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Curves of every genus with many points. II: Asymptotically good families. (English) Zbl 1072.11041
Let \(A^{-}(q):= \liminf_{g\to\infty} N_q(g)/g\), where \(N_q(g)\) denotes the maximum number of rational points over \(\mathbb F_q\) that a curve of genus \(g\) can have. To show that \(A^{-}(q)\) is greater than a constant \(c\) one has to show the existence of a curve over \(\mathbb F_q\) of large genus \(g\) with more than \(cg\) points. The curves used to construct optimal towers do not satisfy this property; Csirik, Wetherell, Zieve [arXiv:math.NT/0006096]. This article is related to the paper [J. Algebra 250, No. 1, 353–370 (2002; Zbl 1062.14027)] by A. Kresch, J. L. Wetherell and M. E. Zieve, among other results, the authors answer a question of Serre by showing that \(A^{-1}(q)\geq d\log q\), where \(d>0\) is a constant. The method of the proof involves degree-2 covering of curves and class field towers. The first step is to produce a sequence of curves \((C_i)\) over \(\mathbb F_q\) with many rational points whose genera grow at most exponentially (e.g. class field towers and Shimura curves). Then it is shown that each curve \(C_i\) is 2-covered by a curve \(B_i\) of genus greater than a constant multiple of the genus of \(C_i\). It happens that \(B_i\) or its quadratic twist has at least as many rational points as \(C_i\). Then the sequence of curves \((B_i)\) gives a positive answer to Serre’s question.

MSC:
11G20 Curves over finite and local fields
14G05 Rational points
14G15 Finite ground fields in algebraic geometry
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