On the Invariants of Towers of Function Fields over Finite Fields
Abstract
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):=(EFn)n 0 is a tower over the field Fq. Then we deal with the following: What can we say about the invariants of E; i.e., the asymptotic number of places of degree r for any r≥ 1 in E, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over Fq with finitely many prescribed invariants being positive, and towers of function fields over Fq, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r≥ 1 with qr a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.