Contemporary Mathematics Volume Introduction to MODULAR TOWERS Generalizing dihedral group modular curve connections

We join Hurwitz space constructions and the universal Frattini cover of a nite group The goal is to form and apply generalizations of the towers X p X p X p of modular curves This generalization relies on the appearance of the dihedral group Dp and its companion group Zp f g in the theory of modular curves We replace Dp by any nite group G and p by any prime dividing the order of G The replacement for Zp f g is the Universal p Frattini Cover of G Diophantine motiviations include an outline for using the Ihara Drinfeld relations for the Grothendieck Teichm uller group Conjecturally this is the absolute Galois group of Q We consider how nding elds of de nition of absolutely irreducible components of Hurwitz spaces can test this There are many applications to nite elds The simplest bounds the exceptional primes to realizing any nite group G as the Galois group of a regular extension of Fp x The structure of Modular Towers connects this problem to other diophantine problems Alternating groups test the modular representation theory that appears in Part II of the paper These give a modular tower di erent from modular curves The classical link here is to theta functions with characteristic Summary To each nite groupG we can attach a projective pro nite group G the universal Frattini cover of G FrJ x xII A B Further for any col lection of r conjugacy classes C of G there is a natural moduli space Its points are equivalence classes of covers of the Riemann sphere P rami ed over r points The particular covers have geometric monodromy group G C is the set of con jugacy classes of branch cycles of the cover We conjoin these two constructions The moduli space construction applies to a natural co nal collection of nite quotients of G This produces arithmetic invariants for the theory of curve cov ers A special case uses a prime p dividing jGj and conjugacy classes C of orders relatively prime to p This p unrami ed lifting invariant G p C is com patible with terminology of Se It also produces a tower of moduli spaces We denote them formally as G p C moduli spaces Informally these are a modular tower This moduli space material is in Part III Arithmetic geometers know a special case the tower of covers X p X p X p of modular curves Points on X p correspond to pairs of elliptic curves with a cyclic p power isogeny Here G is the dihedral group Dp of order p Also r and C is four repetitions of the involution conjugacy class in Dp DFr x Part I especially xI D discusses example problems with ready applications for modular towers These include highly structured versions of the inverse Galois problem and construction of general exceptional covers Part II discusses universal Frattini covers of a nite group Each level of a modular tower comes from a quotient of a universal p Frattini cover Moduli space di culties at a particular level will have a modular representation inter pretation For example this holds for the appearance of a center in the quotient group for a particular level see The Center Hypothesis of xIII B We illustrate these matters with an application to An and cycles This is another classical