-Voloch theory and sets new bounds to the number of Fqrrational points on X . In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Stöhr-Voloch's and Ihara's.

Number of rational branches of a singular plane curve over a finite field

“…Here, we have c 1 ≥ q u + 2(r − 1) and c m−u ≥ q u , and these numbers can be bigger depending on some properties of F and g r n . As shown in [1], the bound (3) improves Hasse-Weil and Stöhr-Voloch bounds in many situations.…”

“…Notation and terminology are standard. Our main references are [1,8,11]. As before, F q denotes a finite field of order q, where q is a power of a prime number p. Let F q denote the algebraic closure of F q .…”

“…, k. Then g r n is defined over F q . By [11,Theorem 2.13] and [1,Theorem 4.4], both bounds (2) and (3) associated g r n can be written in terms of d, t, r i , s i . Denote by Γ t the linear system of all plane curves of degree t. Then Γ t cuts on F a linear series g r ′ n ′ of degree n ′ = td and dimension r ′ = 1 2 t(t + 3) [8, Section 7.7].…”

“…The bound (2) improves the Hasse-Weil bound in many instances (see [11]). In [1] a bound obtained via a variation of Stöhr-Voloch method is presented: if g r n is defined over F q , given positive integers u, m with u < m and gcd(u, m) = 1, there are positive integers c i (depending on i and g r n ) with i = 1, u, m, m − u, such that…”

“…. , κ r−2 ) is a sequence of non-negative integers, which depends also on g r n , such that 0 ≤ κ 0 < · · · < κ r−2 , see [1,Theorem 4.4]. Here, we have c 1 ≥ q u + 2(r − 1) and c m−u ≥ q u , and these numbers can be bigger depending on some properties of F and g r n .…”