Modular
Tower Time Line
This file consists of a
list of papers (by year) for
particular
events in the development of M(odular)T(ower)s.
I've divided these into three periods.
-
Lessons from Dihedral groups – Before '95:
- Construction
and Main Conjectures on MTs
– '95–'04:
- Progress
on the Main MT Conjectures – '05–'09:
Each item connects to a fuller explanation of the history and
significance of the contribution. Three html files provide handy
reminders on basics guiding progress on M(odular) T(ower)s. We refer to
sections in them.
- The R(egular) I(nverse) G(alois) P(roblem),
its literature and relation to Nielsen classes and the MT
conjectures: RIGP.html.
- Nielsen
classes, a genus generalization that separates
sphere covers into recognizable types. Denoted Ni(G,C),
for conjugacy classes C=C1,…,Cr
of a finite group G. The set consists of generating
r-tuples (g1,…,gr),
in (some order in) the classes of C, satisfying the
product-one condition g1…gr=1.
R(iemann)-H(urwitz) gives the corresponding sphere cover genus.
- The B(ranch)
C(ycle) L(emma) tying
definition fields of covers (and their automorphisms) to branch point
locations: Branch-Cycle-Lem.html
These files let us speak of progress on MTs without
great elaboration. Other such files, building on the three above,
explain more what the papers are about. Finally, this page ends with
further references that give results technically related to MTs
in the abstracts to papers.
Individual MTs have an attached prime (usually
denoted p). When the MT data
passes a lift invariant test (coming up in [Se90a]), then the MT
is an infinite (projective) system of nonempty levels
(result stated precisely in [Fr06]
below). Each level is a normal algebraic variety, all levels covering
the classical j-line when r=4,
and an r-3 dimensional generalization of it for
larger r. Indeed, the levels are reduced Hurwitz spaces.
Most modern applications of algebraic equations requires more data than
is given by the moduli of curves of genus, or even of Shimura
varieties. Hurwitz spaces, however, do carry such data, and unlike
rougher use of curve families, they retain the virtue of having moduli
properties.
MTs come with what we call the usual
MT conditions: Each has an
attached group G, and a collection of r
conjugacy classes, C, in G with
p' elements: of orders prime to p.
Further, G is p-perfect: p
divides its order, but G has no surjective
homomorphism to the cyclic group Z/p.
For G a dihedral group, with p
odd and r=4, we are in the case of modular curve
towers. So, MTs generalizes modular curves towers.
Since there are so many p-perfect groups, the
generalization is huge.
The Main Conjectures are these: High tower levels
have general type; and even if all levels have
definition field a finite extension K of Q,
still K points disappear (off the cusps) at high
levels. Bringing particular MTs alive plays on
cusps, as do modular curves. Part III of our TimeLine includes precise
comparison of MT cusps with those of modular curve
towers, consequences of this, and two different methods that have given
substantial progress on the Main Conjectures.
The intimacy between the Inverse Galois Problem and MTs
gives an elementary urgency to identifying their levels. A graphical
device – the sh(ift)-incidence matrix – displays
these cusps, despite only for modular curves having congruence
subgroups at our disposal.
Shimura varieties are another generalization of modular curves. They
also have towers, and primes, etc. The connection of abelian varieties
to MTs has been made in several ways. There is one
easily stated standout: The S(trong) T(orsion) C(onjecture) on torsion
points on abelian varieties implies the rational point conjecture on MTs
(see [CaDe08].
It is, however, by labeling MT cusps that we see
tools for generalizing Serre's Open Image Theorem,
especially through recognizing p-Frattini
covers and using reduced Hurwitz
spaces (defining tower levels). These are examples
that show my web site has help for related topics.
I. Lessons from Dihedral
groups – Before '95:
This section goes from well-known projects to their connection with the
MT program. Its reference to [DeFr94]
– which included e-mail exchanges with Mazur – sets the stage for the
recent-years' division of the project into two branches. The arithmetic
has concentrated in the hands of Pierre Dèbes and his collaborators
Cadoret, Deschamps and Emsalem. The structure of particular MTs
– based on homological algebra and the geometry of the spaces (cusps
and components) – follows my papers and my relation to Bailey,
Kopeliovic, Liu-Osserman, Semmen, Serre and Weigel. The effects of
quoted work of Ihara, Matsumoto and Wewers, all present at my first
talks on MTs, is harder to classify.
