Multiple Kummer extension and the number of prime divisors of degree one in function fields (Q1208220)

Let \(F=\mathbb{F}_ q(t)\) be the rational function field with finite constant field \(\mathbb{F}_ q\), let \(f_ i(t)\in\mathbb{F}_ q[t]\) for \(i=1,\dots,n\), and take \(E=F(f^{1/r}_ 1,\dots,f^{1/r}_ n)\), where \(r\) is a prime dividing \(q-1\). The author investigates the splitting of prime divisors in the extension \(E/F\) and computes the genus of \(E\). As an application he shows that \(A(3)\geq 1/3\) and \(A(5)\geq 1/2\), where \(A(q)\) is defined as follows: Let \(X\) be a smooth projective curve of genus \(g_ X\) over \(\mathbb{F}_ q\) and let \(X(\mathbb{F}_ q)\) denote the number of points of \(X\) defined over \(\mathbb{F}_ q\). Take \(q\) fixed and let \(g_ X\) tend to infinity. Then \(A(q)=\limsup X(\mathbb{F}_ q)/g_ X\).