A smooth curve has a number N of points over a field of order q bounded by the inequality $$N<q+1+2g\sqrt{q},$$ where $g$ is the genus of the curve. Is there some simple condition for reaching a value of N close to the upper bound?
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$\begingroup$ Even in the case of $g = 1$ there does not seem to be any simple condition, see arxiv.org/abs/1807.05255 $\endgroup$– Stanley Yao XiaoCommented Jun 8, 2023 at 12:50
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2$\begingroup$ This is Weil, not Weyl. And there is a huge literature on the subject: I recommend Serre's *Rational points on curves over finite fields". $\endgroup$– abxCommented Jun 8, 2023 at 13:03
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$\begingroup$ Ok, corrected. Thanks. $\endgroup$– AlmCommented Jun 8, 2023 at 13:06
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$\begingroup$ Searching with the right spelling, I found this math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/… $\endgroup$– AlmCommented Jun 8, 2023 at 13:11
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$\begingroup$ The behavior is very different if you are interested in fixing $g$ while $q$ grows, or fixing $q$ while $g$ rows. But I think there isn't a short answer in either case. $\endgroup$– David E SpeyerCommented Jun 11, 2023 at 14:50
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