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:
- How all primes enter on higher rank Modular Towers (necessary to
be able to find "Hecke Operators").
- 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).
- 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).
- 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