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

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We give new arguments that improve the known upper bounds on the maximal number Nq(g) of rational points of a curve of genus g over a finite field Fq 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 Fq 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 Fq with N points must have a low-degree map to another curve over Fq, and often this is enough to give us a contradiction. In particular, we able to provide eight previously unknown values of Nq(g), namely: N₄(5) = 17, N₄(10) = 27, N₈(9) = 45, N₁₆(4) = 45, N₁₂₈(4) = 215, N₃(6) = 14, N₉(10) = 54, and N₂₇(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 F₈ 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 < log₂ q, the difference between the Weil-Serre bound on Nq(g) and the actual value of Nq(g) is at least g/2.