§0. Prelude
on Topics and Methods
Fried's major work has predominated in the geomety and arithmetic of families
of nonsingular projective
curve covers. The absolute Galois group of a perfect field K, G_{K},
is the profinite group of automorphisms of an algebraic closure of K.
He often was
intent on relating such geometric objects defined over K to G_{K}.
The basic geometric objects are normal
varieties over a field K. Variety means that
there is one component only
defined over K,
but that allows that there might be several components over, say, the
algebraic closure of K.
Such a variety has a function field (over K) and therefore the notion
of the normalization
of that variety in its function field. That normalization is unique. A
variety whose singular set has codimension two is already normal.
Throughout there is little loss in just assuming all varieties
are
normal, and usually they are nonsingular. Also, except for considering
reductions of covers modulo primes of a number field (and then we must
also assume the cover is separable),
we usually assume K
is contained in
the complex numbers.
A cover of algebraic varieties is a finite, flat
morphism, φ: X
→ Y.
But. if the domain (X)
and range (Y)
are both nonsingular, then flatness of a finite map is automatic. A
cover (defined over K) is Galois
if
the order of the group of automorphisms (also defined over K) of the
domain that commute with
the map to the range is the same as the degree, n = n_{φ}
of the cover. The assumption that X
and Y are
normal, means that the automorphisms of the cover correspond to the
automorphisms of the function field of X that fix the
function field of Y.
The
notation Z/t denotes the
cyclic group of order t,
and the notation Z_{p}, with
p a prime
denotes the p-adic integers.
Arithmetic
vs Geometric Monodromy of a cover: Every cover has a
(minimal) Galois closure cover. So Galois covers are
cofinal in all covers of a given fixed algebraic
variety. If φ is the covering map. then G_{φ
}refers to the group of automorphisms over
an algebraic closure, K^{alg}.
This is the geometric
monodromy of φ. There is also a Galois closure cover over any
definition field K
of φ: producing the arithmetic
monodromy (group, over K),
G_{φ}^{ar}.
It contains G_{φ }as a normal
subgroup.
There is a canonical permutation representation, T_{φ},
attached to
G_{φ}^{ar},
a faithful, transitive, homomorphism into the symmetric group S_{n}.
Such a representation is primitive
if the subgroup stabilizing an integer is maximal in the group.
It is typical for applications to start with Y absolutely
irreducible: Irreducible over the algebraic closure, K^{alg},
of K, and
usually so is X.
Even if both are absolutely irreducible, significantly, we don't expect
G_{φ}^{ar}
and G_{φ }to be
equal. An example
of great significance is
the (G_{φ}^{ar},
G_{φ}) pair associated
to
modular
curve covers of the j-line.
A regular
realization
of a group G
over a field K
is a Galois extension, L/K(x), with group G, x transcendental
over K,
where elements
of K are
the only algebraic elements in L.
Said with covers, it is where G_{φ}^{ar}
= G_{φ} and then any
quotient of G_{φ} is
automatically regularly realized over K. In most serious applications,
there
is no automatic reduction to equality of G_{φ}^{ar}
and G_{φ}.
Given two groups (G*,
G), a cover
φ (over K),
is an A-G realization of (G*,
G) (over
K) if (G*, G) = (G_{φ}^{ar},
G_{φ}). This then
automatically includes any quotient of G_{φ}^{ar}
as realized over K, where the part given by the G_{φ}^{ar}/G_{φ}
quotient is stationary
under Hilbert
Irreducibility specializations.
It
is easy, therefore, to see that the A-G version of the inverse
Galois problem generalizes both the standard and the regular versions
of the inverse Galois problem.
Covers given by rational functions – morphisms between copies of the
projective
line, P^{1}
– in one variable have
often been the hook that ties Fried's applications to that of many
other
researchers.
Yet, covers beyond these have revealed deeper problems. It
can be
surprising how classical and many of these problems are, at least
partially, answered by producing A-G realizations.
The
monodromy method and its tools: Fried's monodromy method (MM)
uses the categorical equivalence between covers with a fixed range Y and permutation
representations of the fundamental group of Y. That
fundamental group is the topological fundamental group
if the definition fields have characteristic 0. The initial novelty of
the MM was
its application
to problems on reducing polynomials with integer
coefficients modulo primes
p.
Two group theory tools made it possible to describe and create the
covers that produced solution to applications.
Three geometric tools related applications to the classical concern
with
connectedness of moduli.
I. Davenport's Problem
guided early developments
Harold Davenport
(1964) asked to characterize pairs (f_{1},f_{2})
of polynomials in one variable
over Q
having the same ranges mod p
for almost all primes p.
He and Donald
Lewis previously wrote on Issai Schur's 1921 Conjecture.
The hypothesis for that was a polynomial f in one variable
that mapped one-one on infinitely many residue class fields (Davenport and
Lewis called such f
exceptional).
I.a.
Primitive
covers, Arithmetic vs Geometric monodromy and Exceptionality:
Fried's
rephrasing of Schur's conjecture was that f was
a composition over Q
of polynomials. each of which – after an affine change of
variables
over the algebraic closure – were either cyclic (x^{n})
or Chebychev (T^{n}(x), with T^{n}(cos(ϑ))
= cos(nϑ))
polynomials.
Fried recognized
exceptionality as an arithmetic property detected using the fiber
product of the polynomial with itself. That allowed characterizing
exceptionality for general covers (not necessarily of just the
projective line).
Exceptionality
Definition:
Consider a cover φ: X
→ Y (of
normal varieties) over a field K. If K
= F_{q} is a
finite field (with q =
p^{u} elements, p
a prime) then φ is exceptional if and only if it is a
one-one map on F_{qt}
for infinitely many t.
If K is a number field, then φ is exceptional
if and only if it is exceptional over infinitely many residue class
fields. Denote the set of such primes of exceptionality by P_{φ,K}.
Exceptionality for φ is equivalent to one-one on infinitely
many residue class fields, but some of those may be given by primes for
which the reduction is not an exceptional cover.
Exceptionality
Characterization: With φ as above,
form the fiber product of X with
itself, and remove its diagonal component and denote the normalized
result by X_{2}
- Δ. For K a finite field, then φ is exceptional if and only
if X_{2}
- Δ has no absolutely irreducible K component. For K a number
field, exceptionality is characterized completely – using the Cebotarev
Analog for numer fields over finite fields – by the (G_{φ}^{ar},
G_{φ}) pair associated to φ.
These results^{[Fr74]} are a generalization of a
very special case due to Charles
MacCluer^{[Mc67]}. They epitomize mondromy precision
about a cover – a property depending only on the arithmetic/geometric
monodromy.
It was elementary to reduce Schur's conjecture to where f is indecomposable
over Q. The
MM reduced
Schur's Conjecture to where f
is indecomposable over
the algebraic closure. There were two "small"
ingredients in this proof that showed in most applications of the MM.
- That f
is incomposable over K^{alg}
(resp. K)
interprets as G_{φ}
(resp.
G_{φ}^{ar}) under T_{f}
is a primitive representation.
- Because f is
a polynomial and
it is neither cyclic nor Chebychev, then T_{f}
on G_{φ }is not only primitive (true for
any indecomposable cover), it is actually doubly
transitive.
That is, for distinct integers i,
j: there
is g
in G_{φ} with 1 = T_{f}(g)(i) and 2 = T_{f}(g)(j).
Polynomials stand out from general rational functions (in one
variable) in that the inertia generator, g_{∞}
over ∞ has T_{f}(g_{∞})
an n-cycle.
Fried
generalized Schur to arbitrary number fields.^{[Fr70] } Schur
was the
simple opening step to solving Davenport's
problem (over Q)
when f_{1}
was not a composition of lower degree polynomials (indecomposable
hypothesis IH).
That group results of G.
Burnside and I.
Schur played an important role was
a hint that group theory would continue to appear.
I.b.
Davenport and
permutation
representations vs group representations: We
exclude the trivial case of Davenport pairs: f_{1}(x)=f_{2}(ax+b). Fried showed
that
Davenport pairs (f_{1},f_{2})
over any number field K,
satisfying IH,
had G_{f1}
= G_{f2},
and T_{f1}
and T_{f2}
are identical
as group (but not permutation) representations.
That is, the trace of
the permutation matrices T_{f1}
(g) and T_{f2 }(g) for each g in G are the same.
Another ways to say this: The covers are inequivalent by the natural
rank n
bundles defined by the covers are isomorphic.
Most important: That
inertia
generator g_{∞}
over ∞ is necessarily an irrational
cycle in G_{φ}^{ar}.
That is, for some j
prime to n,
g_{∞}^{j} is
not conjugate to g_{∞}
in G_{φ}^{ar}.
For example, this holds if j = -1. Such a j is non-multiplier.
This example showed that the shape of the permutations is
inadequate for understanding the problem, and that the data for that
integer j
requires some delicate understanding of the group.
The conclusion from the BCL
is that if K
= Q,
then f_{1}
is conjugate (by applying an element of G_{Q} to its
coefficients)
to f_{2}.
That contradicts that (f_{1},f_{2})
are over Q.
I.c.
The tools for
describing Davenport pairs over number fields:
Assuming IH,
the only possible degrees of Davenport pairs over any number field were
confined
to a finite list (7, 11, 13, 15, 21 and 31). This required the
classification of doubly
transitive groups, a special case – due to Canter, Curtis and Seitz of the finite simple group
classification that preceded the full classification
result.
While that was new, it was also new to figure the essential data – Nielsen
classes – for any families – Hurwitz spaces
– of covers that might be solutions to Davenport's problem. There was
one exceptional group, but there were infinitely many possible groups
that might have produced Davenport pairs: Affine
groups close to special
(determinant 1) linear groups over finite fields. That only
finitely many did was the first realization of an affirmative solution
to the genus
0 problem. It was an example of the transitivity of BrA
action result. The section prime-squared
degree
does an example computation interpreting a known property of
modular curves from this viewpoint, to show how braid action works on
Nielsen classes.
Applying BrA
to the Nielsen classes that arose for Davenport's problem,
and a crude genus formula for compactified
dimension 1 reduced
Hurwitz spaces, Fried found that pure
transcendental parameters
parametrized the seven possible (they do exist) degrees of Davenport
pairs. This extension of
Riemann's existence
theorem using the BrA
gave a complete
description
of those number fields supporting Davenport pairs.
The BCL
showed that the
irrational cycle over ∞ gave the definition fields of the families.
While there was (except for n
= 7) more than
one family, for each possible degree the families were all conjugate
(by
elements of G_{Q}); an
example of the BrA definition field
result.
Sections II-VII show these developments extended to more general covers
and questions far beyond Davenport's.
II.
The B(ranch)C(ycle)L(emma) and Br(aid)A(ction)
II.a.
Branch cycles and Nielsen
classes of covers: Consider a cover, φ: X
→ P^{1},
of the projective line over C
branched
over r
≥ 3 points z_{1},…,z_{r}
(of degree n)
with its monodromy group G_{φ},
permutation representation T_{φ}
and r
(unordered) conjugacy classes, associated to z_{1},…,z_{r}
= z,
Ç = C_{1}∪
C_{2}∪
… ∪ C_{r}
within that group. There are elements g_{i}
in C_{i} (any ordering of the classes will do) with
these two properties.
- product-one:
Their product, in order
of
their subscripts, is 1.
- generation: g_{1},…,g_{r }
generate G.
Such r-tuples
with these properties are branch
cycles. Denote the collection of such g
= (g_{1},…,g_{r })
by Ni=Ni(G,T, Ç): a
Nielsen Class.
We always assume it is nonempty. Conversely, given r
points on P^{1},
(G,T, Ç), and elements
satisfying generation and product-one, there is a cover (usually
several,
if G is
nonabelian) with those r
points as branch
points, in this Nielsen class.
There are several equivalences of covers with the same branch points.
