Constructing function fields with many rational places via the Carlitz module (Q1598147)

Let \(N_q(g)\) denote the maximal number of prime divisors of degree one over congruence function fields over \({\mathbb{F}}_q\) of genus \(g\), where \({\mathbb{F}}_q\) is the finite field of \(q\) elements. There are several upper bounds for \(N_q(g)\), for instance, Weil's bound and Serre's bound. Several constructions have been made of curves over \({\mathbb{F}}_q\) with many rational points for some values of \(q\) and \(g\). In the paper under review, the author considers cyclotomic function fields \(k(\Lambda _M)\) where \(k = {\mathbb{F}}_q(T)\) is a rational function field over \({\mathbb{F}}_q\), \(M \in {\mathbb{F}}_q[T]\) is any nonconstant polynomial and \(\Lambda _M\) is the \(M\)-th torsion of the Carlitz-Hayes module. Let \(M\) be such that \((M, T^2-T)=1\), \(\deg M \geq 2\) and \(q\geq 3\). Let \(H\) be one of the six subgroups of \(\text{ Gal} (k(\Lambda _M/k)\) in \[ \text{Subgr}:= \{\{1\}, \langle T\rangle, \langle T, T-1\rangle, {\mathbb{F}}_q^\ast, \langle {\mathbb{F}}_q^\ast, T\rangle, \langle {\mathbb{F}}_q,T,T-1\rangle\}. \] Using the explicit genus formulas found by \textit{A. Keller} [Jungnickel, Dieter (ed.) et al., Finite fields and applications. Augsburg, 1999. Berlin. Springer, 293-302 (2001; Zbl 0993.11062)] for \(H\in \text{Subgr}\), the author computes formulas suitable for automatic calculations that give the number of prime divisors of degree one of \(k(\Lambda _M)^H\). By extensive computations for \(q = 4, 8, 16, 32, 64, 3, 9, 27, 5\) and \(25\) he obtains some results beating those given by van der Geer and van der Vlugt in \url{http://www.wins.uva.nl/~geer} and fills some gaps. For some values of \(q\) (\(q=4,32,64\)) there are no new results.