Towards a classification of recursive towers of function fields over finite fields
The authors deal with the question of classification of recursive towers of function fields over finite fields. Recall that a tower of function fields over \(\mathbb{F}_q\) is an infinite sequence \(\mathcal{F}=(F_0,F_1,\dots)\) of function fields \(F_n/\mathbb{F}_q\) such that (1) \(F_0 \subset F_1 \subset \dots\) and for all \(n \geq 0\) the extension \(F_{n+1}/F_n\) is separable of degree greater than \(1\). (2) For some \(r \geq 0\) the function field \(F_r\) has genus greater than \(1\). (3) For all \(n\), \(\mathbb{F}_q\) is algebraically closed in \(F_i\). If one denotes by \(g(F_n)\) the genus of \(F_n\) and by \(N(F_n)\) the number of rational places, the limit \[ \lambda(\mathcal{F})=\lim_{n \to \infty} \frac{N(F_n)}{g(F_n)} \] exists and the main purpose of these constructions is to compare it with \[ \limsup_{g \to \infty} \frac{N_q(g)}{g} \] where \(N_q(g)\) is the maximal number of points of an absolutely irreducible, projective smooth curve of genus \(g\) over \(\mathbb{F}_q\). Asymptotically good towers (i.e. such that \(\lambda(\mathcal{F})>0\)) are difficult to construct and most examples in the literature are of the following type. Definition: \(\mathcal{F}\) is recursive if it can be described by two separable rational functions \(f(T),g(T) \in \mathbb{F}_q(T)\), i.e. if there are elements \(x_i\) with (1) \(F_0=\mathbb{F}_q(x_0)\) is the rational function field. (2) \(F_{n+1}=F_n(x_{n+1})\) and \(f(x_{n+1})=g(x_n)\) for all \(n \geq 0\). (3) \([F_{n+1} : F_n]=\deg f(T)\) for all \(n \geq 0\) (the degree of \(f(T)=a(T)/b(T)\) with \(a,b \in \mathbb{F}_q[T]\) coprime is \(\max(\deg(a),\deg(b))\)). The authors prove then the following theorem which can be seen as a first step towards the classification of recursive towers. Theorem: Let \(\mathcal{F}\) be a tower which can be described recursively by the equation \(f_1(Y)=g_1(X)\) with \(f_1(T),g_1(T) \in \mathbb{F}_q(T)\). Let \(F=\mathbb{F}_q(x,y)\) with \(f(y)=g(x)\) and set \(z=f_1(y)=g_1(x) \in F\). Suppose that \(f(T) \in \mathbb{F}_q(T)\) is another rational function with the properties: (1) \(\deg f(T)=\deg f_1(T)= \deg g_1(T)\). 2) There exist elements \(\bar{x}\) and \(\bar{y}\) in the field \(F\) such that \(\mathbb{F}_q(x)=\mathbb{F}_q(\bar{x}), \mathbb{F}_q(y)= \mathbb{F}_q(\bar{y})\), \(f(\bar{x}) \in \mathbb{F}_q(z)\) and \(f(\bar{y}) \in \mathbb{F}_q(z)\). Then the tower \(\mathcal{F}\) can be described recursively by \[ f(Y)=A \cdot f(B \cdot X) \] for suitable matrices \(A,B \in \text{GL}(2,\mathbb{F}_q)\) acting by Möbius transformation. Then they specialize their theorem under more restrictive hypotheses (Towers of Kummer type, Artin-Schreier type, \(\dots\)) and recover the functions \(f\) for the already known recursive asymptotically good towers.
scientific article; zbMATH DE number 5004386