Give the definition of the cusp and component tree on a Modular Tower. Establish the idea that the surest way to construct the projective sequence of Modular Tower components is to describe a cusp branch given by a g-p' cusp: form a g-p' cusp branch. It is a working hypothesis that a component branch, with all levels defined over some fixed number field  exists if and only if there is a g-p' cusp branch defining it. 

The remainder of this definition introduces the definition that arise in the two most prevalent questions (items I and II below), and how we approach them today. 

I. What is the present route to establishing the Main Conjecture on Modular Towers for r=4?  

Suppose we have a component branch B' (notation from Luminy [Lum, \S3]).  Assume B' is defined (as the \bar M_4 orbit) by a g-p' cusp branch B, which we regard as the \Cu_4 (cusp group orbit) of a g-p' cusp branch formed from a projective system \tilde \bg=\{\bg_k\}_{k=0}^\infty of Nielsen classes. Such a g-p' cusp branch is sufficient, and conjectured to be necessary, to define a component branch defined over a number field K, denoted a \PSC_K. 

Procedure: Apply \Cu_4\circ\sh\circ \Cu_4 (with \sh the shift) to \tilde \bg. Call the result (\tilde \bg)\Cu^{-1}_4. Decide when (\tilde \bg)\Cu^{-1}_4(\tilde \bg) at some level k contains at least three p cusps. There is a characterization of those MTs that fail the Main Conjecture when r=4. None can three p cusps at any level[Lum, Thm. 5.1]. Notice: For any r, the similarly defined union of cups (\tilde \bg)\Cu_r^{-t} (\Cu^{-1}_r applied t times) running over all t give the full component branch of the \MT defined by \tilde \bg.    

For example, the Main Conjecture follows if at some level (\tilde \bg)\Cu^{-1}_4(\tilde \bg) contains a p cusp [MC,\S2]. In this case we call (\tilde \bg)\Cu_4\circ\sh\circ \Cu_4 a spire. If this criterion applies to a g-p' cusp at, say, level k_0, we say the branch has a spire at level k_0. The definitions here were en-spired by this being the case for the modular curve tower \{X_1(p^{k+1}\}_{k=0}^\infty.  Here the g-p' cusp branch is the shift of an H-M cusp branch, and the whole cusp tree comes in one fell swoop from this procedure.   

[MC] considers the Main Conjecture for the prime p=2 (and many other primes as well) for Nielsen classes given by odd order branching and containing an H-M rep. It uses [Lum, Princ. 4.25] (called Frattini Principle 3) and the Fried-Serre Lifting invariant to decide exactly when there is a spire by level 1. Combining the results of [Liu-Oss] and [AltGps], this allows us to point to explicit cases of yes we know, or this is what we need to know about \bar M_4 orbits and lifting-invariants, to prove the Main Conjecture in generality.  

II. When does a g-p' cusp branch define a \PSC_K? 

We know that a g-p' cusp branch for any r defines a \MT. We also know there are special cases (even with r=3) where there is a \MT, and no g-p' cusp branch. In the latter cases, however, the \MT is not a \PSC_K for any K. On the other hand, we know that if there is a uniform in k bound on the \bar M_r orbits of H-M cusps at all levels, then any \MT defined by an H-M cusp branch is a \PSC_K for some number field K (a consequence of [MT1, Thm. 3.28]).  
 
[BF02, Thm. 6.1] shows this.  If every level of a \MT has a \bQ_\ell point, for some prime \ell, then when you compactify the \MT, there will be a projective system of \bQ_\ell points, centered around a cusp branch. [DE] have used the criterion of [MT1, Thm. 3.28] to produce \MT s which have \bQ_\ell points for all \ell not dividing |G|. The point of their program interprets the shift of an H-M cusp branch as a projective sequence of Harbater patchings. 

Among the questions are these? 

1. Will the results about (the shift of) H-M cusps extend to g-p' cusps? For example, is there a Harbater patching interpretation of g-p' cusps, and will the [DE] results extend to these using a version of [MT1, Thm. 3.28]? 

2. Could it be that there are an unbounded number of braid orbits of H-M reps. in the projective system of Nielson classes defined by a \MT?