Bounding the number of rational places using Weierstrass semigroups

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over F_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 Lewittes' bound in [6] 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.