Moduli of relatively nilpotent algebraic extensions

$p $ , and finite-perfect group $G $ , high levels of a $(G,p) $ Modular Tower have no rational points. When $p $ is an odd prime and $G $ is the dihedral group $D_{p} $ we call the towers Hyper-modular [BFr02] proves cases of the Main Conjecture for Modular Towers with $p=2 $ and $G $ an alter-nating group. We use differences between Hyper-modular and general Modular Towers to give new moduli applications. Their similarities give these applications insights successful with modular curves. The universal-Prattini cover of $G $ and acollection of $p’ $ conjugacy classes define aparticular Modular Tower’s levels. The number of com-ponents at alevel and how cusp ramification grows from level to level relate to the appearance of Schur multipliers. [BFr02] applies formulas of the author and Serre to spin covers to handle the case $p=2$. We give here an exposition on going beyond that analysis. As with modular curves, the tower levels are moduli spaces. We use aDemjanenkO-Manin result and aconstruction from [Fri95] to more directly approach rational points than does Falting’s Theorem. 1991 Mathematics Subject Classification. Primary llF32, llG18, llR58, $14\mathrm{G}32$;Secondary $20\mathrm{B}05,20\mathrm{C}25,20\mathrm{D}25,20\mathrm{E}18,20\mathrm{F}34$