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, GK, 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 GK.  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: XY.  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 = nφ 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 Zp, 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 Gφ refers to the group of automorphisms over an algebraic closure, Kalg. 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), Gφar. It contains Gφ as a normal subgroup.

There is a  canonical permutation representation, Tφ, attached to Gφar, a faithful, transitive, homomorphism into the symmetric group Sn. 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, Kalg, of K, and usually so is X.  Even if both are absolutely irreducible, significantly, we don't expect  Gφar and Gφ to be equal. An example of great significance is the (Gφar, Gφ) 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  Gφar = Gφ and then any quotient of Gφ is automatically regularly realized over K. In most serious applications, there is no automatic reduction to equality of Gφar and Gφ.

Given two groups (G*, G), a cover φ (over K), is an A-G realization of (G*, G) (over K) if (G*, G) = (Gφar, Gφ).  This then automatically includes any quotient of Gφar as realized over K, where the part given by the Gφar/Gφ 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, P1 – 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 (f1,f2) 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 (xn) or Chebychev (Tn(x), with Tn(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: XY (of normal varieties) over a field K. If K = Fq is a finite field (with q = pu elements, p a prime) then φ is exceptional if and only if it is a one-one map on Fqt 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 Pφ,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 X2 - Δ. For K a finite field, then φ is exceptional if and only if  X2 - 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 (Gφar, Gφ) 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.
  1. That f  is incomposable over Kalg (resp. K) interprets as Gφ (resp. Gφar) under Tf  is a primitive representation.
  2. Because f is a polynomial and it is neither cyclic nor Chebychev, then Tf on Gφ 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 Gφ with 1 = Tf(g)(i) and 2 = Tf(g)(j).

Polynomials stand out from general rational functions (in one variable) in that the inertia generator, g over ∞ has Tf(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 f1 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:  f1(x)=f2(ax+b). Fried showed that Davenport pairs (f1,f2) over any number field K, satisfying IH, had Gf1 = Gf2, and Tf1 and Tf2 are identical as group (but not permutation) representations.

That is, the trace of the permutation matrices Tf1 (g) and Tf2 (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 Gφar. That is,  for some j prime to n, gj is not conjugate to g in Gφar. 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  f1 is conjugate (by applying an element of GQ to its coefficients) to f2. That contradicts that (f1,f2) 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 GQ); 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: XP1, of the projective line over C branched over r ≥ 3 points z1,…,zr (of degree n) with its monodromy group Gφ, permutation representation Tφ and r (unordered) conjugacy classes, associated to z1,…,zr = z,
= C1∪ C2∪ … ∪ Cr
within that group. There are elements gi in Ci (any ordering of the classes will do) with these two properties. Such r-tuples with these properties are branch cycles. Denote the collection of such g = (g1,…,gr ) by Ni=Ni(G,T, ): a Nielsen Class. We always assume it is nonempty. Conversely, given  r points on P1, (G,T, ), 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] 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, Ur,  minus its discriminant locus. Assume two covers (φ12) have their corresponding (G,T, ) distinct (up to equivalence). Then the covers are distinguishable as deformation inequivalent: No matter how you (continuously)  drag φ1 – following a trail of branch points in Ur – over to the branch points of φ2, the resulting cover φ1' will be inequivalent to φ2.

When r=3, φ1'  is always reduced equivalent to φ1; for discussing families of covers this is the trivial case.

If you drag φ1around a closed path on Ur (based, say, at z0) then you can express branch cycles, g1', for φ1' in terms of branch cycles,  g1, for φ1. There are two paths, generating the fundamental group – the Hurwitz monodromy groupHr of Ur. Denote their homotopy classes, respectively, by: 
Conjugating q1 by sh i-1 times gives the ith twist, qi, i = 1, …, r-1, and there are various relations among these elements coming from Hr being a quotient of the Artin braid group, but none more important than
 The P1 relation q1qr-1qr-1q1= 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, ) have a natural complex analytic parameter (Hurwitz) space H(G,T, ). For absolute or inner equivalence of covers, H is naturally an unramified cover of  Ur 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 tale topology that represent all analytic families of covers in the Nielsen class.   Then, H(G,T, ) 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) = sumi=1r ind(gi), 
with ind(g) the degree of T, deg(T), minus the number of fixed points of T(g).

