Improved upper bounds for the number of points on curves over finite fields
Everett W. Howe, Kristin Lauter · Annales de l’institut Fourier · 2003
We give new arguments that improve the known upper bounds on the maximal number N q ( g ) of rational points of a curve of genus g over a finite field 𝔽 q , for a number of pairs ( q , g ) . Given a pair ( q , g ) and an integer N , we determine the possible zeta functions of genus- g curves over 𝔽 q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- g curve over 𝔽 q with N points must have a low-degree map to another curve over 𝔽 q , and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of N q ( g ) , namely: N 4 ( 5 ) = 17 , N 4 ( 10 ) = 27 , N 8 ( 9 ) = 45 , N 16 ( 4 ) = 45 , N 128 ( 4 ) = 215 , N 3 ( 6 ) = 14 , N 9 ( 10 ) = 54 , and N 27 ( 4 ) = 64 . Our arguments also allow us to give a