Interpret these from their branch cycles assuming those are
prepared from the same classical
generators of the r
punctured projective line. Three stand out.^{[BFr02, § 3.1]}
- Absolute
equivalence relative to T:
The subgroup, N_{Sn}(G, Ç), of
S_{n}
normalizing the image of G under T,
and permuting the classes of Ç, conjugates between branch cycle r-tuples of
covers. Denote Ni/N_{Sn}(G, Ç) by Ni^{abs}.
- Inner
equivalence for T the regular
representation: Regard the covers as Galois covers, with an
explicit isomorphism of the cover automorphism group with G, up to
conjugation by G.
Notation for Ni/G
is Ni^{in}.
- Reduced
equivalence: Any equivalence of P^{1 }covers
can be augmented by reduced equivalence where a cover and its
composition with
an automorphism of P^{1}
are equivalent. There are absolute and inner reduced
spaces.
These equivalences generalize the distinctions between families of
covers that appear in modular
curves.
II.b.
Dragging a cover by
its
branch points: The space of
r
unordered
branch points identifies with projective r space, U_{r},
minus its discriminant locus. Assume two covers (φ_{1},φ_{2})
have their corresponding (G,T, Ç)
distinct (up to equivalence). Then the covers are
distinguishable as deformation
inequivalent: No matter how you (continuously) drag φ_{1}
– following a trail of branch
points in U_{r}
– over to the branch points of φ_{2},
the resulting cover φ_{1}' will
be inequivalent
to φ_{2}.
When r=3, φ_{1}'
is always reduced
equivalent to φ_{1}; for discussing
families of covers this is the trivial case.
If you drag φ_{1}around a closed path on U_{r}
(based, say, at z_{0})
then you can express branch cycles, g_{1}',
for φ_{1}' in terms of branch cycles, g_{1},
for φ_{1}. There are two paths, generating
the
fundamental group – the
Hurwitz
monodromy group – H_{r}
of U_{r}.
Denote their homotopy classes, respectively, by:
- shift:
sh
which maps (g_{1},…,g_{r})
to (g_{2},…,g_{r},
g_{1});
and
- 1st
twist:
q_{1}
which maps (g_{1},…,g_{r})
to (g_{1}g_{2}g_{1}^{-1},g_{1},
…,g_{r}).
Conjugating q_{1}
by sh i-1 times gives the
ith
twist, q_{i},
i
= 1, …, r-1,
and there are various relations among these elements coming
from H_{r}
being a quotient of the Artin braid group, but none more important than
The P^{1}
relation:
q_{1}…
q_{r}_{-1}q_{r}_{-1}…
q_{1}=
1 which does not come from the braid group.
II.c.
The BCL and Moduli
of covers in a Nielsen class: All
covers in a
given Ni(G,T, Ç) have a
natural complex analytic parameter (Hurwitz) space H(G,T, Ç). For absolute
or inner equivalence of covers, H
is naturally an unramified cover of U_{r}
with fibers of cardinality equal to (respectively) the number of
absolute or inner Nielsen classes. So, it is a complex
analytic
manifold.
It is a moduli
space
with objects in the étale topology that represent all
analytic families of covers in the Nielsen class. Then, H(G,T, Ç) generalizes
the moduli of
curves of genus g computed
from the Riemann-Hurwitz
formula for the genus of a cover,
2(deg(T) + g -1) = sum_{i=1}^{r}
ind(g_{i}),
with
ind(g) the
degree of T, deg(T),
minus the number of fixed points of T(g).
Compactifications
of H(G,T, Ç): There are
several compactifications of the Hurwitz space of its reduced version,
used for different purposes. The easiest to form is the union of the
normalizations of P^{r} in
the function fields of the components of H(G,T, Ç). A
normalization of a compactication of U_{r}/ PSL_{2}(C) is also used for
the reduced versions. The result for r = 4 is a
nonsingular curve cover of the classical j-line, but for higher values
of r
typically the spaces have singularity in codimension 2.
Outside these cases, two compactifications are Fried's "specialization
sequence" stratification^{[Fr95, p. 44-45]} and a
generalization of the Deligne-Mumford compactification by Wewers and
Mochizuki, and used by Debes-Emsalem. The results in both cases are
nonsingular spaces which contain well defined information on what
happens to the Riemann surface covers when, as they follow the
process of dragging
the branch points, the branch points coalesce (come
together).
In all cases the phrase a point on a "Hurwitz
space off its cusp locus"
will mean a point on the original version of the space before
compactification.
The geometric and
formula versions of the BCL: The Galois
group of the cyclotomic closure, Q^{cyc},
of Q
(obtained by adjoining all roots of 1) identifies with the
invertible profinite integers Z^{*}.
Any subfield of Q^{cyc}
is a cyclotomic field. Restricting any element σ of G_{Q} to Q^{cyc}
gives a profinite integer n(σ).
The BCL
produces a moduli field
F_{Ni}
– that happens
to be a cyclotomic field – for each equivalence, attached to a Nielsen
class. This is a geometric consequence of the formula version of the BCL. Here is how to
compute it for absolute
equivalence.
Consider the extension Ç_{abs} of each of
the conjugacy classes of Ç
inside N_{Sn}(G, Ç). Extending
classes can end up joining two classes.
- Denote all covers
in the Nielsen class defined over Q^{alg}
by W_{Ni}.
- By applying any element σ of G_{Q}
to W_{Ni}
you get W_{Ni}^{σ}.
- Let Ç_{abs}^{n} be the set of
elements of Ç_{abs} set to the power n, and let I(Ç_{abs})
be those n
in Z^{*}
for which Ç_{abs} = Ç_{abs}^{n}.
Absolute Version of the
BCL: Then F_{Ni}
is the fixed field in Q^{cyc}
of I(Ç_{abs}).
It is the smallest field K'
for
which W_{Ni}
= W_{Ni}^{σ}
for each σ fixed on K'.
A fine
moduli family is one for which any family of covers in the
Nielsen class is determined by the natural map of its parameter space P to H. Simple conditions
on (G,T) guarantee that
there is a strong
moduli family with F_{Ni}
its definition field.
For absolute classes, the fine moduli condition is that the
stabilizer, T(1), is
a self-normalizing
subgroup. That includes the case T
is primitive. Inverse
Galois
Problem applications use inner Hurwitz spaces and the fine
moduli condition is that G
has no center.
Fine
Moduli Version
of the BCL:
F_{Ni}
is the minimal field of definition of the natural family of covers that
determines all other families if fine moduli holds. Further, from
Hilbert's irreducibility, F_{Ni
}is the intersection of all fields of definition of all
covers in W_{Ni}.
The inner version of these BCL
results is obtained by replacing Ç_{abs}
by Ç.
Formula
version of the BCL:
The explicit formula can be applied to particular covers given by their
branch points and branch cycles. A usual application is to decide if
there is any chance that covers in a Nielsen class (or a particular
component of one) could be defined over a particular number field (say,
K). For
this start with a cover φ: X
→ P^{1
}which you assume is over K, so its branch
points, as a set are in K. (Limiting each branch point to be in K
destroys serious
applications.) Any σ in G_{K}
acts through G_{φ}^{ar},
so through ω_{σ} in N(G, Ç). If
σ maps the ith branch point to the jth, then:
ω_{σ}
C_{j}
ω_{σ}^{-1} = C_{i}^{-n(σ)}. ^{[Fr76,
Thm. 5.1] or [Fr12, (5.2)] or [Vo96,]}
So, if this formula fails, then φ does not have definition
field K.
If the ith branch point is in K, then the extension of C_{i} to a
conjugacy class in G_{φ}^{ar}
is K-rational.
For
reduced equivalence when r
= 4, there are two
conditions: A quotient, Q''=Q/<(q_{1}q_{3}^{-1})^{2}>,
of a quaternion
group Q,
acts faithfully on the Nielsen class (as a Klein 4-group); and
neither q_{1}q_{2 }nor q_{1}q_{2}q_{3}
have fixed points in their action on Ni/Q''^{.[BFr02, Thm. 2.9]}
Modular
curves don't have fine (reduced) moduli because Q'' acts trivially on
the dihedral
group Nielsen classes that define them.
For r
> 4, reduced fine moduli fails can only fail at singular points
on
the reduced spaces. These correspond to fixed points of the PSL2(C)
action, the analog of the second r = 4 condition.
At such points, the precise test for failure comes through precise
formulas on branch cycles.
Sometimes such covers do exist (in a given Nielsen class). Sometimes
they don't. There is no analog for the singular points –
corresponding to curves
with automorphisms – on the space of curves of genus
g.
All covers in the Nielsen class are
deformation equivalent if the BrA
is transitive on Nielsen classes. In that case the BCL gives precisely
the
cyclotomic definition field of the unique component of the family.
Applications feature the
main problem.
Cover Separation Problem:
How to recognize when covers in a Nielsen class are in
distinct
components when H_{r}
is intransitive. Then, how to find the definition fields of their
components.
II.d.
Inner vs absolute
Nielsen classes: Given Ni(G,T, Ç) there is a
natural covering map from the space H
^{in} of covers
corresponding to inner equivalence to the space H
^{abs} for absolute equivalence.
Suppose H
'' is a component of H
^{in}. Then the map to its image, H
', in H
^{abs} is a Galois cover with
automorphism group, Aut(H
''/H
') identifying with a subgroup of N_{Sn}(G). When the groups
are equal the elements of N_{Sn}(G) are said to be braidable over H
'. Indeed there is a natural construction of the
space H
^{in} and the map H
^{in} → H
^{abs} based on the fiber product
construction of a Galois cover applied to the situation where fine
moduli holds for H
^{abs}.^{[Fr10, Prop. A.5,
§A.2.4]} Extending this construction is the main aid in
computing the monodromy group of H
^{in} → U_{r},
or its reduced version H
^{in,red} → J_{r},
so important to extending
the OIT.
Here is a special case of the Cover Separation
problem.
Automorphism Separation:
How to recognize when two covers corresponding to points of H
^{in} over H
' are braidable.
Geometry of Reduced equivalence:
Using reduced equivalence preserves the sequence H
^{in} → H
^{abs} → U_{r},
but instead of a cover of r-dimensional nonsingular
complex manifolds,
the result H
^{in,red} → H
^{abs,red} → U_{r}/PGL2(C)
=^{def} J_{r},
consists of ramified covers of r-3-dimensional
spaces, usually singular if r
≥ 5. The case r
= 4 is special. We can understood it directly in terms that compare
the spaces with modular curves using a normal subgroup, Q, of H_{4}.
- sh^{2}
and q_{1}q_{3}^{-1}
generate Q,
with (q_{1}q_{3}^{-1})^{2}
generating its center and acting trivially on all Nielsen
classes.
- H^{*,red}
(* = in or abs) is a nonsingular curve and J_{4}
identifies with the classical j-line
(minus ∞).
- H^{*,red}
is an upper half-plane quotient that naturally compactifies to a
non-singular curve H^{*,red}.
- Points of H^{*,red} ramify
(at most) of order 3 (resp. 2) over j = 0
(resp. j
= 1).
- All other ramification is over j = ∞;
points over j
= 0 (resp. j
= 1,
∞) identify with q_{1}q_{2}
(resp. sh, q_{2})
orbits on Ni*/Q.
- Genus
computation:
Modulo identifying H_{4}
orbits on Ni*, this gives the genus of
H
^{*,red}
components.^{[FBr02, (8.1)]}
II.d.
Geometric/Arithmetic
Monodromy of Hurwitz Space covers: Suppose O is a braid orbit
from its acting on an inner Nielsen class Ni(G, Ç)^{*}
with * = inner or inner, red. Then, we can compute the geometric
monodromy group of the component H_{O}
→ U_{r},
if * is inner or absolute equivalence orbit as the group of
permutations acting on the Nielsen classes in O. Similarly for
reduced equivalence, but J_{r}
substitutes for U_{r}.