[Sh64] G. Shimura, Abelian
Varieties with Complex Multiplication and Modular Functions,
Princeton U. Press, 1964 (with Y. Taniyama), latest edition 1998. ✺ I
studied this source during my two year post-doctoral 67-69 at IAS.
Standout observations: Relating a moduli space's properties to objects
represented by its points, through the Weil co-cycle condition,
especially in canonically finding the definition fields of a tower.
Results from it: The Branch-Cycle-Lem.html
and its early uses (dav-red.pdf,
UMShortStory.html
and HurwMonGG.pdf
for problems not previously considered as moduli-related. Also, a model
for producing automorphic functions
on Hurwitz spaces.
[Se68] J.P. Serre, Abelian
l-adic representations and elliptic curves, New York, Benj.
Publ., 1968. ✺ That the monodromy groups of towers of modular curve
covers have a Frattini property, suggesting the general expectation for
the action of GQ
on a projective system of points over a j-line
point in Q. Use of the p cusps
on modular curves, and their attachment to Tate's p-adically
uniformized elliptic curves to decipher the GQ
action when j is p-adically
"close to" ∞. For K a complex quadratic extension
of Q, the technical point of complex multiplication
is the discussion of 1-dimensional characters of GK
on the Qp
vector space – Tate module, or 1st p-adic étale
cohomomology – of an elliptic curve with complex multiplication by K.
On the 2nd p-adic étale cohomomology it is the
cyclotomic character, while on the 1st there is no subrepresentation of
any power of the cyclotomic character. Roughly: A small part of Abelian
extensions of K are cyclotomic, a result that
generalizes to higher dimensional complex multiplication in [Sh64]. As [Ri90]
emphasizes Serre's book is still relevant, especially for the role of
abelian characters, those represented by actions on Tate modules (from
abelian varieties), and those not.
Results from it: Recognition that the modular curve Frattini property
is inherited from a Frattini property for sequences of dihedral groups;
so the proper generalization of Serre's main result to MTs
should exploit this [Fr06, ?6.3].
The full (and comfortable) completion of Serre's O(pen) I(mage)
T(heorem)
awaited replacement of an unpublished Tate piece by ingredients from
Falting's Thm. [Fa83] (as in [Se97b]).
[Fr78] M. D. Fried, Galois groups
and complex multiplication, Trans. Amer. Math. Soc.
235 (1978), 141–163. MR MR472917 (81c:14015) ✺
Identification of modular curves with Hurwitz spaces, and the
classification of the Schur-conjecture for rational functions as
equivalent to part of [Se68].
Generalizing the Inverse Galois Problem, to geometric-arithmetic
realizations, using Hurwitz spaces.
Down-to-Earth result from it: From the GL2 part
of the OIT, explicit production from each elliptic curve over Q
with non-integral j-invariant, infinitely many
primes p and a degree p2
rational function f decomposing, over the algebraic
closure, into two rational functions of degree p,
with no such decomposition over Q ([GMS03]
and [Fr05, Prop. 6.6]). Abstract
results from it: Over suitable fields, you can achieve any
geometric-arithmetic realizations, giving the first proven group
presentation of GQ.
Then, formulating the likely generalization of Shafarevich's Cyclotomic
Conjecture GQpresentation.pdf.
[Ih86] Y. Ihara, Profinite
braid groups, Galois representations and complex multiplications,
Ann. of Math. (2) 123 (1986), no. 1, 43–106. MR MR825839 (87c:11055) ✺
The similar titles with [Fr78] gives away the similar influence of
Shimura. Both played on using and interpreting braid group actions, a
monodromy action capturing data from curves, versus from abelian
varieties. A moduli interpretation of "complex multiplications"
required to generate the field extension giving the second commutator
of GQ.
Down-to-Earth result from it: Generating the second commutator
(arithmetic) extensions using Jacobi sums derived from Fermat curves.
Abstract result from it: An interpretation of
Grothendieck-Teichm?ller on towers of Hurwitz spaces [IM95].
[Se90a] J.P. Serre, Relèvements
dans Ãn, C. R. Acad. Sci.
