Hilbert 2-class field towers of inert imaginary quadratic function fields (Q2870891)

Let \(q\equiv 3\bmod 4\) be a prime power, \(k=F_q(T)\), let \(F/k\) be an inert imaginary quadratic extension (i.e., \(\infty=(1/T)\) is inert), and let \(\mathrm{Cl}_F\) be the class-group of the integral closure of \(F_q[T]\) in \(K\) (the case of ramified \(\infty\) has been treated by the author earlier [Bull. Korean Math. Soc. 50, No. 3, 1049--1060 (2013; Zbl 1368.11119)]). The analogue of the Golod-Shafarevich theorem for function fields, proved by \textit{R. Schoof} [J. Number Theory 41, No. 1, 6--14 (1992; Zbl 0762.11026)] implies that if \(\mathrm{rank}_4(\mathrm{Cl}(F))\geq 5\), then the Hilbert \(2\)-classfield tower of \(F\) is infinite. The author shows that already the inequality \(\mathrm{rank}_4(\mathrm{Cl}(F))\geq4\), and in certain cases even \(\mathrm{rank}_4(\mathrm{Cl}(F))\geq3\), suffices. Moreover he proves that a positive proportion of inert imaginary quadratic extensions with \(\mathrm{rank}_2(\mathrm{Cl}_F)=r\) and \(\mathrm{rank}_4(\mathrm{Cl}_F)=s\) has infinite Hilbert \(2\)-classfield tower for \(r=3\), \(s=1,2\), and \(r=4\), \(0\leq s\leq3\).