With no loss take * to inner or inner, red.
The collection cover:
Pick a base point u_{0} on U_{r}. There is a unique geometric Galois cover φ_{u}_{0}_{,G,Ç}:
X_{u}_{0}_{,G,Ç}→
P^{1
}– the collection
cover – with
branch points at u_{0}
so that φ_{u}_{0}_{,G,Ç}
factors
through each cover φ: X
→ P_{1}in
O with
branch points in u_{0},
and it is minimal with this property. Denote the geometric monodromy
group of φ_{u}_{0}_{,G,Ç} by G_{u}0. _{
}
The
Monodromy Automorphism
Principle: Denote the inner Nielsen class of φ_{u}_{0}_{,G,Ç }by
Ni_{u}_{0}_{,G,Ç}^{inn}.
Then, H_{r}
acts on Ni_{u}_{0}_{,G,Ç}^{inn}
and if we restrict the action to the orbit O of φ_{u}_{0}_{,G,Ç},
then any q in H_{r}
acts as an automorphism of G_{u}_{0}.
This induces the analogous result for the corresponding reduced orbit
of O.
That leaves three problems to decipher the Arithmetic/Geometric
monodromy of H_{O}
→ U_{r},
or its reduced version.
- Find a good presentation of the collection group of G,Ç, O.
- Find the braid action on it, and identify it as a subgroup
of the
automorphism of G_{u}_{0}.
- Deduce from the above the possibilities for the Arithmetic
monodromy
group of H_{O}
→ U_{r},
or its reduced version.
III.
The Genus
zero problem and covers in positive characteristic
III.a.
The Guralnick/Thompson Genus 0
Problems: John
Thompson,
on becoming aware of the solutions of Davenport's problem,
Schinzel's
problem
and other applications of the MM,
instituted a
program on monodromy groups of rational functions. His goal was to show
the composition
factors of rational function
monodromy in 0 characteristic are alternating or cyclic
groups
(general cases), excluding some long, but finite, list of simple
groups (special cases).
Robert Guralnic,
working with Thompson,
strengthened this conjecture to make stronger statements
about the precise monodromy groups and permutation representations in
the general case. For example, that indicated which groups (and
permutation representations) with
alternating groups as composition factors would appear in long general
series.
He also formulated the correct variant of the general cases for the
extension from rational function covers to genus 1 (resp. genus
exceeding 1) covers of P^{1}.
For Thompson, it was important to figure how precisely they were able
to list monodromy groups and corresponding Nielsen classes for the
special cases. For particular problems, such as the description of
rational function
exceptional
covers over number fields, there were bound to be finitely
many monodromy groups that fell outside the general cases given by the
OIT.
With experience with branch cycles, they could recognize those.
Peter Mueller
explicitly listed the special groups for polynomial monodromy.
About half appeared in the solution
to Davenport's problem. As already appeared in Davenport's
problem,
this was a list of double transitive groups. Therefore the affine groups – close to the core
of general linear groups over finite fields – with primitive monodromy
were immensely easier.
Guralnic and Thompson also recruited many group theorists, experts on
particular pieces of the finite group classification, to complete the
proof of the conjecture. Fried stayed in touch with developments to
make this use of the classification more accessible to
those without extensive group theory backgrounds, who might still use
pieces of the classification.
- By using applications to show how the Aschbacher-O'Nan-Scott
theorem usefully lists series of primitive groups by indexing
their production from series of simple groups.
- By showing that while many series of primitive groups are
outside the training of all but expert group theorists, the series that
arise from practical applications of covers tend not to be.
III.b.
Exceptional
positive characteristic covers: Motivation
for the genus 0 problem in characteristic p came from working
with the correct
version of exceptional
covers over finite fields:
one-one maps on points in finite fields of order p^{t}
for infinitely many t.
The MM
worked there. For polynomials f, unless p divides the
degree of
f, the
result was
exactly the same as in 0 characteristic.
The full result required
extending the MM
method to include noncyclic
ramification groups (over ∞). Fried, Guralnick and Jan Saxl gave a
classification of the possible monodromy groups, again using the
indecomposability hypothesis IH.
They saw that
characteristic p
would have a special place for affine groups over finite fields of the
same characteristic. Here, though, wild ramification made a result like
Mueller's much more difficult.
Outside affine group monodromy, as in Davenport's
problem, there were but finitely many special polynomial
degrees.
They listed those, replicating the use made of the classification but
applied to a more general group theory problem: Groups with a set
theoretic factorization T_{G}(1)xP, with T_{G}(1)
the stabilizer of a letter in T_{G},
and P the
inertia over ∞. The exposition describing groups,
with such factorizations, is a model for how to use the Aschbacher-O'Nan-Scott
Theorem.
Individual works by Steven
Cohen, Mueller and Hendrik
Lenstra-Mike
Zieve
produced the polynomials for those covers, showing they had the
exceptional covering
property.
III.c. Extending Grothendieck's Theorem
to wild
ramification: Alexander Grothendieck considered
the natural map of a
connected family of tame
covers of projective 1-space to the branch point space U_{r}.
One half of his result showed that if that map is constant (the branch
points don't move) then the family is trivial after pullback
by an étale cover of the configuration space. Fried produced a finite
type configuration space over U_{r}
that generalizes the space U_{r}
to wild ramification.
The analogous setup applied to wildly
ramified covers has this conclusion: If the map to the configuration
space is constant, then the family is trivial after pullback by a
finite
(not necessarily étale) cover of the configuration space.
IV.
Serre's
OIT, exceptional Covers
and the Branch Bound Conjecture
Fried developed the Hurwitz space/Modular
Tower approach to generalize
the applications of modular curves. For that to prove its worth, the
generalization had to be considerable, applications had best start
close to traditional uses of modular curves, and the generalizing
techniques were better understood if they extended
aspects of those used on modular curves. In this section k
≥ 0 is a parameter indicating a level, and p
is a prime, which we often assume is odd.
This section starts with the Hurwitz space
reinterpretation of the modular curves X_{0}(p^{k}^{+1})
and X_{1}(p^{k}^{+1}).
It then
reviews pieces of Serre's O(pen) I(mage) T(heorem).
The fundamental ingredient – Frattini covers of finite groups – appears
in two places in the applications: In the sequence of groups giving the
Hurwitz space levels, and also in the monodromy groups of the spaces as
covers of the configuration space. For modular curves the configuration
space is the j-line.
that allows the complete generalization.
There are three kinds of applications. The 1st is the dihdral
version of the expansion
on the Inverse Galois
problem as generalizing many of the major results on modular
curves.
The last two reinterpret the two types of OIT fibers (GL_{2} and CM) over the j-line as giving
detailed solutions to problems on rational functions f(w), in the
complex variable w.
The second of these hinges on using the modular curves X(p^{k}^{+1})
which dominate in the OIT. Every aspect of modular curves
considered here has a Hurwitz space interpretation that generalizes to
Modular Towers.
IV.a.
Modular
curves as Hurwitz spaces: Fried interpreted
modular curves as reduced Hurwitz spaces associated
to
dihedral
groups (G =
D_{pk}_{+1},
group of order 2p^{k}^{+1})
using r
= 4
involution conjugacy classes, denoted Ç_{34}:
The Nielsen class Ni(D_{pk}_{+1,
}Ç_{34}).
Denote the standard degree p^{k}^{+1}representation
of D_{pk}_{+1}
on the cosets of the group generated by any involution by T.
J.P.
Serre's OIT^{[Se68]} can be
interpreted as properties of projective
systems of fibers
of the natural sequence of modular curves {X_{0}(p^{k}^{+1})}_{k≥ 0},
all of which cover the j-line, P^{1}_{j}. The other
prevalent series of modular curves, for each p, is the
collection {X_{1}(p^{k}^{+1})}_{k≥ 0}.
They are distinguished in the two views by their moduli space
properties up to the expected natural equivalences. In the classical
view we have:
- A
point of X_{1}(p^{k}^{+1})
over j',
not equal to ∞, is an elliptic curve – genus 1 curve with a
chosen point as origin – with j
invariant j',
together with a p^{k}^{+1}
torsion point up to isomorphism.
- Respectively, a point of X_{0}(p^{k}^{+1})
replaces the point with the cyclic subgroup generated by that
point.
Closely
aligned Hurwitz spaces:
- H((D_{pk}_{+1,}T,_{
}Ç_{34})^{abs,red}
is a Hurwitz space of genus 0 covers (like the computation for the (Z/p)^{2}
semidirect product with Z/2
case). For p
odd, rational functions automatically represent such covers.
- Then, H((D_{pk}_{+1,}Ç_{34})^{in,red}
is the inner space of Galois closures of those covers, with an explicit
– up to inner automorphism – identification of its automorphism group
with D_{pk+1}.
Choose
1 or -1 as canonical generators (up to
conjugation by the generator of Z/2)
of its normal order p^{k}^{+1}
given its
isomorphism with Z/p^{k}^{+1}.
In commenting, we avoid j
= 0 or 1 for which the corresponding elliptic curves have automorphisms
beyond multiplication by -1, special cases treated separately.^{[Fr78,
(2.10)]}
Moduli properties:
The Hurwitz spaces H((D_{pk}_{+1,}T,_{
}Ç_{34})^{abs,red
}and H((D_{pk}_{+1,}Ç_{34})^{in,red}
are fine moduli from the standard inner
(resp. absolute) criterion applied to the dihedral
group (resp. in its embedding by T).
Not so for the reduced spaces because of trivial Q'' action
in this case.[Fr95,]
Jacobian map:
Any local family of covers in the Hurwitz space view maps to
a
corresponding family in the modular curve view. For example, associate
a point on H((D_{pk}_{+1,}Ç_{34})^{in,red
}with the jacobian – which has both an origin and inherits
the dihedral group action – of the genus 1 curve corresponding to the
point. Then, the image of the origin by represented by a canonical
generator gives the order p^{k}^{+1}
in the modular curve view.^{[Fr78, §2]
}IV.b.
Aspects of the Open
Image Theorem: For
any algebraic
j'
outside j
= ∞, and for any
projective sequence of points x_{p,k}_{+1}
on X_{1}(p^{k}^{+1}),
consider the group H(x_{k}_{+1})
of the Galois closure of the field extension Q(j',x_{p,k}_{+1})/Q(j').
With I_{2} the 2x2 identity matrix, the arithmetic
(resp. geometric) monodromy of the (absolutely
irreducible) cover φ_{p,k}: X_{1}(p^{k}^{+1}) →
P^{1}_{j} is GL_{2}(Z/p^{k}^{+1})/<±I_{2}>
(resp. SL_{2}(Z/p^{k}^{+1})/<±I_{2}>),
as in the Hurwitz
space view.
Adding
a choice of isomorphism class to a point of X_{1}(p). If we assume we
have added such a choice, then it is possible to speak of a
projective system of generators of p^{k}^{+1}
division points on an elliptic curve with j-invariant j'. That then
allows looking at the images under elements of G_{Q(j')} of
projective systems. Thereby, we may consider the image of G_{Q(j')} in GL_{2}(Z_{p}),
with Z_{p}
the
p-adic integers. Serre, though, rarely spoke in terms of moduli. While
the moduli interpretation of X_{1}(p)
seems to have been in the air in the late '60s, [O71] seems to be one
of the earliest actual references to it in papers.
Serre's Final OIT
Result: This divided algebraic points j' outside j = ∞ into two
types for which, for each p,
the following holds.
- GL_{2}-type: H_{p,j}_{'}
(resp. the image of G_{Q(j')}) is
an open subgroup of GL_{2}(Z_{p})/<±I_{2}>
(resp. GL_{2}(Zp)).
- CM-Type:
H_{p,j}_{'}
(resp. the image of G_{Q(j')}) is
essentially abelian, stemming from a classical description of the
class fields of complex quadratic extensions of Q.
