manYPoints – Table of Curves with Many Points
Entry details for q = 131 = 13, g = 3
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Lower bound Nmin = 32

Submitted by C. Ritzenthaler
Date 04-07-2009
Reference Jean-Pierre Serre
Résumé des cours de 1983-1984
Annuaire du College de France (1984), 79-83. Reprinted in Vol. 3 of Jean-Pierre Serre, OEvres, Collected Papers. Springer- Verlag, New York (1985).
Comments
This number of points is reached by the Fermat quartic x^4+y^4+z^4=0 (see J-P. Serre, Rational points on curves over finite fields, Lectures given at Harvard University, 1985. Notes by F. Q. Gouvêa, p.64-68).

Its Weil polynomial is (X^2+6*X+13)^3.
Tags Explicit curves

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Upper bound Nmax = 32

Submitted by C. Ritzenthaler
Date 05-26-2009
Reference Karl-Otto Stöhr, José Felipe Voloch
Weierstrass points and curves over finite fields
Proc. London Math. Soc. (3) Vol. 52, No. 1 (1986) 1–19
Comments
This is the so-called Voloch's bound "2q+6" for genus 3 curves (q <> 8 and 9). See loc. cit. Proposition 3.1 and also "Curves of genus 3 over small finite fields" by J. Top Prop.2.1 item b)
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