Bounding the number of rational places using Weierstrass semigroups

Details Bounding the number of rational places using Weierstrass semigroups Bounding the number of rational places using Weierstrass semigroups

2007-10-25

arxiv.org/abs/0710.4662v1

Journal of Pure and Applied Algebra 213 (2009), no. 6, 1152-1156

Mathematics

Algebraic Geometry

16 pages, 3 tables

Scientific paper

10.1016/j.jpaa.2008.11.013

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields.

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