Compactifications of  H(G,T, ): 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 Pr in the function fields of the components of H(G,T, ). A normalization of a compactication of Ur/ PSL2(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, Qcyc,  of  Q (obtained by adjoining all roots of 1)  identifies with the invertible profinite integers Z*. Any subfield of Qcyc is a cyclotomic field. Restricting any element σ of GQ to Qcyc gives a profinite integer n(σ).

The BCL produces a moduli field FNi – 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  abs of each of the conjugacy classes of  inside NSn(G, ). Extending classes can end up joining two classes.
Absolute Version of the BCL: Then FNi is the fixed field in Qcyc of I(abs). It is the smallest field K' for which WNiWNiσ 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 FNi 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: FNi 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, FNi is the intersection of all fields of definition of all covers in WNi.

The inner version of these BCL results is obtained by replacing abs by .

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: XP1 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 GK acts through  Gφar, so through ωσ in N(G, ). If  σ maps the ith branch point to the jth, then:
ωσ Cj ωσ-1 = Ci-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 Ci to a conjugacy class in Gφar is K-rational.

For reduced equivalence when r = 4, there are two conditions: A quotient, Q''=Q/<(q1q3-1)2>, of a quaternion group Q,  acts faithfully on the Nielsen class (as a Klein 4-group); and neither q1q2 nor q1q2q3 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 Hr is intransitive. Then, how to find the definition fields of their components.

II.d. Inner vs absolute Nielsen classes: Given Ni(G,T, ) 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 NSn(G). When the groups are equal the elements of NSn(G) are said to be braidable over H '. Indeed there is a natural construction of the space  H in and the map H inH 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 → Ur, or its reduced version H in,redJr, 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 inH abs → Ur, but instead of a cover of r-dimensional nonsingular complex manifolds, the result H in,redH abs,redUr/PGL2(C) =def Jr,  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 H4. 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, )* with * = inner or inner, red. Then, we can compute the geometric monodromy group of the component HO Ur, if * is inner or absolute equivalence orbit as the group of permutations acting on the Nielsen classes in O. Similarly for reduced equivalence, but Jr substitutes for Ur. With no loss take * to inner or inner, red.

The collection cover: Pick a base point u0 on Ur. There is a unique geometric Galois cover φu0,G,: Xu0,G,P1 – the collection cover – with branch points at u0 so that φu0,G, factors through each cover  φ: XP1in O with branch points in u0, and it is minimal with this property. Denote the geometric monodromy group of φu0,G, by Gu0. 

The Monodromy Automorphism Principle: Denote the inner Nielsen class of  φu0,G, by Niu0,G,inn.  Then, Hr acts on Niu0,G,inn and if we restrict the action to the orbit O of φu0,G,, then any q in Hr acts as an automorphism of  Gu0. This induces the analogous result for the corresponding reduced orbit of O.  

That leaves three problems to decipher the Arithmetic/Geometric monodromy of  HO Ur, 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 P1.

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.
  1. 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.  
  2. 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 pt 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 TG(1)xP, with TG(1) the stabilizer of a letter in TG, 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 Ur. 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 tale cover of the configuration space. Fried produced a finite type configuration space over Ur that generalizes the space Ur 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 tale) 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 X0(pk+1and  X1(pk+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 (GL2 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(pk+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 = Dpk+1, group of order 2pk+1)  using r = 4 involution conjugacy classes, denoted 34: The Nielsen class Ni(Dpk+1,  34). Denote the standard degree pk+1representation of Dpk+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 {X0(pk+1)}k≥ 0, all of which cover the j-line, P1j. The other prevalent series of modular curves, for each p, is the collection {X1(pk+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:
Closely aligned Hurwitz spaces:
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((Dpk+1,T, 34)abs,red and H((Dpk+1,34)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((Dpk+1,34)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 pk+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 xp,k+1 on X1(pk+1), consider the group H(xk+1) of the Galois closure of the field extension Q(j',xp,k+1)/Q(j').

