  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.
