On the asymptotic theory of towers of function fields over finite fields

Official URL: http://192.168.1.20/record=b1477110 (Table of Contents)

In this thesis we consider a tower of function fields F = (Fn)n≥0 over a finite field Fq and a finite extension E=F0 such that the sequence E := E. F = (EFn)n≥0 is a tower over the field Fq. Then we study invariants of E, that is, the asymptotic number of the places of degree r in E, for any r≥1, if those of F are known. We give a method for constructing towers of function fields over any finite field Fq with finitely many prescribed invariants being positive. For certain q, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions of function fields. Moreover, we show the existence of towers over a finite field Fq attaining the Drinfeld-Vladut bound of order r, for any r≥1 with qr a square. Finally, we give some examples of recursive towers with various invariants being positive and towers with exactly one invariant being positive.