Applying
Modular Towers to the Regular Inverse Galois Problem
This paper introduces two major uses of Modular Towers:
- How they encode a key question on the R(egular) I(nverse) G(alois) P(roblem).
- How they introduce towers of spaces whose properties plausibly
generalize those of modular curves.
The Braid monodromy method captures the correspondence between
solutions of the RIGP for a given group G and rational points on any
one of a sequence of Hurwitz
space components HG={Hi}, i running over an indexing set IG.
The correspondence is most effective if you can establish properties of
the set HG. Technical point used below: If a normal variety has a K point, then that K point must lie on a component
defined over K. All Hurwitz
spaces are normal varieties.
The monodromy method is based on the B(ranch) C(ycle) L(emma)
in that regular realizations of G put constraints on the conjugacy
classes C={C1,…,Cr}
for which you have regular realizations. For example, if you insist on
realizations over Q, then (as
in the previous reference), C
MUST consist of a rational union of conjugacy classes guaranteeing the
(inner) Hurwitz space H(G,C)in
with its natural map to an open subset of Pr has definition field Q. Further, H(G,C)in
MUST have an absolutely irreducible Q
component. Here are two archetypes that guide what we can expect of
this correspondence.
- You can replace G by an
explicit cover group G', so
that with C consisting of all conjugacy classes in G', with each class repeated
suitably (but not given explicitly) often, then running over just the
rational unions of such C
gives absolutely irreducible Hurwitz spaces H(G,C)in
.
- If G=An, and C= C3r
, then the indexing set IAn, C3 (analog to IG
above, but add in that the support of the conjugacy classes is just in
class of 3-cycles C3), has this property: For
each r ≥ n (resp. r =
n-1) there are two indices i+
and i- (just one value of r) that map(s) to r.
What the first means: There
infinitely many absolutely irreducible Hurwitz spaces over Q whose rational points give
realizations of G.
What the second means:
That we know exactly how to list those absolutely irreducible Q components if we restrict to
3-cycle covers. Further, for the second, when you get that explicit,
you see deeper connections between group theory and arithmetic
geometry. In this case, the connection goes all the way back to
Riemann's greatest accomplishment, the production of explicit
nondegenerate functions, identified by their evenness and
oddness.
Both results apply to make explicit statements about the absolute
galois group GQ of Q. Yet, despite the seeming
limitation on the second result, we see being explicit gives it
many applications related to 3-cycles, and so many chances to
understand the RIGP and the braid monodromy method better.
Existence of Modular Towers Forced:
We get even moreso from M(odular)
T(ower)s. Let K be any
number field. Suppose we ask about the entire collection of p-Frattini extensions of a p-perfect group G (has no Z/p
quotient but p divides |G|), say one that has been neatly
handled by the braid monodromy method. You find at first there is
nothing in the method that seems to differentiate between the conjugacy
classes for G and those for
any one of its covers.
Still, Thm. 4.4 of this paper applies the BCL in a profinite way to
consider finding any bound r0
on the number of branch
points – saying nothing about which conjugacy classes we use – for K regular realizations. It
concludes that you can't get all those p-Frattini extensions, unless you
restrict to p' conjugacy
classes (elements in them have order prime to p).
Main MT Conjecture: Thm.
4.4 continues: Even then there must exist a MT with a K point at every level. One
conjectured MT property is easy to understand, it is that a MT
shouldn't have a K point at
every level, or K points
disappear at high levels. [mt-overview.html] gives an overview of many aspects of
MTs that are like, and others that are unlike, those for modular curve
towers. As of 2006 many MTs have been shown to have the K point disappearance property. We
show this by identifying p
cusps on the MT.
Useful MTs: As [ser_gal.pdf, §7],
explains, the dihedral group part of the Main MT Conjecture is
equivalent to statements about torsion on hyperelliptic Jacobians. Yet,
the dihedral case (or its slight generalization to p supersingular groups) are but a
bit of the territory opened by MTs. The difference from p supersingular and
general p-perfect is easily
stated. In the latter case many p-Frattini
extensions of G have
non-trivial p Schur
multipliers, while in the former case that never holds. This paper is
the first to point out the general effect of this, and this paper's
Obstruction Lem. 3.2 and Schur Multipliers Result 3.3 are quoted often
later. The latter shows how a Schur multiplier at one level replicates
to higher levels, a major theme in the latter papers.
What makes MT levels often very different from modular tower levels is
the appearance of obstruction
and of what [lum-fried0611594pap.pdf,
§3.2] calls o(nly)p'
cusps. The former means that sometimes tower levels will have
components with nothing above them. The latter signifies cusps that
have no analog on modular curve towers. The existence of these
geometric objects capture a great deal of why the Inverse Galois
Problem is so difficult. Better yet, MTs captures these objects in a
way by which we can compare them with modular tower cusps.
The three Frattini Principles of [lum-fried0611594pap.pdf] are tools
for investigating which MTs might satisfy Serre's Open Image Theorem,
as §6 of that paper discusses with appropriate examples.