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 x² + 1, x² + x + 1, or x² + x + 2 for some x ∈ ℤ. If r is even, we have N_b(2) = b + 2μ + 1, except for N₄(2) = 10 and N₉(2) = 20. If r is odd and b is not special, N_b(2) = b + 2μ + 1. If r is odd and b is special, we have either N_b(2) = b + 2μ or N_b(2) = b + 2μ−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 Fq. 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.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_FG2.html