With I2 the 2x2 identity matrix, the arithmetic (resp. geometric) monodromy of the (absolutely irreducible) cover φp,k: X1(pk+1) → P1j is GL2(Z/pk+1)/<I2> (resp. SL2(Z/pk+1)/<I2>), as in the Hurwitz space view.

Adding a choice of isomorphism class to a point of X1(p). If we assume we have added such a choice,  then it is possible to speak of a projective system of generators of pk+1 division points on an elliptic curve with j-invariant j'. That then allows looking at the images under elements of GQ(j') of projective systems. Thereby, we may consider the image of GQ(j') in GL2(Zp), with Zp the p-adic integers. Serre, though, rarely spoke in terms of moduli. While the moduli interpretation of X1(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. 
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 GL2 part: Those j' which were not algebraic integers. For that case, two statements are shown for almost all p about the H(x1) 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 GL2 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  … → Xn+1Xn →  … → X0P is eventually p-Frattini if there is a value k for which the  monodromy group of  Xn+1   → P is a p-Frattini cover of the monodromy group of XnP for nk. 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 xk in Xk over z', the decomposition group H(xk) 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  X0(pn+1) as a cover of the j-line. With k0 = 0 for p > 3 (resp. k0 = 1 for p =3; k0 = 2 for p = 2), it satisfies
SL2(Z/pk+1)/<I2>→  SL2(Z/pk0+1)/<I2> is a p-Frattini cover, for k > k0.
So, if for j', H(xp,k0+1) = GL2(Z/pk0+1)/<I2>, then H(xp,k+1) = GL2(Z/pk+1)/<I2> for all kk0.[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 GL2(Z/pk0+1)/<I2>. Such a full fiber has arithmetic monodromy  GL2(Zp)/<I2>. 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 commentp cusps on modular curves, and their attachment to Tate's p*-adically uniformized elliptic curves to decipher the GQ action when j is p*-adically "close to" ∞ 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 GK on the Qp vector space – Tate module, or 1st p-adic tale cohomomology – of an elliptic curve with complex multiplication by K. On the 2nd p-adic tale 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 tale 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 Np.  Then, there is an integer k = kp,Np, K so that any regular realization of Dpk+1, with kkp must have more than Np 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 C2

Pierre Dbes and Fried proved the conjecture is equivalent to showing that for no integer d' does there exist for each k a pk+1-power cyclotomic point – the absolute Galois group of K acts as if the point and its multiplies, over K, are pk+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 Np = 4, then a proof follows from a general principle, The Genus Growth Theorem, and the proof that the  spaces,  H((Dpk+1, 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 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 : wf(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 Pf = PK,f. Then, if f = f2of1, a composition of two lower degree functions over K, then PK,fPK,f2PK,f1

The converse is trickier. Suppose we start with  f1 and f2 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 Ej' with j-invariant j'. Only for the complex multiplication j-invariants, does an elliptic curve have an isogeny Ej'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((Dp, 34)abs,red lying over j' obtained from modding  Ej' 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((Dpk+1, 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 Pf 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 Pf, 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 Gp, the semi-direct product of Vp = (Z/p)2 and Z/2 (written multiplicatively, elements that are +1 or -1). To understand why this example is so important,  recognize that Gp 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 p2) degree exceptionals is a variant on the involution realization Nielsen classes: Ni(Gp,T, 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 Gp and Z/2, the semidirect product of Z/p and Z/2.