Starting from David Hilbert, the CM-Type was understood to be a
production of abelian extensions of Q(j') from the values
of functions arising from complex
multiplication elliptic curves. Much
of Serre's book
concentrated on this part of the OIT. It motivated
much of Goro Shimura's
work^{[ShT61]} from which Fried learned Andre Weil's idea of
moduli field, adapting it for Hurwitz spaces.
It also
included a large chunk of the GL_{2} part: Those
j'
which were not
algebraic integers. For that case, two statements are shown for almost
all p about
the H(x_{1})
decomposition group action on (Z/p)^{2}.
It acts
irreducibly and, using Tate's geometric form of an
elliptic curve at a prime (say, p*,
not to be confused with p)
dividing
the denominator of j', it
contains a transvection: conjugate to
a matrix that acts like translation by the vector (0,1).
Even as one must pick a rational number carefully for its nth roots to
generate an abelian extension (generated by the nth roots of 1;
rather than a solvable extension) of Q,
one must pick j'
carefully among all complex quadratic integers to get the CM-type.
A satisfactory proof, many years later, of the
other GL_{2} cases corresponding to algebraically
integral j'
awaited Falting's proof of the
Mordell Conjecture^{[Se97b]}. So, there was nothing
explicit about it.
IV.c.
Modular
curves form a
Frattini system: The
OIT brought attention to the significance of Frattini properties of
towers. A projective system of (irreducible) covers
… → X_{n}_{+1}
→ X_{n}
→ … → X_{0}
→ P is eventually p-Frattini
if there is a value k
for which the monodromy group of X_{n}_{+1}
→ P
is a p-Frattini
cover of the monodromy group of X_{n}
→ P for n ≥ k. If this applies
for k = 0,
then the system is p-Frattini.
Suppose K
is a number field.
This "eventually Frattini" condition automatically implies a
weak version of the OIT. Suppose for z' in P(K), and x_{k}
in X_{k}
over z', the decomposition group H(x_{k})
equals the whole arithmetic monodromy group of the cover.
Arithmetic
Frattini Result: Then the decomposition group for a
projective system of points on the whole fiber of the system over z' will be the
projective limit of the decomposition groups of the system.
Refer to such a fiber (over z') in the
conclusion of the Frattini result as a full fiber.
Apply Hilbert's
Irreducibility to
find a dense set of z'
in P over
which the fibers are full. If P
is unirational over K,
then we may assume the dense set is in K or even ordinary
integers.
Close inspection of [Se68, IV-23, Lem. 3 and IV-28, exer. 3] reveals
this important group lemma on the geometric monodromy
of X_{0}(p^{n}^{+1})
as a cover of the j-line.
With k_{0}
= 0 for p
> 3 (resp. k_{0} = 1 for p
=3; k_{0} = 2 for p
= 2), it
satisfies
SL_{2}(Z/p^{k}^{+1})/<±I_{2}>→
SL_{2}(Z/p^{k}^{0+1})/<±I_{2}>
is a p-Frattini
cover, for k
> k_{0}.
So, if for j', H(x_{p,k}_{0+1})
= GL_{2}(Z/p^{k}^{0+1})/<±I_{2}>,
then H(x_{p,k}_{+1})
= GL_{2}(Z/p^{k}^{+1})/<±I_{2}>
for all k
≥ k_{0}.^{[FrH14,
Lem. 3.4 ]}
By applying Hilbert's
Irreducibility,
for a fixed p,
we find a dense set of z'
(even of algebraic integers) for which the decomposition group for z' is GL_{2}(Z/p^{k}^{0+1})/<±I_{2}>.
Such a full fiber has arithmetic monodromy GL_{2}(Z_{p})/<±I_{2}>.
Better yet, by using a universal
Hilbert subset we can find a dense set of z' for which, for
all but a finite number of p,
the fiber over z'
is full.
Close to ∞ comment:
Serre's earliest result on the values of j' that give full
fibers interpret as follows. Suppose for some prime p*, j' is
suitably "close to" ∞ in the p*-adic topology.
Then we expect the full fiber result for all but a finite number of p.
Although Serre was more explicit for modular curves, as a result of
having Tate's interpretation of nonintegral j-invariants, this is the
style of the production of universal Hilbert subsets. It is also the
model for generalizing that part of Serre's results to other systems of
moduli space covers.
p-cusp comment:
p cusps on
modular curves, and their
attachment to Tate's p*-adically
uniformized elliptic curves to decipher
the G_{Q} action
when j is p*-adically "close
to" ∞ matches with the general idea of p-cusps on Modular
Towers.
Complex
multiplication comment: Assume K is a complex
quadratic extension of Q.
Complex multiplication
is the discussion of 1-dimensional characters of G_{K}
on the Q_{p}
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.^{[Sh64]}
Serre's 1967 book is still relevant^{[Ri90]},
especially for
the
role of abelian characters represented by actions on Tate
(l-adic)
modules from abelian varieties. Reminder: As Deligne showed in
the
early '70s, even nonsingular hypersurfaces in projective space can have
étale cohomology which is not in the category of l-adic modules
generated by abelian varieties. Yet, it is elementary that the category
of l-adic
modules generated by abelian varieties is contained in that generated
by Jacobians of compact Riemann surfaces.
IV.d.
Involution
realizations of dihedral groups:
Branch
Bound Conjecture
– Dihedral Version:
Fix a prime p
> 2, a
number field K,
and choose any N_{p}.
Then, there is an integer k
= k_{p}_{,Np, K} so
that any regular realization of D_{pk}_{+1},
with k ≥ k_{p }must
have more than N_{p}
branch points.
The
dihedral groups and symmetric groups are the first groups – regularly
realized – as Galois groups in an algebra course. Each dihedral group
and symmetric group is generated by two elements, so over the complexes
both are realized as covers of the sphere with a very small number of
branch points.
The problem – from the BCL:
If you use elements of order a power of p,
than any bound on the branch points for dihedral groups forces a
realization where the Nielsen class has all its conjugacy classes that
of the involution. Denote this C_{2}
Pierre Dèbes
and
Fried proved the conjecture is equivalent to showing that for no
integer d'
does there exist for each k
a p^{k}^{+1}-power
cyclotomic point – the
absolute Galois group of K
acts as if the point and its multiplies, over K, are p^{k}^{+1}
roots of 1 – on a dimension d'
hyperelliptic
Jacobian.^{[DFr94, Subsects 5.1–5.2]}
The Sheldon Kammieny
and Barry Mazur results
and conjectures
are akin, but they are about
disproving existence of p-power
torsion over K,
not about cyclotomic
points.^{[KM92]}
If you take N_{p}
= 4, then a proof follows from a general principle, The Genus Growth Theorem,
and the proof that the spaces, H((D_{pk}_{+1},_{
}Ç_{34})^{in,red},
are modular curves
whose (well-known) genuses grow with k.^{[Fr78,
§2], [Fr05, §6.1-6.2]} From Mazur's Theorem, much more is
known in this case, as every Q
point on all of these spaces is known. A rougher, but similar,
statement is known as Mazur-Meryl, for any number field.^{[Mer06]}
There is no reason to doubt the Modular Tower
conjectural generalizations – allowing Nielsen classes with any value
of r
– despite uncertainty over a suitable higher dimensional replacement
for Falting's Theorem. Still, there has been steady
progress on the case r
= 4, consisting of projective systems of upper half-plane quotients.
That is why that program concentrates on
this case for the novel examples that generalize pieces of Serre's
OIT.
Especially, the multiple
Harbater-Mumford component example
shows that a slight change in the group immediately produces new
territory that tests difficulties hard to weed out in the
Siegel upper-half space case parametrizing Abelian varieties of a given
genus. The Siegel formulations end up with limited statements about
random
values of the parameter.
IV.e. Rational
function applications using the OIT: Fried
interacted with the OIT to describe rational functions f sought
in many applications. These applications also show another point about
using reduced classes. An f
over Q
automatically yields infinitely many examples by composing with Q
linear fractional transformations. Yet, to be significantly different,
you would want examples to be reduced inequivalent.
1.
Find all (f,K) with the Schur cover property
(another name for an exceptional covers): For infinitely many
residue fields of K,
f : w → f(w) = z, with w running over the
residue
field (including ∞), is one-one.
2. Find all (f,K), with f indecomposable
over K,
but f is
a composition
of degree > 1 rational functions over the complexes.
Indecomposable
rational
function statement: Rational functions f over a
given number field K,
that are also exceptional,
but not equivalent to polynomial
covers, are "essentially" those appearing from the two kinds
of fibers
on modular curves that Serre distinguished.^{[Fr78,
Thm. 2.1]
}Designate the primes of exceptionality of f by P_{f}
= P_{K,f}.
Then, if f
= f_{2}^{o}f_{1},
a composition of two lower degree functions over K, then P_{K,f}
= P_{K,f2}
∩ P_{K,f1}. _{
}
The converse is trickier. Suppose we start with f_{1}
and f_{2}
exceptional over K.
Then,
concluding that the intersection of their exceptional sets is nonempty
requires using the fiber
product formulation of exceptionality.^{[Fr05,
Rem. 3.7]}
For prime
degree covers, exceptional covers have cyclic or dihedral
groups
as geometric monodromy, and arithmetic geometric quotient is a subgroup
of (Z/p)^{*},
from absolute
equivalence.
In addition if they are rational function
covers, the arithmetic/geometric monodromy quotient is understood.
For
polynomials, compositions of either cyclics or Chebychevs, the primes
of exceptionality are given by the Chebotarev decomposition law in a
cyclotomic
field.
For rational functions, the primes of exceptionality come from explicit
class field theory, except in the special case of Redyi
functions (ramified over just two places conjugate over K). The
abstract understanding of
complex
multiplication fibers and monodromy properties work so much better
here, rather than working with explicit equation
manipulation.^{[GMS03],} ^{[Fr05,
Subsect.
6.2, esp. Prop. 6.6] }
Comments on the prime
degree
exceptionals
in #1: By applying the fiber product definition of
exceptionality, we find the Nielsen class of a prime (say, p)
degree exceptional f is that
of involution
realizations of dihedral groups with k = 0 and r = 4.
Consider an elliptic curve E_{j}_{'}
with j-invariant
j'. Only
for the complex multiplication j-invariants,
does an elliptic curve have an isogeny E_{j}_{'}
→ E' of
degree p
over Q(j'). That is also
the definition field for the degree p rational function
f, whose
cover corresponds to a
point x in
H((D_{p},_{
}Ç_{34})^{abs,red
}lying over j'
obtained from modding
E_{j}_{'}
and E' out
by their canonical "multiplications by -1."
The Galois closure of the cover from f has
definition field generated, over Q(j'), by
the coordinates of a point x'
in H((D_{pk}_{+1},_{
}Ç_{34})^{in,red}
over x. Such
a point represents the coordinates of the unordered pair {w,-w} where w is a p-division point
giving the isogeny. Excluding primes of bad reduction of the cover, the
exceptional set P_{f}
is the set of primes p' of Q(j'), for which x'
mod p' generates
a nontrivial extension of the residue field. ^{[Fr78, Thm.
2.1]}
Complex multiplication gives a formula tool to compute P_{f},
using "ray classes" (arithmetic progressions). As with cyclic
and
Chebychev polynomials, the
iterations of f
that would invert the map on a residue class field would similarly
come from
'Euler's Theorem.'
Comments
prime-squared
degree
exceptionals
and their contributions to #2: Consider G_{p},
the semi-direct
product of V_{p}
= (Z/p)^{2}
and Z/2
(written multiplicatively, elements that are +1 or -1). To understand
why this example is so important, recognize that G_{p}
is the collection group
for the dihedral group Nielsen class just above.
Therefore the computation below is actually producing the
arithmetic/geometric monodromy groups of modular curves from the
Hurwitz space viewpoint according to the Monodromy
Automorphism Principle.