Paris 311 (1990), 477–482. ✺ This suggested a general context for
viewing mysterious and previously inaccessible central Frattini
extensions of groups, yielding to the braid technique – in this case a
formula for deciding if a regular realization of An
extends to the Spin cover Spinn
(what Serre calls Ãn) of An.
A braid orbit O
in Ni(An,C),
with C of odd-order elements, passes the (spin)
lift invariant test if the natural (one-one) map Ni(Spinn,C)
→ Ni(An,C)
maps onto O. Main Result: If the genus attached to
Ni(An,C)
is 0, then the test depends only on the Nielsen class and not on O.
Results inspired by it: Formulation of the main connectedness result on
Hurwitz spaces CFPV.html.
Classification and application of Frattini central extensions of
centerless groups [Fr02, § 3 and
4].
[Se90b] J.-P. Serre, Revètements à
ramification impaire et thèta-caractèristiques, C. R. Acad.
Sci. Paris 311 (1990), 547–552. ✺ Example result: A formula for the
parity of a uniquely defined half-canonical class
on any odd-branched Riemann
surface cover of the sphere. It is the sum mod 2 of an invariant
depending only on the Nielsen class of the cover, and the spin lift
invariant mentioned in [Se90a].
Result from it: Production of Hurwitz-Torelli automorphic functions on
specific Hurwitz spaces through the production of even theta-nulls [Fr09a, § 6.2].
[DeFr94] P. Dèbes and M. D.
Fried, Nonrigid situations in
constructive Galois theory, Pacific Journal 163(1994), 81–122. ✺
Example result: Formulation of the Main MT
conjecture for dihedral groups like this. Suppose, for some prime p,
there are Q regular realizations of all the
dihedral groups {Dpk}k=0∞
with some bound r0 on their
number of branch points. Then (equivalently), the Branch-Cycle-lemma
implies there is an even integer r1
(≤ r0) and for each k,
there is a dimension (r1-2)/2
hyperelliptic Jacobian (over Q) with a μ(pk)
point for each k (≥ 0). The Involution
Realization Conjecture says the last is impossible: There is
a uniform bound as n
varies on μ(n) torsion points on hyperelliptic
Jacobians of a fixed dimension, over any given number field. (The only
proven case, r1=4, is the
Mazur-Merel result
bounding torsion on elliptic curves.) If a subrepresentation of the
cyclotomic character occurred on the p-Tate module
of a hyperelliptic Jacobian (see [Se68]),
the Involution Realization Conjecture would be blatantly false.
Result from it: Formulation of the general Main MT
conjecture [FrKop97]. Still
missing result: Find μ(n) torsion points on any
hyperelliptic Jacobian for all – even infinitely many – n s.
Back to Top
II. Construction and Main
Conjectures on MTs –
'95–'04:
The extension of a p-perfect finite group G
called its universal p-Frattini
cover has kernel a pro-free pro-p group of
finite rank. Mod out by the kernel's commutator subgroup to get its abelianization.
Both extensions have characteristic series of quotients, respectively, ΓG,p=
{Gk}k=0∞
and ΓG,p,ab ={Gk,ab}k=0∞.
We can use either series to define a tower of Hurwitz spaces.
Either finding rational points on tower levels, or proving they don't
exist, makes sense whichever series we use. Clearly, however, it is
more challenging to find rational points on level k
defined from ΓG,p. Similarly,
it is more affirmative of the Main Conjecture, if we find there are no
rational points on some level defined from ΓG,p,ab.
That is how the progress appears below. Results supporting the Main MT
conjectures are about the abelianized towers. Results providing points
on tower levels are for the series ΓG,p
defining the full tower.
[Fr95] M. D. Fried, Introduction
to Modular Towers: Generalizing dihedral group–modular curve
connections, Recent Developments in the Inverse
Galois Problem, Cont. Math., proceedings ofAMS-NSF Summer
Conference 1994, Seattle 186(1995),111–171.
✺ Formulation of the MT levels, based on the
characteristic quotients of the universal p-Frattini
cover of a finite group. Properties of the universal p-Frattini
cover that translate to fine moduli. A generalization of the lift
invariant, proof that it is a braid invariant, and how it acts under
the absolute Galois group. A criterion for finding Q
components – called
Harbater-Mumford – of Hurwitz spaces, so
guaranteeing existence of projective sequences of absolutely
irreducible Q components on MTs,
beyond those of modular curves.
