Moduli of relatively nilpotent extensions

Gives the most precise available description of the p-Frattini module for any p-perfect finite group G=G_0 (Thm. 2.8), and therefore of the groups G_{k,ab}, k \ge 0, from which we form the abelianized M(odular) T(ower). \S 4 includes a classification of Schur multiplier quotients, from which we figure two points (see the html file http://www.math.uci.edu/~mfried/paplist-mt/rims-rev.html):

1. Whether there is a non-empty MT over a given Hurwitz space component at level 0; and
2. whether all cusps above a given level 0 o-p’ cusp are p-cusps. The diophantine discussions of \S 5 remind how Demjanenko-Manin worked on modular curve towers, showing why we still need Falting’s Thm. to conclude the Main MT conjecture when the p-Frattini module has dimension exceeding 1 (G_0 is not p-super singular). By 2009 there was a successful resolution of the Main Conjecture when the MT levels (reduced Hurwitz spaces) have dimension 1. http://www.math.uci.edu/~mfried/paplist-mt/MTTLine-domain.html reviews all inputs and results of the Modular Tower program starting with Books of Serre and Shimura.