Preprint C98/2010
Towers of Function Fields over Cubic Finite Fields

DDIC

**Keywords: **
towers of function fields | cubic finite fields | limits of towers | generalized Zink?s bound | completions.

In this work we consider two towers of function fields over finite fields with cubic cardinality. For the first of these towers (which was introduced by Bassa, Garcia and Stichtenoth) we explicitly calculate the genus of each one of its steps, using the ramification theory of Artin-Schreier extensions of function fields. For the second tower (which was introduced by Ihara) we prove by structural arguments that its limit is
greater or equal than generalized Zink�s lower bound, showing in this way its asymptotic goodness; for the proof we use the machinery of completions. We also exhibit the relations between these towers and the tower introduced by Bezerra, Garcia and
Stichtenoth.