The Nielsen
class of (odd) prime-squared
(say p^{2})
degree exceptionals is a variant on the involution realization Nielsen
classes: Ni(G_{p}_{,}T, Ç_{34})^{abs},
with T the
coset representation on a copy of Z/2.
The key here is that T is not a primitive representation: There is a
group contained properly between G_{p}
and Z/2,
the semidirect product of Z/p and Z/2.
The
nontrivial element of Z/2
acts as multiplication by -1 on V_{p}.
Understand the group multiplication by representing elements
of G_{p}
as 2x2 matrices M(a, v):
the first row
is (a v), a = ±1 and v in V_{p};
the second row is (0 1).
The Nielsen
class of (odd) prime-squared
(say p^{2})
degree exceptionals is a variant on the involution realization Nielsen
classes: Ni(G_{p}_{,}T, Ç_{34})^{abs},
with T the
coset representation on a copy of Z/2.
The key here is that T is not a primitive representation: There is a
group contained properly between G_{p}
and Z/2,
the semidirect product of Z/p and Z/2.
Therefore an element in the Nielsen class, up to conjugation by M(1 v) has
this form:
M(
v_{2},v_{3})
=
^{def} (
M(-1
0),
M(-1
v_{2}),
M(-1
v_{3}),
M(-1
v_{4}))
with these
equivalences,
generation
⇔
v_{2} and
v_{3}
span
V_{p} and
product-one
⇔
v_{4} = v_{3}- v_{2}.
From Riemann-Hurwitz the genus, g_{p}, of
covers in this family appears from
2(
p^{2}
+
g_{p} - 1)=
4(
p^{2}
- 1)/2.
The
degree
p^{2}
covers of the sphere in this family have
g_{p} = 0.
The
right side of the formula appears as follows. For each of the r
entries in a branch cycle description in a particular Nielsen
representative, count each of its orbits on the cosets of of Z/2 (in the
permutation T),
multiplied by the (length - 1) of that orbit. Now take the total count
of those over all branch cycles.
Computing the geomtric and arithmetic monodomy of The action of a braid
q
on M(v_{2},v_{3})
is as if you apply an element of SL_{2}(Z/p) to V_{p}.
You need only check this for the two generators q_{2}
and the shift (sh).
For example, (up to inner equivalence) the shift has this effect:
M(v_{2},v_{3})
→
M(v_{3}- v_{2},v_{3}- 2v_{2});
represented by the 2x2 matrix with rows (-1 1) and (-2 1) of
determinant 1.
The action of those two generators generates SL_{2}(Z/p), but M(v_{2},v_{3})
and M(-v_{2},-v_{3})
are inner equivalent by conjugation by M(-1
0). From
this we have the Hurwitz approach to figuring out the monodromy of the
cover of the j-line
by the modular
curve classically denoted X(n). We
restrict to over Q
for simplicity, but it works over any number field similarly.
Start with any non-complex multiplication j invariant in Q. Then, consider
any particular odd prime p.
In this Nielsen class there will be one rational function f = f_{p}
in the Nielsen class over Q,
up to absolute reduced equivalence.
Degree p^{2} Exceptional Theorem:
Part A.
Given j,
for almost all p
(by the OIT), the arithmetic monodromy group of the cover from f_{p}
will be the semidirect product of G_{p}
and GL_{2}(Z/p) with their
natural copies of Z/2
identified. For such a p,
since GL_{2}(Z/p) acts irreducibly
on V_{p}
the induced representation T^{ar}
is primitive. Thus, f_{p}
is decomposable over Q^{alg},
but indecomposable over Q.^{[Fr05,
Prop. 6.3]}
Part B. For
∞-ly many primes l
(avoid those of bad reduction) f_{p}
mod l has
the same property: indecomposable over Z/l, but decomposable over its
algebraic closure. For those same primes f_{p}
mod l is
exceptional over Z/l.^{[Fr05,
Prop. 6.6] }
The point in Part B is to consider those l for which the
Frobenius in the Arithmetic monodromy of f_{p}
mod l is
representated by a matrix that is not diagonalizable mod l. By
taking an
especially good j'
of the j
invariant in Q,
one based on Andrew Ogg's
elliptic curves^{[O67]}, an idea of Serre^{[Se81,
Thm. 22]}
suggests reading off those corresponding l of exceptionality
from a zeta function a 'la Langland's conjectures.^{[Fr05,
§6.3.2]} So, the Chebotarev density Theorem gives the ∞-ly
many primes l
for any given values of j
and appropriate corresponding p.
Here, though this is non-abelian number theory. Those values of l will no longer
fall in anything like a union of arithmetic progressions.
IV.f. Conclusions on
A-G dihedral-related
monodromy:
As previously, the solution of Davenport's problem was the guide to
drawing conclusions, based on two aspects. What new would BrA acting on
Nielsen classes divulge, and would the covers
beyond those
revealed as solutions of Problems #1 and #2 be limited in the style of
the solution of Davenport's
Problem a la the Genus
0 problem?
BrA action on Ni(G_{p}_{,
}T,
Ç_{34})^{abs,red} and Weil's pairing:
There are p
- 1 braids orbits, corresponding to the same number of absolutely
irreducible components on H(G_{p}_{,}T,_{
}Ç_{34})^{abs,red},
each conjugate over the field generated by the pth roots of 1.
Each has geometric monodromy SL_{2}(Z/p)/<±I_{2}>
over the j-line,
and arithmetic monodromy GL_{2}(Z/p)/<±I_{2}>.
The arithmetic/geometric monodromy quotient from roots of 1
traditionally comes from the Weil
pairing on p-division
points on an elliptic curve into the pth roots of
1. There is a seemingly different Hurwitz interpretation that
comes from the lift invariant based on the small Heisenberg group.
This is only a
slight modification of the argument in the example multiple HM
components, where the Weil pairing is on a hyperelliptic
Jacobian.
The Guralnick-Müller-Saxl
completion of Problems #1 and #2:
The monodromy of rational covers given above being based on dihedral
groups does not violate the genus 0 problem. Precluding all but
finitely further degrees for covers that where solutions (over number
fields) to Problems #1 and #2 was taken on in [GMS03, Thm. 1.4]. The
method was parallel to the precise description of exceptional
polynomials over finite fields, based on the description
of primitive polynomials from the classification of finite simple groups.
Their
description of the covers of degrees not included above used a simple
device: Divide the rational functions f into three types: The geometric
Galois closure of f has genus g_{f }=
0, 1 or > 1. The first two cases go back to the Schur Conjecture
and
the work above. The 3rd case consists of a finite list of degrees. The
role of Fried in [FrGS91] was taken here by Müller, who also provided
the precise arithmetic touches who showed that the potential degrees in
this case did actually produce example solutions of Problems #1 and #2.
V. Galois
Sratification and Poincaré series
James Ax
and
Simon Kochen
proved a version of
Artin's conjecture that Q
hypersurfaces in projective d^{2}-space
of degree
d have p-adic points.
Their result: For
a given d
there is a finite list, P_{d},
of prime exceptions to the
result. Their device was to construct an ultraproduct of p-adic rings. They
showed these ultraproducts were isomorphic to ultraproducts
of formal power series over finite fields of order p. That isomorphism
left unsolved finding a meaningful elimination of quantifiers for
general diophantine statements over related collections of
fields and rings.
V.a. Galois
Stratifications: The archetype was to do this over almost
all finite
fields. For this Fried formulated the category of Galois
Stratifications. He
combined the MM
and a general version of Chebotarev's density theorem writ large
for an explicit geometric elimination procedure based on this category
that supported the geometry of fiber products.
Why Elimination of
Quantifiers works:
Projections on coordinate axes effectively map Galois stratification to
Galois stratifications, but they don't map elementary statements to
elementary statements.
In the course of doing this, he observed that versions of Chebotarev
and of Hilbert's
Irreducibility Theorem were essentially the
same. More than any other contribution of Fried, Galois stratification
gave meaning to the title "Field
Arithmetic" and the significance of Hilbert's theorem to
characterizing fields by their diophantine properties.
V.b.
Projective
Groups, and PAC and Projective Fields: A
profinite
group G is
projective if any homomorphism of it onto – a (group) cover – of
A
– extends to any
(profinite) cover B
→ A, of A.
If the extension can be taken to be a cover of B whenever B is a quotient of G, then G is said to have
the embedding property,
and the combination of projective and the embedding property is called superprojective.
Ax had discovered – but Gerhard
Frey named – the pseudo-algebraically closed
fields K:
every
absolutely irreducible variety over K has a nonsingular
K
point.
Ax
noted that G_{K}
is a projective group.
So we say K
is projective. In this K
is akin to the algebraic closure of finite fields and the field of
cyclotomic numbers, though neither is pseudo-algebraically closed.
Given any projective profinite group G^{*},
it is easy to construct fields K having G_{K}
isomorphic to G^{*}.^{[ExProjF]}
Yet, the location of such K
was another matter. Ax
thought that pseudo-algebraically closed subfields of Q^{alg}
were rare in that the only one that
would be Galois over Q
would be Q^{alg}
itself. Fried and Moshe
Jarden,
however, produced such fields in great abundance including that such
fields could have the Hilbertianity property.
Then, Fried, Jarden
and Dan
Haran used
these ingredients to extend the Galois stratification elimination of
quantifiers to statements over various
collections of fields inspired by the Chebotarev density theorem. These
are the Frobenius Fields: PAC fields K with G_{K}
having the embedding property.
These fields not only support Chebotarev's famed field crossing argument,
but that property essentially characterizes them. The geometric
construction of Frobenius subfields in Q^{alg},
Galois over Q,
fell to applying Hilbert's irreducibility to covers
of Hurwitz spaces. This generalized a previous result:
Regular Inverse
Galois
for PAC fields: Every
finite group has a regular realization over any PAC
field K.
Indeed, such realizations correspond to K points on an
infinite collection of spaces indexed by pairs given by the
group and K-rational
collections of its generating conjugacy classes. Key though was showing
that among them were absolutely irreducible Q (and so K)
components.
The more general result produced absolutely
irreducible covers of absolute Hurwitz spaces by inner Hurwitz spaces
attached to each finite group in some cofinal collection of finite
groups. One result was fields that illuminated how G_{Q} could
appear as varying
kinds of extensions of known groups by known
groups.
V.c.
Poincaré
series with Cohomology coefficients: Further,
to such statements over a given finite field, Galois Stratifications
provided coefficients –
that one could specialize at any prime p, by applying the
appropriate power of the Frobenius
– for Poincaré
series.
So, each p
produced a power series with integer coefficients. The
explicitness extended to show these series were explicitly
computable rational functions. This relied on the cohomology of Bernard Dwork and on
Enrico Bombieri's
explicit computation of bounds on that cohomology.
Later
Jan Denef
and Francois Loeser
used this result to form Chow
motive coefficients – formal sums of étale cohomology
groups of nonsingular projective varieties – for Poincaré series
attached to many p-adic
problems. The
pure Galois stratification method applied to every prime p. The
Denef-Loeser refinement produced canonical series (but valid only for
almost all primes). Their series supported such sophisticated notions
as an Euler
Characteristic, inherited from the Chow coefficients.
By
contrast, Fried used the additivity of these series, with Tate twist
(powers of cyclotomic characters) coefficients to consider construction
and detection of series, with infinitely many 0 coefficients,
attached to well-known diophantine problems. The subject – Monodromy
Precision – generalizes the whole topic of exceptional
covers
and Davenport pairs. The chain of ideas used an umbrella generalization
of
exceptionality and Davenport pairs called pr-exceptional
covers.
For these, a MacCluer-type theorem holds.^{[MonPrec]}
So
there was no error term in applying Chebotarev Density to characterize
such covers by their arithmetic monodromy. For example, consider two
Poincaré series for a Davenport pair.
