## 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

Zbl 1062.14027
Full Text:

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