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Pointless curves of genus three and four. (English) Zbl 1116.14010
Aubry, Yves (ed.) et al., Arithmetic, geometry and coding theory (AGCT 2003). Selected papers of the European school “Algebraic geometry and information theory” and the 9th international conference “Arithmetic, geometry and coding theory”, Luminy, France, May 19–23, 2003. Paris: Société Mathématique de France (ISBN 2-85629-175-9/pbk). Séminaires et Congrès 11, 125-141 (2005).
Summary: A curve over a field \(k\) is pointless if it has no \(k\)-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field \(\mathbb{F}_q\) if and only if \(q < 26\), that there exist pointless smooth plane quartics over \(\mathbb{F}_q\) if and only if either \(q < 24\) or \(q = 29\) or \(q = 32\), and that there exist pointless genus-4 curves over \(\mathbb{F}_q\) if and only if \(q < 50\).
For the entire collection see [Zbl 1072.11001].

14G05 Rational points
11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus
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