- P_{U}(f_{1},f_{2}) for counting
points in the union of the ranges.
- P_{I}(f_{1},f_{2}) for counting
points in the intersection of the ranges.
Coefficients in
P_{U}(f_{1},f_{2}) - P_{I}(f_{1},f_{2}) corresponding to
finite fields in which the ranges of f_{1}
and f_{2}
are the same are 0.
VI.
Presentations
of G_{Q} from
transitive BrA
VI.a. An
equivalent to the Regular Inverse Galois Problem:
Fried
and Helkmut Voelklein
attached to each centerless finite group G
and subfield K
of the
algebraic numbers an infinite collection V_{G,K} of
absolutely
irreducible Hurwitz spaces. There
were two significant properties of the Nielsen classes, after their
being nonempty (elements had to exist satisfying product-one and
generation).
- They included all those conjugacy class collections
Ç
that were rational
unions over K. Then, K regular
realizations of G
corresponded one-one K
points on one of the varieties in V_{G,K}.
- To even have any chance for K points required
that some variety in the collection V_{G,K} had
at least one K
component, a conclusion guaranteed by transitive BrA.
VI.b.
Perfect
groups and the
L(ift)I(nvariant):
A p-perfect
group G is
one with order divisible by p,
but with no cyclic order p
quotient. A perfect
group is p-perfect
for all allowed p.
Such a group has a special Frattini cover: the universal
central extension, ψ_{G}:
R_{G}
→ G.
Denote the kernel of ψ_{G}
by U_{G}. The
lift invariant
LI, is a BrA
invariant, and it is computed from the Galois closure of the cover.
It maps elements in a Nielsen class, Ni=(G,T, Ç), to U_{G,Ç}, a quotient
of U_{G}.
Example: When the order of U_{G} is
prime to the
order of all elements in Ç, then U_{G,Ç} = U_{G}.
Compute the lift invariant of (g_{1},…,g_{r}) by selecting the
unique element ĝ_{i}
in R_{G
}lying over g_{i} having the same
order. Then, form the product g_{1}^{…}g_{r}.
That works with the first example developed by Fried and Serre. When G is
the degree n
alternating group, and all elements in Ç
have odd order, then U_{G,Ç} has
order 2. That case has a precise formula for the LI when the genus g_{G,Ç} of
the covers in the Nielsen class is 0. When all the elements of Ç are
3-cycles, the genus is 0 when r
= n - 1,
and the LI
is n
-1 mod 2. See the higher genus
case.
When you can't lift to have the same order, remove the
ambiguity in the lifts by forming a quotient of U_{G,Ç}.
While that seems like a formula, there is no general procedure for
computing the LI.
In general, R_{G}, for
perfect G
finding the maximal central quotient of the universal Frattini cover of
G, is
nontrivial to
compute.
VI.c. Conway-Fried-Parker-Voelklein
and
Harbater-Mumford components:
CFPV states
that for a given Nielsen
class Ni=(G,T, Ç), the number
of components of the Hurwitz space is precisely the number of elements
in U_{G,Ç}
under an inexplicit assumption called with high multiplicity.
CFPV Statement: While
the best results are harder to state, this applies, with a fixed G, if
all conjugacy classes appearing in Ç, do so with high multiplicity.
For, however, the present applications – excluding to the OIT – such as
the presentations
of G_{Q},
the high multiplicity assumption was removed almost immediately by
results that appeared with the topic of Modular Towers.
Harbater-Mumford Components:
A representative g
= (g_{1},…,g_{r })
in a Nielsen
class where r
≥ 4 is even and g_{2i-1}=g_{2i}^{-1},
i=1, …,r/2,
is a Harbater-Mumford (HM) representative. David Mumford
considered total
degeneracy of curves over a p-adic disk, and Michael Artin
introduced the idea of total degeneracy of covers to David Harbater,
who applied it to p-adic
production of covers with a given monodromy group.
The
braid orbit of an HM rep. is an HM orbit, and its corresponding Hurwitz
space component is an HM component. HM representatives (and their
orbits) always have lift invariant 0.
HM
Theorem:^{[Fr95, Thm. 3.1]}
Part 1: For a given Nielsen class with a rational union of conjugacy
classes, the collection of HM components has definition field Q.
Part 2: The condition that Ç is HM-gcomplete
(includes that
the Nielsen class contains an HM rep. and Ç contains each
conjugacy class at least four times) guarantees there there is
precisely one HM component. So that component has definition field Q.
The proof of the HM Theorem used a compactification scheme called a specialization sequence.
Stefan Wewers,
and separately Dèbes and Michel
Emsalem
developed a version of the Deligne-Mumford compactification to prove it
in a more classical setting. The HM Theorem explicitly replaces the
Nielsen classes used by CFPV
in the most
well-known applications of it.
As important as general results was providing explicit Nielsen/Hurwitz
spaces addressing these points:
HM orbits don't suffice to handle these topics for these reasons:
- The strongest comparison with modular curves – and also
the place where our present diophantine results work – is when r = 4, and the HM
Theorem criterion won't hold.
- HM components always have trivial lift invariant.
- The Davenport parameter properties rarely hold for HM
components.
VI.d.
Quotients of
the Absolute Galois Group: By using a cofinite
collection of finite groups within the projective
limit of all finite groups, Fried-Voelklein result could rely on an
extreme case of the CFPV:
U_{G,Ç} is
the trivial group. The first applications presented G_{Q} as
extensions of a
known
group by a known group. Both results assume that K is a PAC subfield
of Q^{alg}.
Denote the profree
group on a countable
set of generators by F_{ω}
and refer to a G_{K}
isomorphic to it as ω-free.
Theorem
A:
Then, K is
Hilbertian
if and only if G_{K}
is
isomorphic to F_{ω}.
In that case G_{K}
is in a short exact sequence
1→ F_{ω} →
G_{K}
→ ∏_{n=2}^{∞}
S_{n}
→ 1: The product of copies of the symmetric group S_{n},
by F_{ω}.
Theorem
B:
Parallel to Theorem A, K
is RG-Hilbertian
if and only if each finite group is a quotient of G_{K}
(realized as a Galois group over K).
Further, there are RG-Hilbertian PAC fields that are not Hilbertian.
These results made obvious the right
generalization of Shafarevic's Cyclotomic field conjecture:
(*)
Shafarevic Generalization:
An Hilbertian subfield of the algebraic numbers should have profree
absolute Galois group if and only its Galois group is projective.
New
realizations of simple group series as Galois Groups:
Serre
didn't use the braid monodromy (rigidity) method. This is necessary to
achieve
anything significant in regular realizations except for the few lucky
groups for which three conjugacy classes happen to satisfy the rigidity
criterion. The difference shows almost immediately in considering the
realizations of Chevalley groups of rank exceeding one.
Serre's
book records just three examples of Chevalley groups of ranks exceeding
one having known regular realizations at the time of his book.
When Thompson and Voeklein
initiated a program to produce (many series of) simple groups as
absolute Galois groups
over Q,
the only known simple groups so realized, outside of cyclic and
alternating groups, were rank 1 affine
groups.
They chose varieties appearing in the Fried-Voelklein series with
transcendental parametrizing parameters akin to those that
solved Davenport's
problem.^{[VolRef]}
They did this for simple groups of arbitrarily high
rank, also using conjugacy classes – that as a collection – were
rational over Q,
guaranteeing regular realizations of the groups. Clearly only
technically expert group theorists could have gone that
far. Yet, that didn't produce regular realizations
of even all simple groups, much less those that arose in the
Modular
Tower Conjecture.
VII. The inverse Galois
problem generalizes modular curve
thinking
VII.a.
The Universal Frattini
Cover: Assume G
is p-perfect.
Like the universal
central extension, but with a much bigger kernel,
there is the universal Frattini
cover U_{G}
→ G.
The kernel is pro-nilpotent, so a product of its p-Sylows. Each such
p-Sylow
is a pro-free pro-p
group whose finite rank (minimal number of
generators – m_{p}) is often
non-obvious, despite an explicit
computation for it.^{[Fr02, Thm. 2.8] }
That allows
forming:
- the universal p-Frattini cover of
G: U_{p}→
G
extending to any cover of G
with p-group
kernel;
- the universal abelianized p-Frattini
cover of G: U_{p,
ab}→
G
(with kernel isomorphic to Z^{mp})
extending to any cover of G
with abelian p-group
kernel; and
- U_{p,k,
ab}
→ G,
the universal extension
of G with
abelian kernel of exponent p^{k}, k = 0, 1, 2, …, for
which U_{p,1,
ab} is an indecomposable Z/p[G] module (no
nontrivial Z/p[G] summand) and the
kernel U_{p,k+1,
ab}
→ U_{p,k,
ab} is a copy of U_{p,1,
ab}.
Among simple (noncyclic) groups there are no present
realizations over Q (including regular realizations) of any of the
covers U_{p,1,
ab}
→ G (k=1). Example: We
know precisely these groups for G
= A_{5},
p = 2, 3
and 5.^{[Fr95, Part B] }
A full Frattini quotient
of U_{p,
ab}→
G is a
quotient E
→
G with
kernel isomorphic to (Z_{p})^{m'}, 1 ≤ m' ≤ m_{p}.
There are then corresponding groups E_{p,k,
ab} →
G,
k = 0, 1,
2, ….
Example: The universal central extension of A5, has kernel Z/2. It is a
Frattini quotient, but its only extension to a full Frattini quotient
is to U_{2,
ab}
→ A_{5},
for which m_{2}
= 5.^{[Fr95, Prop. 2.9 and (2.8)]} This phenomenon,
is common, for simple groups.
For, however, groups G
with normal p-Sylow subgroup, we expect (proper) full Frattini
quotients. For example, suppose (Z/p)^{m}xH = G is p-perfect, with m ≥ 2: H is p' and it acts
without fixed points. Then, there is an action of H on (Z_{p})^{m} making (Z_{p})^{m} x H as a full
Frattini quotient of U_{p,
ab}→
G and m_{p} = (m-1)(p^{m})+1.^{[FrJ,
p. 195]}
Even in this last case, the use of the universal p-Frattini
cover allows computing the lift invariant of elements in a Nielsen
class. Modular Towers specifically needs to know, at each level and for
a given prime p, the maximal G
invariant quotient of the kernel of U_{p,
ab}→
G for
identifying components attached to different lift invariants, as in
the small Heisenberg group.
VII.b.
The
Modular
Tower
conjecture: Early on the Modular
Tower conjecture had this form: Generalize the dihedral
group Branch
Bound Conjecture by taking
the
inputs to be any finite p-perfect
G. Then,
replace D_{pk+1} by
U_{p,k, ab}
→ G,
still using the bound N_{p}
on the number of branch points.
Suppose Ç consists
of p' (orders
relatively prime to p)
classes of G.
Then, each class in Ç
lifts to a unique conjugacy class in G_{p,k}.
This is an easy case of the Schur-Zassenhaus Lemma. So, there is no
ambiguity in the notation for Nielsen classes such as Ni=Ni(U_{p,k, ab}, Ç)*, with * an
equivalence, indexed by k
= 0, 1, 2, …. and
their corresponding Hurwitz spaces.
Comment
on Modular Towers with absolute Nielsen classes: Suppose
the permutation representation T
attached to an absolute Nielsen class is from cosets of a p' group G(1): T is p'. then
Schur-Zassenhaus allows forming towers based on absolute classes: Use
the conjugacy class of the p'
lift of G(1)
to each level. This applies to the modular curves X0(pk+1), for p odd,
using that G(1)
is Z/2.
For simplicity, restrict to * =
inner or inner, reduced: The main ideas of p-cusp growth, and
computing the genuses of tower levels still apply for absolute classes
under the T
is p'
assumption, but the formulas are trickier.
The following is an application of the BCL; the major
point is that Ç is
p'.