[FrKop97] M. D. Fried and Y. Kopeliovic,
Applying
Modular Towers
to the Inverse Galois Problem, Geometric Galois
Actions II Dessins d'Enfants, Mapping Class Groups and Moduli 243,
London Mathematical Society Lecture Note series, (1997) 172–197. ✺
Suppose G is a group with many known regular
realizations; maybe An
semidirect product some finite abelian group: see RIGP.html, § IV.1.
Consider, for some prime p for which G
is p-perfect if there are regular realizations of
the whole series of ΓG,p,ab
over some number field K.
Thm. 4.4 asks this under the condition that all such have a uniform
bound on the number
of branch
points – saying nothing about which conjugacy classes we use.
Conclusion: Such regular realizations are only possible by restricting
to p'
conjugacy
classes. Even then, there must exist a MT over K
with a K
point at every level. This geometric Fontaine-Mazur analog [Fr06b], generalizes for each such G
the Involution Realization Conjecture [DeFr94]
for dihedral groups.
[We98] S. Wewers, Construction
of Hurwitz spaces,
Thesis, Institut für Experimentelle Mathematik 21 (1998), 1–79. ✺
Develops a compactification of Hurwitz spaces based on the stable-compactification
theorem. Allows a standard comparison – contrasting with the
group theoretic use of specialization sequences in [Fr95, Thm. 3.21] – for labeling
Harbater-Mumford cusps as lying on Harbater-Mumford components. Both
compactifications are compatible with the MT
construction (they form natural projective systems), and they support
that only for primes dividing |G| can the system
have bad reduction.
[BaFr02] P. Bailey and M. D.
Fried, Hurwitz
monodromy, spin separation and higher levels of a Modular Tower,
Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. 70
of AMS (2002), 79–220. arXiv:math.NT/0104289 v2 16
Jun 2005. ✺ Computes everything of possible comparison with modular
curves about level one (and level 0) of
the M(odular) T(ower) attached to A5
and four repetitions
of the conjugacy class of 3-cycles. Shows the Main MT Conjecture for
it: No K points at high levels (K
any number field). Level 0 has one component of genus 0, while level
one has two components, of genus
12 and genus 9. Concludes with a conceptual
accounting of all cusps, and all real points on the whole tower (none
on the genus 9 component).
Visualization of cusps on both levels uses a new pairing on them, based
on Nielsen classes: The sh(ift)-incidence
matrix (§ 4 of CSCshInc.pdf
has many examples, including level 1 of modular curves). A version of
the spin cover (extending the domain of use of [Se90a])
obstructs anything beyond level 1 for the genus 9 component. Much is
made of this argument: Any prime l of good
reduction, for which there are Z/l
points at each level of a MT, would automatically
give the trivial power of the cyclotomic character acting on a Tate
module, as disallowed in [Se68].
[Fr02] M. D. Fried, Moduli of relatively
nilpotent extensions, Communications in Arithmetic
Fundamental Group, Inst. of Math. Science Analysis, vol. 1267, RIMS,
Kyoto,
Japan, 2002, pp. 70–94. ✺ Moduli of relatively nilpotent
extensions, Inst. of Math. Science Analysis 1267,
June 2002, Communications in Arithmetic Fundamental Groups, 70–94. ✺
Gives a precise description of the p-Frattini
module for any p-perfect G (Thm. 2.8), and
therefore of the sequence ΓG,p,ab.
§
4 labels Schur multiplier types, especially those called antecedent.
Example: In MTs where G=An,
the antecedent to the level 0 spin cover affects MT
components and cusps at all levels ≥ 1 (see [Fr06]).
[Sem02] D. Semmen, The
Frattini module and p'-automorphisms of free pro-p groups,
Comm. in Arith. Fund. Groups, Inst. Math/Sci Analysis 1267
(2002), Kyoto University, RIMS (2002), 177–188. ✺ Striking challenges
to the Inverse Galois problem arise by using any one p-perfect
group, and
analyzing characteristic p-Frattini extensions and
the components of their corresponding Hurwitz spaces. In lieu of the CFPV.html result
and [Fr95, Thm. 3.21], the most
serious phenomenon in unexplained Hurwitz space components – making it
difficult to identify definition fields – comes from nonbraidable outer
automorphisms of groups. Such have occurred at several level 1 MTs,
producing two separate Harbater-Mumford components.
