Preprint A49/2001
On Tame Towers over Finite Fields

Henning Stichtenoth | Garcia, Arnaldo

**Keywords: **
towers of curves | rational points | finite fields

We consider towers of function fields over finite fields having only tame ramifications.We introduce two basic sets:the ramification locus and the completely splitting locus of the tower.The tower will have a positive limit(for the ratios of number of rational points over the genus)if the ramification set is finite and the completely splitting set is nonempty.Several towers consisting of degree two extensions is considered.One of them represents equations for the curves
X_0(2^n)(parametrizing elliptic curves with level structure).
The determination of the completely splitting locus in this tower gives the first description of the coordinates of the supersingular points of X_0(2^n) and this is done in terms of the Deuring's polynomial(describing supersingular elliptic curves in Legendre's normal form).