The nontrivial element of Z/2 acts as multiplication by -1 on Vp. Understand the group multiplication by representing elements of Gp as 2x2 matrices M(a, v):  
 the first row is (a v), a = 1 and v  in Vp; the  second row is (0 1).
The Nielsen class of (odd) prime-squared (say p2) degree exceptionals is a variant on the involution realization Nielsen classes: Ni(Gp,T, 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 Gp 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(v2,v3) =def (M(-1 0), M(-1 v2), M(-1 v3), M(-1 v4)) with these equivalences,
generationv2 and v3 span Vp and product-onev4v3- v2

From Riemann-Hurwitz the genus, gp, of covers in this family appears from
2(p2 + gp - 1)= 4(p2 - 1)/2. The degree p2 covers of the sphere in this family have gp = 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(v2,v3) is as if you apply an element of SL2(Z/p) to  Vp. You need only check this for the two generators q2 and the shift (sh). For example, (up to inner equivalence) the shift has this effect:
 M(v2,v3) → M(v3- v2,v3- 2v2); represented by the 2x2 matrix with rows (-1 1) and (-2 1) of determinant 1.

The action of those two generators  generates SL2(Z/p), but M(v2,v3) and M(-v2,-v3) 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 = fp in the Nielsen class over Q, up to absolute reduced equivalence. 

Degree p
2 Exceptional TheoremPart A. Given j, for almost all p (by the OIT), the arithmetic monodromy group of the cover from fp will be the semidirect product of Gp and GL2(Z/p) with their natural copies of Z/2 identified. For such a p,  since GL2(Z/p) acts irreducibly on Vp the induced representation Tar is primitive. Thus,  fp is decomposable over Qalg, but indecomposable over Q.[Fr05, Prop. 6.3]

Part B. For ∞-ly many primes l (avoid those of bad reduction) fp mod l has the same property: indecomposable over Z/l, but decomposable over its algebraic closure. For those same primes fp 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 fp 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(Gp, T, 34)abs,red and Weil's pairing: There are p - 1 braids orbits, corresponding to the same number of absolutely irreducible components on H(Gp,T, 34)abs,red, each conjugate over the field generated by the pth roots of 1. Each has geometric monodromy SL2(Z/p)/<I2> over the j-line, and arithmetic monodromy GL2(Z/p)/<I2>.

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-Mller-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 gf = 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 Mller, 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 Poincar series

James Ax and Simon Kochen proved a version of Artin's conjecture that Q hypersurfaces in projective d2-space of degree d have p-adic points. Their result: For a given d there is a finite list, Pd, 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 ontoa (group) cover –  of A –  extends to any (profinite) cover BA, 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  GK 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 GK isomorphic to G*.[ExProjF] Yet, the location of such K was another matter. Ax thought that pseudo-algebraically closed subfields of Qalg were rare in that the only one that would be Galois over Q would be Qalg 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 GK 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 Qalg, 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 GQ could appear as varying kinds of extensions of known groups by known groups.  

V.c. Poincar 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 Poincar 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 tale cohomology groups of nonsingular projective varieties – for Poincar 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 Poincar series for a Davenport pair. Coefficients in PU(f1,f2) - PI(f1,f2) corresponding to finite fields in which the ranges of f1 and f2 are the same are 0. 
VI. Presentations of GQ 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 VG,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).
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, ψG: RGG. Denote the kernel of ψG by UG. 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, ), to UG,a quotient of UG.

Example: When the order of UG is prime to the order of all elements in , then UG, = UG. Compute the lift invariant of (g1,…,gr) by selecting the unique element ĝi in RG lying over gi having the same order. Then, form the product  g1gr

That works with the first example developed by Fried and Serre. When G is the degree n alternating group, and all elements in  have odd order, then UG, has order 2. That case has a precise formula for the LI when the genus gG, of the covers in the Nielsen class is 0. When all the elements of 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 UG,. While that seems like a formula, there is no general procedure for computing the LI. In general, RG, 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, ), the number of components of the Hurwitz space is precisely the number of elements in UG, 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 , do so with high multiplicity.

For, however, the present applications – excluding to the OIT – such as the presentations of GQ, the high multiplicity assumption was removed almost immediately by results that appeared with the topic of Modular Towers.