Here are techniques for computing p-Frattini
extension outer automorphisms. Then, in cases from [BaFr02,
? 9] (especially where G=A4,
p=2 and the reduced Hurwitz space components have
genus 1) it identifies
the non-braidable outer automorphism.
[DeDes04] P. Dèbes and B.
Deschamps, Corps
ψ-libres et thèorie inverse de Galois infinie, J. für die
reine und angew. Math. 574 (2004), 197–218. ✺ If the arithmetic Main MT
were wrong, then there would be a finite group G
satisfying the usual
conditions
for p and C so that for some
number field K, the corresponding MT
would have a K point at every level. Using
compactifications of the MT levels (as in [We98]), for almost
all primes l
of K, this would give a projective system of OK,l
(integers of K completed at l)
points on cusps. The Main Results here considered what MTs
(and some generalizations) would support ZK,l
points for almost all l (inverse to [BaFr02]) using Harbater
patching.
Back to Top
III. Progress on the Main MT
Conjectures – '05–'09:
As with modular curves, the actual MT levels come
alive by recognizing moduli properties attached to particular
(sequences) of cusps. It often happens with MTs
that level 0 of the tower has no resemblance to modular curves, though
a modular curve resemblance arises at higher levels.
Level 0 of alternating group towers illustrate: They have little
resemblance to modular curves. Yet, often level 1 starts a subtree of
cusps that contains the cusptree of modular curves. We can see this
from a (preliminary) classification of cusps [Fr06,
?3]. By striving for appropriate generalizations of Serre's O(pen)
I(mage) T(heorem) to MTs, present MT
projects are entwining the general theory of abelian varieties with
properties of finite simple groups.
[We05] T. Weigel, Maximal
l-frattini quotients of l-poincare duality groups of dimension 2,
volume for O.H. Kegel on his 70th birthday, Arkiv der
Mathematik--Basel, 2005. ✺ [Se97a, I.4.5] extends the classical notion
of Poincar? duality to any pro-p group. Especially
it was applied to the pro-p completion of the
fundamental group of a compact Riemann surface of any given genus. This
paper uses the extended notion, intended for groups that have pro-p
groups as extensions of finite groups. Main Result: The p-Frattini
cover ΓG,p (and ΓG,p,ab)
is a p-Poincar? duality group of dimension 2.
Result from it [Fr06, Cor. 4.19]
(compare with the statement in [Se90a]):
Assume the usual MT
conditions, and let RG,p
be the maximal p central Frattini extension of G.
Then, there is a (nonempty) abelianized MT over the
Hurwitz space component corresponding to O if and
only if the natural (one-one) map – compare with [Se90a]
– Ni(RG,p,C)
→ Ni(G,C) is onto O.
[DeEm05] P. Dèbes and M.
Emsalem, Harbater-Mumford
Components and Hurwitz Towers, J. Inst. of
Mathematics of Jussieu (5/03, 2005), 351–371. ✺ Continuing the results
of [DeDes04], based on [We98], ties together the notions of
Harbater_Mumford components and the points on cusps that correspond to
them, connecting several threads in the theory. As an application, they
construct, for every projective system Gn,
a tower of
corresponding Hurwitz spaces, geometrically irreducible and defined
over Q (using the criterion of [Fr95,
Thm. 3.21]), which admits
projective systems of points over the Witt vectors with algebraically
closed residue field of Zp,
avoiding only those p dividing some |Gn|.
Applied to MTs and the sequence ΓG,p,
the results are much stronger. This is done explicitly using
Harbater-Mumford cusps (as in [We98]),
with the primes dividing |G|
and the cyclotomic extension defined by the orders of elements in C
to consider. [Fr06, Fratt.
Princ. 2] says existence of a g-p' cusp defines a
regular realization of ΓG,p
over any algebraic closure of Q in the Nielsen
class, and likely this is if and only if. The approach to more precise
results has been to consider a Harbater patching converse:
Identify the type of a g-p' cusp that supports a
Witt-vector realization of ΓG,p.
