An Overview of Modular Towers and ( the Meaning of ) Profinite Geometry
For progress on the program see MTTLine-domain.html

Denote the Riemann sphere uniformized by a variable z by P1z. We can assign to covers of the r-punctured sphere P1z  ⁄ {z1,…,zr} (or to covers ramified over {z1,…,zr} = z a group G and a set of conjugacy classes C=(C1, …, Cr) of G. Riemann's Existence Theorem (RET) inverts this by producing such (G,C) covers through geometric representations of the fundamental group P1z  ⁄ {z1,…,zr}. We call the space of unordered r-tuples the configuraton space for branch point covers, and denote it by Ur.

The elementary Branch Cycle Lemma gives a necessary condition for an arithmetic (G,C) representation of the fundamental group, one factoring through the absolute Galois group of Q. For every finite group G and prime p dividing |G|, there is a universal p-Frattini cover pḠ of G [London4-Weigel.pdf, 1–4]. The natural map pḠ → G factors through any extension of G by a p group. Indeed, pḠ  is the projective limit of groups Gk (we suppress p) having this property: If H → G is a covering homomorphism with p group kernel, then the natural map Gk→ G factors through it, and Gk is minimal with this property.

Now we suppose G is p-perfect (if has no Z/p quotient) and also C consists of p' conjugacy classes (all their elements have order prime to p). From (G,C, p) we get a system of inner Hurwitz spaces {H(Gk,C)}k=0∞. Each H(Gk,C) is an affine algebraic variety and an unramified cover of Ur. An action of the linear fractional transformations PSL2 (over the complexes C) forms another version of these spaces – {H(Gk,C)rd}k=0∞ – called reduced spaces for which the configuration space is Jr=Ur/PSL2.

DEFINITION OF MODULAR TOWER: A Modular Tower (MT) attached to (G,C, p) is a projective sequence of (absolutely irreducible) components of the spaces  {H(Gk,C)}k=0∞ (or of {H(Gk,C)rd}k=0∞) [London4-Weigel.pdf, 5].  These spaces can be completed uniquely (to normal varieties) over Pr (respectively, Pr /PSL2). We call these completed spaces compact Hurwitz spaces (for which we use notation like {H^(Gk,C)}k=0∞).

MEANING OF PROFINITE GEOMETRY: The compact (reduced) Modular Tower for G a dihedral group Dp (p odd) and C four repetitions of the conjugacy class of involutions, is the classical modular curve sequence ... → X1(pk+1) → X1(pk) → ... → P1j. Notice: Implicit in this case is that there is just one component of the level k space.

Often, however, there can be several kinds of components coming from having incompatible cusp (see below) types. FS-Lift-Inv.html has a completely homological description of MTs. This also explains cohomologically, and combinatorially, the cusps – components of H^(Gk,C)   ⁄ H(Gk,C) –  for which there is also a reduced version. The phrase Profinite Geometry derives from this use of gadgets attached to p-Poincaré duality (see Comment on #1 below). 

HOW MTS ARISE: Many famous conjectures of algebraic number theory, like the Fontaine-Mazur conjecture, postulate that if you limit the ramification of extensions of Q to a finite number of primes, then the maximal extension will arise residually in a natural way. Here is the analog of this for MTs [London4-Weigel.pdf, 5-6].  Suppose you seek regular realizations of all the groups {Gk}k=0∞ , but you only those realizations with no more r0 branch points. You may take your choice of p-perfect group (say A5 with p=2, or the Monster, or even the dihedral group of order 10, with p=5) and r0 even three trillion.

Your challenge: In each case realize each Gk with no more than three trillion branch points, but you can use any conjugacy classes in any Gk you like.

Here is why I bet you can't do it. [fried-kop97.pdf,  Thm. 4.4] shows that if you can, then there must be a collection of p' conjugacy classes in G defining a MT {H(Gk,C)}k=0∞ for which each level has a Q point. This applies the Branch Cycle Lemma agreeing with our meaning for Profinite Geometry.

Further, existence of Q points at every level of even one MT contradicts the Main Conjecture(see below), and so also the Strong Torsion Conjecture. Just, however, proving you can't do this for dihedral groups would vastly generalize Mazur-Meryl. In the course of proving cases of the Main Conjecture, we also see many modular curve-like aspects of general  MTs.  Both [lum-debes09-05-06-pap.pdf] and [oberwolf-friedrep06-16-06.pdf] have more modern expositional discussions of the Main Conjecture.