Main MT Proposition:^{[FrK98]
[De05]} Assume for some N_{p}
(dependent on G
and p),
and for each k,
there is a K-regular
realization of U_{p,k, ab}
with no more than N_{p}
branch points. Then, for there is a p' set, Ç, of
cardinality r' ≤ N_{p},
of classes in G,
for which the Hurwitz space, H(U_{p,k, ab}, Ç)^{in}
(or H(U_{p,k, ab}, Ç)^{in,red}) off
its cusp locus,
has a K
point k =
0, 1, 2, ….
Main MT Conjecture: For the
spaces H(U_{p,k, ab}, Ç)^{in }for
k large
dependent on fixed (G,
Ç, p) as
above.
- They have general type: high powers of their
canonical
bundles embed them in projective space.
- They have no K
points (off the cusp locus).
- Further, these statements should hold also for any full
Frattini
quotient, E
→
G,
of U_{p,
ab}→
G.
Genus Growth Theorem:
For r = 4,
#1 and #2 are equivalent and also equivalent to showing that the genus
of the
compactification of H(U_{p,k, ab}, Ç)^{in,
red }(or of H(E_{p,k, ab}, Ç)^{in,
red}) exceeds 1 for large k.^{[Fr06,
Thm. 5.1]}
Modular Tower
Definition: The objects called (abelianized)
Modular Towers are projective sequences of (irreducible) components on
the collections H(U_{p,k, ab}, Ç)^{in }(or
H(E_{p,k, ab}, Ç)^{in}),
or their reduced versions, k
= 0, 1, 2, …..
Proof
of the Main Conjecture for r = 4: If r = 4,
these upper half plane quotients are only modular curves when G is closely
related to the dihedral group and the conjugacy classes are
involutions.
Two approaches have proved the conjecture when r =
4.
Method
1:
Precise formulas for the genus of reduced Hurwitz spaces – based on BrA
– when r =
4 allow bounding the genus away from 1 for
the level k
spaces by referencing new properties of the cusps. Fried
produced many examples with the crucial p-cusps
that force the genuses of tower levels to rise quickly and explicitly.
That gives
levels beyond which there would be only finitely many K
points.^{[Fa83]}
Given only finitely many K
points at high levels, such points at every level imply there is a
projective sequence of K
points on the tower. Reduce the tower modulo some prime of good
reduction (its characteristic is prime to |G|), producing a
projective sequence of points on the reduced tower all defined over a
fixed finite field F_{q}. That
gives a line on a Jacobian for which
the action of a power of the Frobenius over F_{q} is
trivial. This
is contrary to the Riemann
hypothesis for abelian varieties over finite
fields. That says the eigenvalues of the Frobenius have absolute value q^{1/2}.
Method 2:
Tamagawa and Cadoret later proved the result in general for r = 4, but
with no explicitness result.
No Tate Character
Theorem in Dimension 1: Suppose χ: G_{K}
→ Z_{p}* is
that acts through p-torsion
on an abelian variety A
and it does not appear as a
subrepresentation on any Tate module.^{[Se68],
[DeFr94], [BaFr02]}. Then, for A varying in a
1-dimensional family
over a curve S
defined over K,
there is a uniform bound on the elements A_{s},
the fiber over s ∈ S(K), through which the action is through χ (the
analog of a cyclotomic
point). In particular, this gives the Main MT conjecture when
r = 4.
The
MT
conjecture and the
Inverse
Galois Problem: If the nontrivial
center condition for fine moduli on inner spaces holds, then K points on tower
levels correspond to regular realizations of the G_{k}
defining the kth
tower level. Since the group G_{k
}is automatic from the level 0 group, two facts
assure that the nontrivial center condition does not affect
the main conjecture.
- When G_{0}
is p-perfect
and centerless, then so is G_{k},
k ≥ 0.^{[BFr02,
Prop. 3.21]}
- If there is a p part of the center of G_{0}
= G,
then there is a p-Frattini cover G → G_{c},
with G_{c}
p-perfect and trivial p center, with
corresponding classes Ç_{c}. The main conjecture
holds for the tower for (G_{c},Ç_{c})
if and only if it holds for (G,Ç).^{[Fr06,
Prop. 3.3]}
VII.c.
Cusps and
Explicit Tower Levels: The ramification indices of cusps
on a
reduced braid orbit Ni', are the lengths
of the orbits of q_{2}
on Ni'. The explicitness in Method 1 appears in displaying
the precise ramification of the cusps over j = ∞.
The three cusp types for r
= 4 are
indicated through
the entries of a representative g
= (g_{1},g_{2},g_{3},g_{4}).^{[Fr6,
§5.2.1]}
- p-cusps:
a
cusp for which that orbit length is divisible by p.
Equivalent to p
divides the order of the middle product, (g_{2}g_{3})mp, g_{2}g_{3}.
- g-p'
cusps
– group p'
– which include the q_{2}
orbits of HM elements. The elements g1 and g2 (resp. g3 and g3) both generate
p' groups.
- o-p'
cusps – only p'
– neither p
nor g-p'.
It is the growth of the ramification of p-cusps that it is
the main ingredient guaranteeing the growth of the genus of tower
levels corresponding to a projective system { _{k}O}_{k=0}^{∞}
of braid orbits. Denote a projective system of cusps through
a projective system of representatives { _{k}g}_{k=0}^{∞}.
If p^{u}
exactly divides ( _{k}g)mp, u ≥ 1, then p^{u}^{+1}
exactly divides ( _{k+1}g)mp.^{[FK97,
Lift. Lem. 4.1]} Denote the cusp associated with _{k}g by _{k}p and the
ramification of _{k+1}p/_{k}p
by e( _{k+1}p/_{k}p).
p-growth Frattini Principle[BFr02,
Lemma 2.23] : If the p
part of the center of G
is trivial, for p
odd (resp. p
= 2) and k
≥ 0 (resp. k
>> 0), then e( _{k+1}p/_{k}p) = p. That is the
expected conclusion in most cases, even p=2.
Explicitness Addition ^{[Fr06, Princ.
3.5]}: This details how to drop the condition on the center
of G. It
also gives the details for the case p = 2. The
complications for p=2
are from figuring what are the orbit lengths, nondecreasing with k, of Q'' on _{k}g. All Q''
orbits on a given braid orbit have the same length, 1, 2 or 4. Two
extremes that give the expected conclusion:
- Q'' is trivial on the Nielsen class of _{k}g for all k (as with modular
curves).
- Q'' action has 4 Nielsen classes as its orbit on _{k'}g some k' (and then it
does so for all k
≥ k').
Expectation
there will be p-cusps at some level: Suppose at no level
is there a p-cusp.
Then, we know
precisely, for large k,
what would be the relative covers from level
k+1 to
level k of
the modular tower level components: Either if is a degree p polynomial cover
or it is a rational function
ramified of order p
at precisely two places.^{[Fr06,§ 5] or [Fr06b, Thm.
6.1, Cor. 6.2]} Once a p-cusp
appears at some level, the genus rises rapidly at higher
levels.^{[Fr06, Fratt. Princ. 1, Princ. 3.5]}
There may be no p-cusps,
say, at level 0; only o-p' and g-p'
cusps. Then, there is a lifting invariant criterion for the
existence of p-cusps
above the o-p' cusps.^{[Fr6, Princ. 4.24, Fratt. Princ. 3]}
The p-cusps
and g-p'
are exactly
the types of cusps that fall on modular
curves (indeed, the g-p'
are just HM reps.), but no analog of the o-p type.
For all
values of r,
there are generalizations of the types above. For the g-p'
type, the lift invariant of an BrA
orbit will always be trivial. At this time, though these
methods for proving the Main Conjecture do not extend to r ≥ 5.
Both Method 1 and 2 hit one piece of non-explicitness hidden
in Falting's
result.^{[Fa83]} Say, in Method 1, once
the genus exceeds 1, that guarantees
only finitely many K
points at each level, forcing a projective system of K points along the
levels, so
eventually none. But there is no method yet for giving a
bound on that
finite number, and so for finding level k_{0}
beyond which there are no K
points.
From the subtle differences between the inner and absolute spaces, even
in the modular curve, case, it is the inner spaces for which the
conjectures have been made. Still, the genus formulas for r = 4, apply to
both absolute and inner spaces. For absolute spaces there is an extra
consideration in computing the p
ramification in the absolute case over cusps.
Specific formulas handle this.^{[BFr02, §3.3.1]
}
VII.d.
Invariants that
distinguish Tower Level components: Grothendieck
prophesied that
it would be difficult to untangle phenomena involving questions on
abelian varieties of dimension d
> 1, at least along larger codimension loci in the space of such
varieties, because of all the correspondences coming from jacobians of
curves. The Hurwitz space approach tames much of this concern by
detecting the effect of such correspondences through the distinction
between components.
Item
#2 aims to show that Modular Towers are a device to handle the issue
Grothendieck raised in service of l-adic representations for which the
goal was to produce results like those of the OIT. Such a big project
required an example, as encompassing as is modular curves,
which
revealed new phenomena. Item #2 produces the l-adic
representations from projective systems of points on a projective
system of spaces whose components and definition fields at different
levels we understand. RETURN
When a component with trivial LI
has transitive BrA,
then that component has definition field Q.
As with the use of the BCL,
when the BrA
is transitive on all elements
with a given lift invariant, the definition field of the component with
that lift invariant is
in a cyclotomic field.
VII.e. Four systems of
Nielsen
classes: The LI
invariant has been tested in challenging particular cases. In each,
the conjugacy classes Ç are Q
rational. RETURN
- Alternating
groups and theta nulls: G
= A_{n}, Ç = Ç_{3r}
, r ≥ n ≥ 5 (g_{G,Ç} ≥
1; r
3-cycle conjugacy classes) for absolute
spaces parametrizing degree n covers or the inner spaces
parametrizing their Galois closures.
- Multiple
Harbater-Mumford Components: G_{pk+1},_{
}is the semidirect product of
Z/3
(generated by an element α) and (Z/p^{k+1})^{2},
where the lift invariants run over all values in the kernel of
ψ_{G}: R_{Gpk+1}
→ G_{pk+1},
which is isomorphic to Z/p^{k+1}.[Fr06,
§ App. A.2]
Here Ç is two repetitions each of the conjugacy class
of α and its inverse.
- G = An, n
RETURN In the alternating and 3-cycle case above, when r = n - 1, that unique
value of the LI
corresponds to a stronger result: There is precisely one Hurwitz space
component (BrA
is transitive), so its definition field is Q. There are several
inverse Galois applications to just this "one" case and its
generalization to r
≥ n
(below).
[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].
Each cover in each case has a corresponding natural half-canonical
divisor class obtained by taking 1/2 of the differential divisor of the
covering map. Serre computed the parity of its linear system.
For Item #1, there is one component, but the lift invariant is the
covers in those components
separate according to whether their supporting theta functions are
respectively even
or odd.
The significant difference
is that their corresponding theta-nulls can only be nonzero if the
theta functions are even. Then, those theta-nulls are analogous
to automorphic functions on the Hurwitz space components, even though
those components are not homogenous spaces.
Small Heisenberg
group: For Item #2, there are two extreme cases: when the LI is 0, and when
the LI is
relatively prime to p
(p'). The
components with p'
LI are
all
conjugate over the cyclotomic field generated by a p^{k+1}
root of 1. Also, the BrA
is transitive on all elements with a given p' lift invariant.
There are, however, K_{p}p^{k} components with 0 LI : K_{p}= (p-1)/6
(resp. (p+1)/6)
if 3 divides p-1
(resp. p+1).
So far, the cover
separation problem remains a mystery for these components, all of which
are Harbater-Mumford.
The remaining components in Item #2, as k varies,
extrapolates between these two results. The analogy with Serre's Open
Image Theorem for modular curves regards modular curves as the
case where G
is the dihedral group: semidirect product of Z/2 acting as
multiplication by -1 on Z/p^{k+1}.
