A note on optimal towers over finite fields
Let \(B_n(F)\) be the number of places of degree \(n\) of \(F/\mathbb{F}_q\) and \(g(F)\) be the genus of \(F\). Let \(F=(F_0,F_1,F_2,\cdots)\) be a tower of a function fields over \(\mathbb{F}_q\) and \[ \lambda(F)=\lambda(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_1(F_i)/g(F_i). \] The author defines \(\Delta_n(F)=\Delta_n(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_n(F_i)/g(F_i)\) and studies the exact values of \(\Delta_n(F)\), and upper and lower bounds for \(\Delta_n(F)\). Using the results on \(\Delta_n(F)\), it is proved that for all \(n \geq 2\), \(\Delta_n(F/\mathbb{F}_q)=0\) and \(\lambda_n(F\mathbb{F}_{q^n}/\mathbb{F}_{q^n})=\sqrt{q}-1\) if \(q\) is a square and \(F\) is an optimal tower function fields over \(\mathbb{F}_q\) where \(F\mathbb{F}_{q^n}=(F_0\mathbb{F}_{q^n},F_1\mathbb{F}_{q^n},F_2\mathbb{F}_{q^n},\cdots)\) is defined to be composite tower of \(F\) over \(\mathbb{F}_{q^n}\). In particular, it is shown that the composite tower \(F\mathbb{F}_{q^n}\)is not optimal. Moreover, simple criterion whether a tower is optimal or not and many new recursive towers of finite ramification type are given.
scientific article; zbMATH DE number 5302213