On explicit towers over finite fields

Arnaldo Garcia
Institute of Pure and Applied Mathematics (IMPA)

A tower of curves over a finite field k is an infinite sequence of curves C_m and of maps C_m C_(m-1), both defined over k, with genus g(C_m) growing to infinity with m .
A tower has a fundamental invariant called its limit that measures the asymptotic behaviour of the ratios of the number of rational points over k of the curve C_m by its genus g(C_m).
For applications, the tower should be explicit; i.e., the curves C_m in the tower are explicitly given by polynomial equations with coefficients in the finite field k.
We shall survey some explicit constructions of towers of curves over finite fields with “good limits”, most of them obtained together with Stichtenoth, and we will relate them with results obtained by Ihara, Tsfasman-Vladut, Elkies, Serre, Geer-Vlugt and Zink.
We finish with a new tower (work in progress with Bassa-Beelen-Stichtenoth) over finite fields with q^n elements, for any odd integer n>1, with a “very big limit”.

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