Towers of Function Fields over Non-prime Finite Fields

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties

On the Invariants of Towers of Function Fields over Finite Fields

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, 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\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.Comment: 23 page

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

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

A note on subtowers and supertowers of recursive towers of function fields

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields.Fil: Chara, María de Los Ángeles. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaFil: Navarro, H.. Universidad del Valle; ColombiaFil: Toledano, Ricardo Daniel. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; Argentin

Explicit towers of Drinfeld modular curves

We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.Comment: 10 pages. For mini-symposium on "curves over finite fields and codes" at the 3rd European Congress in Barcelona 7/2000 Revised to correct minor typographical and grammatical error

Asymptotics for the genus and the number of rational places in towers of function fields over a finite field

We discuss the asymptotic behaviour of the genus and the number of rational places in towers of function fields over a finite field

Good families of Drinfeld modular curves

In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach on how to obtain explicit defining equations for some of these towers and in particular give a new explicit example of an optimal tower over a quadratic finite field