HOW WE UNDERSTAND MTS FROM THEIR CUSPS: Reduced compact Hurwitz spaces have cusps. You can view them combinatorially as orbits of a cusp group (unless r=4, a specific cyclic subgroup of a braid group) acting on the defining Nielsen class. A MT (with all levels nonempty) must have several projective systems of cusps (a cusp branch).  The first division of cusps into three types p, g(roup)-p' and o(nly)-p' ([Lum; §3.1.2] for r=4; [London2-AltGps.pdf, p. 8, and App. B2 in general]). This is quite easy to use in one sense. It is usually not difficult – a piece of cake for GAP – to describe all cusps on all Hurwitz space components defined by a Nielsen class. The more difficult problem is placing the cusps within absolutely irreducible components. 

HOW MTs SHOULD RESEMBLE MODULAR CURVE TOWERS:

  1. Main MT Conjecture: Assuming a MT has all levels nonempty and uniformly defined over a number field K (a K MT), high levels (H(Gk,C)rd for k large) should have no K points.
  2. Relation to the Strong Torsion Conjecture (STC): The STC is a conjecture, generalizing the famous Mazur-Meryl result on elliptic curve, bounding the K (a number field) torsion points on abelian varieties of dimension d. According to Cadoret, the STC implies the Main Conjecture.
  3. Cusps should control the geometry of MTs: Three Frattini Principles (FPs) are at the heart of explicit analysis of MTs, each principle governs the behavior of a different type of cusp [Lum, Princ. 3.5, 3.6 and §4.5]. 
  4. Some systems of MTs should resemble modular curve towers: This should happen even though dihedral groups are such special cases of finite groups.
  5. Low MT levels should produce never before seen results: For any finite (nonabelian) simple group G (even A5) and p any prime dividing |G|, its first characteristic Frattini cover G1has never has a Q regular (or even ordinary, even for p=2) realization as a Galois group. The Q rational points on any level 1 MT for G are exactly about regular realizations of G1.
Comments on the list above:
#1: For r=4 infinitely many MTs have been shown to satisfy the conclusion of the Main Conjecture Main-MT-Conj.html, including those given by pure-cycle Nielsen classes with all conjugacy classes having odd-order elements (automatically G=An), and covers in the class have genus 0 (see oneorbit.pdf). These results build on results of Liu and Osserman). This is based on the Fried-Serre lifting invariant FS-Lift-Inv.html using Weigel's p-Poincaré Duality result [London4-Weigel.pdf, 7–11].

#2: Little is known about the STC. Each MT case of the Main Conjecture is proven by explicit means that therefore explicitly reflects on the STC. For example among the Liu-Osserman examples, an infinite number of cases of level 0 reduced Hurwitz spaces are genus 0 curves. From [Lum, Thm. 5.1]
we have only to show there are at least two p cusps (and one other cusp) at some level. Further, in these particular cases, the Main Conjecture fails without this. For example, in these cases, when p=2, there are no p cusps at level 0, but the lifting invariant produces them at level 1.

#3: Additional on the three FPs: FP1 says a projective systems of p cusps define increasingly higher powers of p ramification; FP2 says a g-p' cusp at level 0 always defines a MT (nonempty at all levels); and  FP3 is an if and only if condition for an o-p' cusp at one level to have only p cusps above it. 

#4: We can expect modular curve-like attributes of a MT that has a cusp sub-tree – called a spire – equivalent to a modular tower cusp tree. An infinite number of the Liu-Osserman examples have been shown to have this property. Also, a system of MTs can resemble the usual systems of modular towers if there are Hecke-Like operators and cusp forms and other modular curve-like gadgets. [Lum, §6] develops two very similar looking cases that start, respectively with the groups on Z/2 and Z/3 acting on a rank 2 free group. The first case is the full system of modular curves in disguise [London4-Weigel.pdf, 12–13]. Using that presentation, the second case is a whole new system of MTs (all consisting of curves, none are modular curves) that seem to have all the expected properties [London4-Weigel.pdf, 15]. 
#5:  [Lum, §6.4]  shows there are two genus 1 components H+ and H-  at level 1 (among six total components) in a particular MT with G=A5 with  this property. The only possible change for the regular realization of G1(A5) with infinitely many PSL2-inequivalent covers with four branch points (no matter what are the conjugacy classes) is for H+ to have infinitely many Q points. The Liu-Osserman cases provide manifold examples like this.  The production of MT levels that have properties radically different from modular curves comes from the appearance of Schur multipliers of subquotients of the groups Gk. Semmen ([lum-semmen04.pdf] and [Lum, Prop. 3.12] – MTs with g-p' cusps at every level) uses density theorems to show the existence of such Schur multipliers to produce cusps of various types on MT levels. These give precise challenges to the STC. 

Mike Fried 3/30/2009 mfri4@aol.com mfried@math.uci.edu