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Improved upper bounds for the number of points on curves over finite fields

Everett W. HoweKristin E. Lauter — 2003

Annales de l’institut Fourier

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...

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