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Improved upper bounds for the number of points on curves over finite fields. (English) Zbl 1065.11043
Ann. Inst. Fourier 53, No. 6, 1677-1737 (2003); corrigendum ibid. 57, No. 3, 1019-1021 (2007).
Summary: 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 $${\mathbb F}_q$$, 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 $${\mathbb F}_q$$ 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 $${\mathbb F}_q$$ with $$N$$ points must have a low-degree map to another curve over $${\mathbb F}_q$$, 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 non-computer-intensive proof of the recent result of Savitt that there are no genus-$$4$$ curves over $${\mathbb F}_8$$ having exactly $$27$$ rational points. Furthermore, we show that there is an infinite sequence of $$q$$’s such that for every $$g$$ with $$0<g<\log_2 q$$, the difference between the Weil-Serre bound on $$N_q(g)$$ and the actual value of $$N_q(g)$$ is at least $$g/2$$.

##### MSC:
 11G20 Curves over finite and local fields 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry
Magma
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