The Main Conjecture of Modular Towers and its Higher Rank Generalization:
Published pdf file lum-fried0611594pap.pdf

The genus of projective curves  discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli Mg of genus g curves. Yet, modern applications require a data variable (function) on such curves. The resulting spaces are versions, depending on our need from this data variable, of Hurwitz spaces. A Nielsen class is a set defined by r ≥ 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog.

The Main Conjecture:  Using Frattini covers of G, every Nielsen class produces a projective system of related Nielsen classes for any prime p dividing |G|. A nonempty (infinite) projective system of  braid orbits in these Nielsen classes is an infinite (G,C) component (tree) branch. These correspond to projective systems of irreducible (dim r-3) components from {H(Gp,k(G),C)}k=0. Such a component branch is a  (G,C,p) M(odular) T(ower). The classical modular curve towers
{Y1(pk+1)}k=0 (simplest  case: G is dihedral, r=4, C are involution classes) are an avatar.

Any specific Nielsen class can support several distinct MTs, and §6.3 gives examples to illustrate this.

The (weak) Main Conjecture says, if G is  p-perfect, there are no rational points at high levels of a MT. When r=4, MT levels (minus their cusps) are upper half plane quotients covering the j-line.

Cusp types and Cusp tree on a Modular Tower: If you compactify the tower levels, you get complete spaces with cusps. The MT approach allows identifying these cusps using fairly elementary finite group theory. It should be no surprise that this generalization of modular curves uses cusps to make progress. Here are our topics.

• Identifying MTs from g-p', p and Weigel cusp branches using the MT  generalization of spin structures
• Listing cusp branch properties that imply the (weak) Main Conjecture and extracting the small list of towers that could possibly fail the conjecture.
• Formulating a (strong) Main Conjecture for higher rank MTs (with examples): almost all primes produce a modular curve-like system.
It is surprising how effectively this approach is able to identify significant properties of the cusps, by more elementary methods than traditionally used by say people who work on Siegel upper half-space (or Shimura varieties), even when such spaces are closely related to MT levels. It is the use of finite group theory, rather than reductive groups that makes this possible.

Illustrating Examples: It behooves that our examples use group theory accessible to any researcher interested in modular curves and their generalizations. Yet, they are MTs that aren't towers of modular curves;  examples understandable without terrific effort that successfully reveal their modular curve-like properties. It is especially useful that our examples hit the edge of unsolved aspects of the Inverse Galois problem.

§6 compares a rank 2 modular tower that gives all modular curves in a natural way, to another rank 2 modular tower that seems at first very similar. The first case starts with Z/2, the second with Z/3, acting on a rank two lattice. Things especially interesting:
1. How all primes enter on higher rank Modular Towers (necessary to be able to find "Hecke Operators").
2. How the Z/2 case gives all modular curves and the role of the universal Heisenberg obstruction (§6.2. Modular curve comparison for Serre's OIT).
3. Using the sh-incidence matrix in the case p=2 (§6.4.2. Graphics and Computational Tools: sh-incidence) to see why the Z/3 case isn't of modular curves (Prop. 6.12).
4. An analysis of precisely why the level 0 and level 1 spaces have more than one component (§6.3. F2×s Z/3, p = 2: Level 0, 1 components).
Item #3 is despite those reduced Hurwitz spaces having many seemingly modular curve properties. There are two components at level 0 of the Hurwitz space, but only one of those supports a (nonempty) MT. The existence of two H(arbater)-M(umford) components at level 1 is made much of (§6.4.5. Level 1 of (A4,C±32, p = 2). In the Z/3 case a result of Serre's implies a component of the reduced Hurwitz spaces carries a function, a θ-null, defined by the moduli problem. A full discussion of the θ-nulls on spaces of odd-cycle covers is in §6 of Alternating groups and moduli space lifting Invariants: description and properties of spaces of 3-cycle covers.

Mike Fried 12/10/07