## Sharp Bound on the Number of Rational Points on Curves with Genus 2

An explicit formula for the maximal number of rational places in a global function field over **F**_{b} with genus 2 was established by Serre in [1], [2], [3].

Let *μ* = ⌊2⌋ and *b* = *p*^{r} with *p* prime. Call *b* “special” if it divides *μ* or if it is of the form *x*^{2} + 1, *x*^{2} + *x* + 1, or *x*^{2} + *x* + 2 for some *x* ∈ ℤ. If *r* is even, we have *N*_{b}(2) = *b* + 2*μ* + 1, except for *N*_{4}(2) = 10 and *N*_{9}(2) = 20. If *r* is odd and *b* is not special, *N*_{b}(2) = *b* + 2*μ* + 1. If *r* is odd and *b* is special, we have either *N*_{b}(2) = *b* + 2*μ* or *N*_{b}(2) = *b* + 2*μ*−1, depending on whether 2 – *μ* is greater than ( − 1)/2 or not.

### References

[1] | Jean-Pierre Serre. Nombres de points des courbes algébriques sur F_{q}.In Séminaire de Théorie des Nombres 1982–83, Exp. 22. Univ. de Bordeaux I, Talence, 1983. |

[2] | Jean-Pierre Serre. Résumé des cours de 1983–1984. In Annuaire du Collège de France, pages 79–83. 1984. |

[3] | Jean-Pierre Serre. Rational points on curves over finite fields. Lecture notes, Harvard University, 1985. |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Sharp Bound on the Number of Rational Points on Curves with Genus 2.”
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