[De06] P. Dèbes, Modular
towers: Construction and diophantine questions,
Luminy
Conference on Arithmetic and Geometric Galois Theory), vol. 13,
Seminaire et Congres, 2006. ✺ This exposition starts with expositions
on [DeFr94], [FrKop97, [DeDes04]
and [DeEm05]. One worthy goal
would replace the use of full Hurwitz spaces in a tower with lower
dimension, maybe even 1, spaces with the potential of giving regular
realizations of some of the groups in, say, the series ΓG,p.
For this there is the result of Cadoret [Ca06]:
Such new varieties – curves or surfaces – are obtained as subvarieties
of
the HM-components
by specializing all branch points but one or two. To preserve
irreducibility requires an (intricate) transitivity condition of some
braid action, achievable with a specific list of restrictions by some
groups G.
[Fr06] M. D. Fried, The Main
Conjecture of Modular Towers and its higher rank generalization,
in Groupes de Galois arithmetiques et differentiels (Luminy 2004; eds.
D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13
(2006), 165–233. ✺ Cusp types and Cusp tree on a Modular Tower: If you
compactify the tower levels, you get complete spaces, with cusps lying
on their boundary. The MT
approach allows identifying these cusps using elementary finite group
theory. They are of three types (? 3.2.1): p-cusps,
g(roup)-p' and o(nly)-p'.
Modular curve towers have only the first two types, with the g-p'
cusps on them the special kind called shifts of
H(arbater)-M(umford). Let O be a braid orbit on Ni(G,C).
- There is a full MT over the Hurwitz
space component corresponding to O if O
contains a g-p' representative (no need to check
central Frattini extensions).
- When r=4, there is a small list of
possibilities for MTs that could fail the Main
Conjectures (Thm. 5.1). Example modular curve generalization: If O
contains a H-M cusp that is also a p-cusp, the Main
Conjectures hold explicitly for any MT over O.
- Generalizing the precise MT criterion
given in [We05, Princ. 4.23]
gives a lift invariant criterion for p-cusps to lie
above o-p' cusps.
[LO08] F. Liu and B. Osserman, The
Irreducibility of
Certain Pure-cycle
Hurwitz Spaces, Amer. J. Math. # 6, vol. 130
(2008), 1687–1708. ✺ Showed the absolute Hurwitz
spaces of pure-cycle (elements in
the conjugacy class have only one length ≥ 2 disjoint cycle) genus 0
covers have one connected component. There is a conspicuous overlap
with the 3-cycle result of [Fr09a,
Thm. 1.3], the case of four 3-cycles in A5.
The impression from [LO08; §5] is that all these Hurwitz spaces are
similar, without significant distinguishing properties. [Fr09b, §5], however, dispels this
impression. First by noting that subsets of these Nielsen classes can
have differing inner Hurwitz spaces, varying in having one or two
components. Then, by detecting seriously diverging behaviors in their
level 1 cusps.
[CaDe08] A. Cadoret and P.
Dèbes, Abelian obstructions in inverse
Galois theory, Manuscripta Mathematica, 128/3
(2009), 329–341. ✺ If a finite group G has a
regular
realization over Q, then the abelianization
of its p-Sylow subgroups has order (pu)
bounded by an expression in their index m in G,
the branch point number r and the smallest prime l
of good reduction of the cover. This is a new constraint for the
regular inverse Galois problem. To whit: If pu
is large compared to
r and m, the covers branch
points must coalesce modulo some prime l; an l-adic
measure of proximity to a cusp on the corresponding Hurwitz space.
A striking conjecture: Some expression in r and m,
independent of l, bounds pu.
This follows from the S(trong) T(orsion) C(onjecture) on abelian
varieties, and it gives forms of the Main MT
conjecture.
[CaTa09] A. Cadoret and A.
Tamagawa, Uniform
boundedness of p-primary torsion of Abelian Schemes, preprint
as of June 2008. ✺ Let χ: GK
→ Zp*
be a character, and A[p∞](χ)
the p-torsion on an abelian variety A
on which the action is through χ-multiplication. Assume χ does not
appear as a subrepresentation on any Tate module of any abelian variety
(see [Se68, [DeFr94]
and [BaFr02]). Then, for A
varying in a 1-dimensional family over a curve S
defined over K, there is a uniform bound on |As[p∞](χ)|
for s ∈ S(K). In particular,
this gives the Main MT conjecture when r=4.