Harbater-Mumford Components: A representative g = (g1,…,gr ) in a Nielsen class where r ≥ 4 is even and g2i-1=g2i-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  is HM-gcomplete (includes that the Nielsen class contains an HM rep. and  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 Dbes 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: 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: UG, is the trivial group. The first applications presented GQ as extensions of a known group by a known group. Both results assume that K is a PAC subfield of Qalg. Denote the profree group on a countable set of generators by Fω and refer to a GK isomorphic to it as ω-free.

Theorem A: Then, K is Hilbertian if and only if GK is isomorphic to Fω. In that case GK is in a short exact sequence
1→ Fω → GK → ∏n=2 Sn → 1: The product of copies of the symmetric group Sn, by Fω.

Theorem B: Parallel to Theorem A, K is RG-Hilbertian if and only if each finite group is a quotient of GK (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 UGG. 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 – mp) is often non-obvious, despite an explicit computation for it.[Fr02, Thm. 2.8]

That allows forming: Among simple (noncyclic) groups there are no present realizations over Q (including regular realizations) of any of the covers Up,1, ab  → G (k=1). Example: We know precisely these groups for G = A5, p = 2, 3 and 5.[Fr95, Part B]

full Frattini quotient of Up, abG is a quotient EG with kernel isomorphic to (Zp)m', 1 ≤ m' ≤ mp.
There are then corresponding groups Ep,k, abG,  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 U2, ab  → A5, for which m2 = 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)mxH = G is p-perfect, with m ≥ 2: H is p' and it acts without fixed points. Then, there is an action of H on (Zp)m making (Zp)m x H as a full Frattini quotient of Up, abG and mp = (m-1)(pm)+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 Up, abG 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 Dpk+1 by Up,k, ab  → G, still using the bound Np on the number of branch points. 

Suppose consists of p' (orders relatively prime to p) classes of G. Then, each class in  lifts to a unique conjugacy class in Gp,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(Up,k, ab, )*, 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 is p'.

Main MT Proposition:[FrK98] [De05] Assume for some Np (dependent on G and p), and for each k,  there is a K-regular realization of Up,k, ab with no more than Np branch points. Then, for there is a p' set, ,  of cardinality r' ≤ Np, of classes in G, for which the Hurwitz space, H(Up,k, ab, )in (or H(Up,k, ab, )in,red off its cusp locus has a K point k = 0, 1, 2, ….

Main MT Conjecture: For the spaces H(Up,k, ab, )in for k large dependent on fixed (G, , p) as above.
  1. They have general type: high powers of their canonical bundles embed them in projective space.
  2. They have no K points (off the cusp locus).
  3. Further, these statements should hold also for any full Frattini quotient, EG, of Up, abG.
Genus Growth Theorem: For r = 4, #1 and #2 are equivalent and also equivalent to showing that the genus of the compactification of H(Up,k, ab, )in, red (or of H(Ep,k, ab, )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(Up,k, ab, )in (or H(Ep,k, ab, )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 Fq. That gives a line on a Jacobian for which the action of a power of the Frobenius over Fq 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 q1/2

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: GKZp* 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 As, 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 Gk defining the kth tower level. Since the group Gk is automatic from the level 0 group, two facts assure that the nontrivial center condition does not affect the main conjecture.
  1. When G0 is p-perfect and centerless, then so is Gk, k ≥ 0.[BFr02, Prop. 3.21]
  2. If there is a p part of the center of G0 = G,  then there is a p-Frattini cover GGc, with Gc p-perfect and trivial p center, with corresponding classes c. The main conjecture holds for the tower for (Gc,c) if and only if it holds for (G,).[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 q2 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 = (g1,g2,g3,g4).[Fr6, 5.2.1] 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 { kO}k=0 of braid orbits.  Denote a projective system of cusps through a projective system of  representatives { kg}k=0.  If pu exactly divides ( kg)mp, u ≥ 1, then pu+1 exactly divides ( k+1g)mp.[FK97, Lift. Lem. 4.1] Denote the cusp associated with kg by kp  and the ramification of  k+1p/kp
 by e( k+1p/kp).