RETURN
[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.
VIII. Concluding Remarks on "Field
Arithmetic"
Arithmetic refers to the basic operations of calculations within a
field. "Field
Arithmetic" tested and differentiated between collections of fields
based on the
general results that hold for various types of covers in the fibers
over points in their base. Though class field theory, abelian varieties
and the classical study of the abelian extensions of a given number
field influenced it, it took an entirely different approach to
investigating Q^{alg}
and its automorphism group G_{Q}.
Especially in Fried's case, not tending to
extend the base field Q
to, say, positive characteristic.
VIII.a.
Hilbert's
Irreducibility
Theorem: For x
in Q denote
the group of automorphisms of the minimal Galois extension Q(x)^{^}/Q containing x by G(x). Refer to x in Q^{alg}
as solvable
(over Q)
if Q(x)^ is in a field
generated by iterating the operation of
adjoining roots (including roots of 1) of previously known numbers. By
the middle of the 1800s, Galois was accepted as having
shown that knowing G(x) is sufficient to
deciding if x
is generated by radicals.
The standard form of the Hilbertian
property for a field K is equivalent to this: If
φ: X
→ P^{1} is a Galois
cover, then there are infinitely many z in P^{1}(K)
for which the decomposition group of the fiber is the same as the group
of φ. The RG-Hilbertian
property has the same conclusion, but it applies only to
those φ (regular over
K) for
which the components of X
are absolutely irreducible over K.
That this was only a crude statement about the nature of x is understood
from the amount of attention given to finding "explicit" generators
(which would also be solvable) of abelian extensions of
specific solvable numbers. As when j' = x is a complex
quadratic number.
To no less than Hilbert it was a mystery quite what to say about the
nature of general algebraic numbers. He posted his irreducibility
theorem as a tool to provide algebraic numbers generating
extensions of given fields beyond abelian.
IG Conclusion:
If G is
regular over Q,
then there are infinitely many x,
with pairwise Q(x)^{^}s
disjoint over Q,
and G(x) = G.
Add to these the following statements, with the first often attributed
to Hilbert.
IG Conjecture:
Show the IG conclusion holds for every finite group G.
IG Realization:
Beyond solvable groups, and symmetric groups, the only successes with
the IG Conjecture come from applying the irreducibility theorem to
regular covers. Also, the method is some version of Thompson-Voelklein.
In applying the irreducibility theorem, the field Q(x)^{^}
often has evidence of the Nielsen class from which it came if it
appears as a decomposition field of a regular
cover. For example, undergraduate algebra courses produce G(x_{p,k})
= D_{pk+1}
for any fixed p with Q(x_{p,k})^
ramified at a bounded set of
primes for all k,
but the ramification index (over Q)
is unbounded at some of those primes.
Even for dihedral groups – allowing no restriction on the hyperlliptic
jacobian dimension – no one has shown the IG conclusion
holds for any fixed p
> 2, and all k
with uniformly bounded ramification indices at
all primes of Q(x_{p,k}).
It is an error to consider A-G
realizations of pairs (G*,G) as so much
harder than the problem of producing every finite group as a Galois
group over Q.
As in the original OIT,
and even with the lattitude where K
is a PAC field, connecting G and G*/G through an A-G realization gives
a guide from how much more control we have over the geometric part.
VIII.b.
Properties
of
fields and understanding the algebraic numbers:
This approach considered fields L_{B}
intermediate between Q
and Q^{alg}
according to properties that interpolated between classical
conjectures and the techniques around Chebotarev's Density Theorem
and Hilbert's
Irreducibility Theorem. The aim was to find
fields L_{B}
whose arithmetic properties would distinguish them by what they said
about G_{Q}.
In addition to L_{B}
being Galois over Q,
we might want to "know" the following about L_{B}:
- Generators for it over Q
and its Galois group;
- properties of generators of its absolute Galois group;
and
- for various collections of covers C' → C, with C absolutely
irreducible over L_{B,}
the possible decompositions groups of its fibers over points of C in L_{B}.
Shafarevich's
conjecture:
The cyclotomic closure, Q^{cyc},
of Q is
perfect for #1. Its generators are all the roots of 1, its Galois group
over Q is the profinite invertible integers, but it is only a
conjecture that G_{Qcyc}
is ω-free, and without some form of the field crossing argument #3 is
untouchable.
The field of totally
real numbers: An α in Q^{alg}
is totally real if all its conjugates over Q are real. For the
collection of all such numbers, Q^{tr},
G_{Qtr}
is freely
generated by involutions (all conjugates of complex conjugation). While
Q^{tr}
is not PAC, it is just as good, for it is always possible to decide if
a non-singular absolutely irreducible Q variety has a real point, and
that is all we need to locate points of Q^{tr}.
Also, it is not Hilbertian, but it has a property almost as good: real-Hilbertianity.
The alternating
closure, Q^{alt},
of Q: This is the
composite of all Galois extensions of Q having some A_{n},
n ≥ n_{0}
(n_{0}
arbitrary) as group. So, its Galois group over Q is a product of An
groups.
Therefore, Q^{tr}
has the kind of properties that bode well for #3. Another result that
is helpful: Adjoin the square root of -1, and you have a Galois
extension of Q that is PAC. Yet, we know little of the Galois
group of Q^{tr}/Q,
and much more about the group of Q^{alt}/Q. Still, for good
reasons an attempt to prove that Q^{alt}
is PAC failed. Combining properties – if only we could – of these
suggestions would produce a canonical L_{B }with
which independent researchers could focus on deeper properties
of Q^{alg}.
VIII.c.
Hilbert
Irreducibility
and Weil's Decomposition Theorem: Ax and Kochen, Rafael Robinson, Peter Roquette, and
others, in their use of ultraproducts of fields of differing
characteristics used an idea they called nonstandard primes.
Fried noted that these were equivalent to the "arithmetic
distributions" used in two famous results that applied over any number
field K,
with ring of integers O_{K}.
- Carl
Ludwig
Siegel's characterization of affine curves over K with infinitely
many points with coordinates in O_{K}.
Not only did they have to have a projective normalization of
genus 0, but, even then, they had only certain special forms.
- Andre Weil's
thesis which – modulo a polarization result – established that on any
abelian variety the abelian group of its points over K was
finitely generated.
Fried also introduced the notion of Universal Hilbert
subsets: Infinite
subsets S_{uni}
in a number field K,
such that for any cover φ: X
→ P^{1} over K, the fiber over
all but finitely many values in S_{uni}
is irreducible over K.
The notion works equally by restricting the conclusion just to a
special set of covers χ, denoting the universal set by S_{uni,χ}.
Fried
used the distributions – as they appeared in Weil's thesis – to show
the equivalence of the results of Spindzuk and Weissauer. He
also
used them to give a standard proof of Weissauer's result using
nonstandard primes that fields with a product formula are Hilbertian.
Fried also proved a result that extended Kuyk's proof that the
cyclotomic closure of a number field K is
Hilbertian.
Theorem (Weissauer):
Assume K
is Hilbertian. Consider a nontrival algebraic
extension M/K. If the finite
algebraic extension M_{1}/M is not contained
in the Galois closure of M/K,
then M_{1}
is Hilbertian.
Weissauer
had intuited this result from using non-standard primes, but Fried used
neither these nor distributions. Dan Haran generalized the proof to
show the following.
Haran's Diamond
Theorem:^{[FrJ, 2nd Ed., Thm. 13.8.3] }Again,
K is Hilbertian. Assume M_{i}/K,
i=1,2, are Galois, and M/K
is contained in neither, but is in M_{1}^{.}M_{2}.
Then, M is
Hilbertian.
VIII.d.
The still mysterious
and
often forgotten Solvable Closure, Q^{sol},
of Q: The iterative
calculations of algebra given by taking roots (including all roots of
1) fall within Q^{sol}.
Less than 200 years ago it was still conceivable that Q^{sol}
= Q^{alg}.
Today it is still difficult to regard quotients of the
automorphism group of Q^{sol
}as
a natural dividing line between abelian (or nilpotent) extensions of Q
and extensions with simple (non-abelian) extensions. Here are some
reasons.
- Unlike the abelian (or nilpotent) closure of Q, Q^{sol}
is not the composition of a finite number of proper Galois extensions
of Q:
this would contradict Haran's Diamond Theorem since it is
clearly not Hilbertian.
- Yet, every proper finite extension of Q^{sol}
is Hilbertian^{[FrJ,
2nd Ed., Ex. 13.9.5]} and projective.
- There is great evidence that for K, projective and
hilbertian, G_{K} is F_{ω},
but Iwasawa conjectured that so is G_{Qsol}.
- Yet, we don't know if Q^{sol}
is PAC^{[FrJ, 2nd Ed., p. 752]}, though the abelian
(and nilpotent) closures are known not to be.
This is the field that summarizes several hundred years of mathematics
before the modern era, that concluded with Galois' characterization of
the elements it contains. As the use of "easy," centerless, solvable
groups for Nielsen
classes forming modular towers shows, there is nothing easy
about solvable groups.
VIII.e.
Simple groups among
all
finite groups and Schinzel's problem: For f, g in C[x], Schinzel’s problem
was to describe those cases when f
(x) − g(y) factors
nontrivially as a polynomial in two
variables. In Schinzel's problem – unlike Davenport's problem
– no
distinction was made about the definition field.
Yet, if we restrict to cases where f
is indecomposable (IH), the
full set of nontrivial solutions to Schinzel’s
problem that do not inherit reducibility is the list of Davenport
pairs, except we must extend the branch points and changes of variable
to
C.^{[Fr73,
Thm. 1],
[Fr12, thm. 4.1]
}
The most trivial cases are where g(x) = f(h(x)) for some
polynomial h(x).
Should it happen that (f_{1},
g_{1})
is a nontrivial solution of Schinzel's problem, then for any
nonconstant polynomial pair (f_{2},g_{2}),
the pair formed by compositions (f_{1}(f_{2}),
g_{1}(g_{2}))
is also. We say it inherits
reducibility if one of the degrees of f_{2} or g_{2}
exceeds 1. We call a solution (f,
g) to
Schinzel's problem newly
reducible if it does not inherit reducibility.
Neither Schinzel's down-to-earth problem, nor Davenport's,
suggest embracing the arithmetic of covers, the simple group
classification or Hurwitz spaces. Yet, once beyond the IH hypothesis, it
and Davenport's problem diverge through for both – with no loss – T_{f}
and T_{g} have the same
degree and the same monodromy. Anything, however, that would force
equality
of the representations – as holds for Davenport's conditions
– would also violate the
newly reducible condition.
The paradigm for considering composite functions f differs, as much
in its own way, as the Universal
Frattini cover does from the data from the classification.
The monodromy G_{f}_{
} is a subgroup
of the wreath product – in the notation above – of the monodromy G_{f}_{1}
and G_{f}_{2}
(or G_{g}_{1}
and G_{g}_{2}).
So, while the monodromy method is still as suitable, the
group problem is to guarantee that subgroup is so small that
the stabilizer of a letter in the representation T_{g}
is intransitive in the representation T_{f}.
Two problems stand out in finding Schinzel pairs. Both have one
long-known example where both f
and g have
degree
4.
- Avanzi-Zannier-Gusic:^{[AZ03],
[Gu10]} Find Schinzel pairs of the form (f(x), ζf(x)) with ζ a
nontrivial root of 1. This is a special case of "locating" singular
points on reduced Hurwitz spaces.
- The
(n,m)-problem:
Given f_{1} (degree n) and g_{1}
(degree m)
with disjoint branch points (outside ∞), do there exist f_{2}
and g_{2} so that (f_{1}(f_{2}),
g_{1}(g_{2}))
is a Schinzel pair?
It is not that the classification is now irrelevant. Rather, the
entwining of groups in split extensions, with both groups possibly
solvable, requires new thinking.