Further observations:
- The §5.2 result says: If you have a p-Frattini
cover of G with kernel having Zp
as a quotient, then the corresponding version of a MT
has a level with no K points. Examples of [Fr06, §6.3] show that whether or not
this applies to the whole abelianized p-Frattini
tower depends on G.
- [Fr09b, Prop. 5.15]
displays a MT level where the genus exceeds 1. So, [Fa83] implies this level has, for any
K, but finitely many rational points. While more
explicit than this §5.2 result, ultimate bounds for either method
depend on conjectures like those in [CaDe08].
[Fr09a] M. D. Fried, Alternating groups
and moduli space lifting Invariants, description and
properties of spaces of 3-cycle covers, Arxiv #0611591.
01/04/09 To appear in Israel J. ✺ The paper's main theorem strengthens
a theorem of Fried-Serre on deciding when sphere covers with odd-order
branching lift to unramified Spin covers. Each
component of such a Hurwitz space carries a canonical half-canonical
class. In many cases, including the 3-cycle cases, it uses [Se90b] to separate components
precisely according to the evenness or oddness of these half-canonical
classes. Example corollaries: To produce Hurwitz-Torelli
automorphic functions on Hurwitz spaces, and to draw Inverse Galois
Conclusions (see §4 of RIGP.html).
[Fr09b] M. D. Fried, Connectedness of families
of sphere covers of An-type,
Preprint as of June 2008. ✺
Gives stronger results in Liu-Osserman cases, by considering the inner
(rather than absolute) Hurwitz spaces. Prop. 5.15 uses the sh-incidence
matrix to display cusps, elliptic fixed points, and and genuses of the inner
Hurwitz spaces in
two infinite lists of [LO08]
examples. (? 4 of CSCshInc.pdf
compares these examples with applying sh-incidence
to level 1 of modular curve towers.) In one there are two level 0
components (conjugate over a quadratic extension of Q),
and for the other just one. Further,
applying #3 of [Fr06], the
nature of the 2-cusps in the MTs over them are very different. None
have 2-cusps at level 0. For the list with level 0 connected, the tree
of cusps, starting at level 1, contains a subtree isomorphic to the
cusp tree on a modular curve tower: it has a spire.
For the other list, there are 2-cusps, though not quite like those of
modular curves.
A spire makes plausible a version of Serre's O(pen) I(mage) T(heorem)
on such MTs. The goal of recognizing p-cusps
is a key to showing high MT levels have general
type, and gives an approach to the Main Conjectures for r≥
5.
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Extra Bibliography
[Ca06] A. Cadoret, Harbater-Mumford
subvarieties of Hurwitz moduli spaces of covers,
Math. Annalen, 2006.
[Fa83] G. Faltings,
Endlichkeitsätze Über
Zahlkörpern, Invent. Math. 73 (1983), 349–366.
[Fr05] M. D. Fried, The
place of exceptional covers among all diophantine relations,
J. Finite Fields 11 (2005) 367–433.
[Fr06b] M. D. Fried, Regular
realizations of p-projective quotients and modular curve-like towers,
Oberwolfach report #25, on the conference on pro-p
groups, April (2006), 64–67. Also available at the
conference archive.
[GMS03] R. Guralnick, P. Müller
and J. Saxl, The rational function analoque of a question of
Schur and exceptionality of permutations representations,
Memoirs of the AMS 162 773 (2003), ISBN 0065-9266.
[IM95] Y. Ihara and M. Matsumoto,
On Galois actions on profinite completions of braid groups,
Recent Developments in the Inverse Galois Problem, Cont. Math.,
proceedings ofAMS-NSF Summer Conference 1994, Seattle 186(1995),173–200.
[Me90] J.-F. Mestre, Extensions
règulières de Q(t) de groupe
de Galois Ãn, J. of Alg. 131(1990),
483–495.
[Ri90] K. Ribet, Review of
Abelian l-adic representations and elliptic curves by J-P. Serre.
Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1,
214–218.
[Se97a] J.P. Serre, Galois
cohomology, Springer-Verlag Berlin, 1997. MR1466966
(98g:12007)
[Se97b] J.P. Serre, Unpublished
notes on l-adic representations, I saw a
presentation during Oct. 97 at Cal Tech, but there are various pieces
produced by various notetakers over a long period of time.
Mike
Fried,
Thursday, March 12, 2009