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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
Software:
Magma
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