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New bounds on the maximum number of points on genus-4 curves over small finite fields. (English) Zbl 1317.11064
Aubry, Yves (ed.) et al., Arithmetic, geometry, cryptography and coding theory. 13th conference on arithmetic, geometry, cryptography and coding theory, CIRM, Marseille, France, March 14–18, 2011 and Geocrypt 2011, Bastia, France, June 19–24, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7572-8/pbk; 978-0-8218-9027-1/ebook). Contemporary Mathematics 574, 69-86 (2012).
Summary: For prime powers $$q<100$$, we compute new upper and lower bounds on $$N_q(4)$$, the maximal number of points on a genus-4 curve over a finite field with $$q$$ elements. We determine the exact value of $$N_q(4)$$ for $$17$$ prime powers $$q$$ for which the value was previously unknown.
For the entire collection see [Zbl 1248.11004].

##### MSC:
 11G20 Curves over finite and local fields 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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