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+1p/kp) = 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 kg. All Q'' orbits on a given braid orbit have the same length, 1, 2 or 4. Two extremes that give the expected conclusion: 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 k0 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 are Q rational. RETURN
  1. Alternating groups and theta nulls: G = An, = 3r , rn ≥ 5 (gG, ≥ 1; r 3-cycle conjugacy classes) for absolute spaces parametrizing degree n covers or the inner spaces parametrizing their Galois closures. 
  2. Multiple Harbater-Mumford Components: Gpk+1, is the semidirect product of  Z/3 (generated by an element α) and (Z/pk+1)2, where the lift invariants run over all values in the kernel of ψG: RGpk+1Gpk+1, which is isomorphic to Z/pk+1.[Fr06, App. A.2] Here is two repetitions each of the conjugacy class of α and its inverse. 
  3. 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 rn (below).

[Se90a] J.P. Serre, Relvements dans n, 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 n) 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, Revtements ramification impaire et thta-caractristiques, 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 pk+1 root of 1. Also, the BrA is transitive on all elements with a given p' lift invariant.

There are, however, Kppk components with 0 LI : Kp= (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/pk+1.

[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 Qalg and its automorphism group GQ. 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 Qalg 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 → P1 is a Galois cover, then there are infinitely many z in P1(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(xp,k) = Dpk+1  for any fixed p with Q(xp,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(xp,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 LB intermediate between Q and Qalg 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  LB whose arithmetic properties would distinguish them by what they said about GQ. In addition to LB being Galois over Q, we might want to "know" the following about LB:
  1. Generators for it over Q and its Galois group;
  2. properties of generators of its absolute Galois group; and
  3. for various collections of covers C' → C, with C absolutely irreducible over LB, the possible decompositions groups of its fibers over points of C in LB.
Shafarevich's conjecture: The cyclotomic closure, Qcyc, 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 GQcyc is ω-free, and without some form of the field crossing argument #3 is untouchable. 

The field of totally real numbers:  An α in Qalg is totally real if all its conjugates over Q are real. For the collection of all such numbers, Qtr, GQtr is freely generated by involutions (all conjugates of complex conjugation). While Qtr 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 Qtr. Also, it is not Hilbertian, but it has a property almost as good: real-Hilbertianity.

The alternating closure, Qalt, of Q: This is the composite of all Galois extensions of Q having some An, nn0 (n0 arbitrary) as group. So, its Galois group over Q is a product of An groups.

Therefore, Qtr 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 Qtr/Q, and much more about the group of Qalt/Q. Still, for good reasons an attempt to prove that Qalt is PAC failed. Combining properties – if only we could – of these suggestions would produce  a canonical LB with which independent researchers could focus on deeper properties of Qalg.

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 OK.  
Fried also introduced the notion of Universal Hilbert subsets: Infinite subsets Suni in a number field K, such that for any cover  φ: X → P1 over K, the fiber over all but finitely many values in Suni 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 Suni,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 M1/M is not contained in the Galois closure of M/K, then M1 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 Mi/K, i=1,2, are Galois, and M/K is contained in neither, but is in M1.M2. Then, M is Hilbertian.

VIII.d. The still mysterious and often forgotten Solvable Closure, Qsol, of Q: The iterative calculations of algebra given by taking roots (including all roots of 1) fall within Qsol. Less than 200 years ago it was still conceivable that Qsol = Qalg.

Today it is still difficult to regard quotients of the automorphism group of Qsol as a natural dividing line between abelian (or nilpotent) extensions of Q and extensions with simple (non-abelian) extensions. Here are some reasons.
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 (f1, g1) is a nontrivial solution of Schinzel's problem, then for any nonconstant polynomial pair (f2,g2), the pair formed by compositions  (f1(f2), g1(g2)) is also. We say it inherits reducibility if one of the degrees of f2 or g2 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 – Tf and Tg  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 Gf  is a subgroup of the wreath product – in the notation above – of the monodromy Gf1 and Gf2 (or Gg1 and Gg2).  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 Tg is intransitive in the representation Tf.

Two problems stand out in finding Schinzel pairs. Both have one long-known example where both f and g have degree 4. 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.