On the Invariants of Towers of Function Fields over Finite Fields

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 deal with the following: What can we say about the invariants of E ; i.e., the asymptotic number of the 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 DrinfeldVladut bound of order r, for any r ≥ 1 with q a square, see [1, Problem-2]. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.