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:
- 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.
- 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.
- 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].
- Some systems of MTs
should resemble modular curve towers: This should happen
even
though dihedral groups are such special cases of finite groups.
- 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