Good Towers of Function Fields

Abstract

In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory). In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any non-prime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields.

Section 2 of this article supersedes the corresponding section of arXiv:1110.6076 Final version of this manuscript will appear in Algebraic Curves and Finite Fields: Codes, Cryptography, and other Emergent Applications (H. Niederreiter, A. Ostafe, D. Panario, and A. Winterhof, eds.), de Gruyter, Berlin