On towers and composita of towers of function fields over finite fields (Q1266427)

Let \({\mathcal F}=(F_1\subset F_2 \subset \cdots)\) be a tower of algebraic function fields \(F_i/{\mathbb F}_q\). A tower is said to be asymptotically good, if \(\lambda({\mathcal F})=\lim_{i\rightarrow\infty}N(F_i)/g(F_i)\) is non-zero. It is said to be asymptotically optimal, if \(\lambda\) is equal to the Drinfeld-Vladut bound \(\sqrt{q}-1\). Explicit constructions of asymptotically optimal towers for \(q\) a square were given by \textit{A. Garcia} and \textit{H. Stichtenoth} [Invent. Math. 121, 211--222 (1995; Zbl 0822.11078) and J. Number Theory 61, 248--273 (1996; Zbl 0893.11047)]. In the paper under review the authors give very simple constructions of asymptotically good towers for \(q\) not a prime. For \(q=4\) and \(q=9\) they are even optimal. The field extensions in the tower are tamely ramified Kummer extensions. In the second half of the paper composita of towers are considered, i.e. if \({\mathcal F}=(F_1\subset F_2 \subset \cdots)\) is a tower and \(E/F_1\) is some finite extension, the compositum tower is \({\mathcal E}=(E\cdot F_1\subset E\cdot F_2\subset\cdots)\). Then \(\lambda({\mathcal E})\leq \lambda({\mathcal F})\) [J. Number Theory 61, 248--273 (1996; Zbl 0893.11047)]. If \(\mathcal F\) is asymptotically good and satisfies some further conditions the authors obtain a lower bound for \(\lambda({\mathcal E})\) (depending on \(\lambda ({\mathcal F})\)). For certain cases they get precise formulas for \(\lambda({\mathcal E})\).