Nonrigid situations in constructive Galois theory: Connect to the paper pdf file

This is one of the papers referenced in §I of the M(odular) T(ower)s Time Line, where it connects it to other papers in that progression. Two results had bigger impact than might be evident in the paper.
  1. Formulation of the M(odular) T(ower) program for dihedral groups:
  2.  Covers over the reals and real points on Hurwitz spaces:
I. Formulation of the M(odular) T(ower) program for dihedral groups:

The formulation of the Main MT conjecture for dihedral groups goes like this. Suppose, for some (odd) prime p, there are Q regular realizations of all the dihedral groups {Dpk}k=0∞ with some bound r0 on their number of branch points. Then (equivalently), the Branch-Cycle-lemma implies there is an even integer r1 (≤ r0) and for each k, there is a dimension (r1-2)/2 hyperelliptic Jacobian (over Q) with a μ(pk) point for each k (≥ 0). The corresponding involution realizations are regular extensions of Q(z) with group Dpk and r1 branch points, each with the involution conjugacy class of  Dpk corresponding to an inertia generator.

§5.1–§5.2 consider the Involution Realization Conjecture, which says the last statement is impossible: There should be a uniform bound as n varies on μ(n) torsion points on hyperelliptic Jacobians of a fixed dimension, over any given number field.

(The only proven case, r1=4, is the Mazur-Merel result bounding torsion on elliptic curves.) If a subrepresentation of the cyclotomic character occurred on the p-Tate module of a hyperelliptic Jacobian (see [Se68]), the Involution Realization Conjecture would be blatantly false: One hyperelliptic Jacobian would produce a projective system of regular involution realizations of dihedral groups.

Result from it: Formulation of the general Main MT conjecture [FrKop97]. There is a still-missing result: Find μ(n) torsion points on any hyperelliptic Jacobian for all – even infinitely many – odd n s. The discussion at [CaTa09] compares the two methods that have been developed to prove the Main Conjecture for general MTs. That notes Cadoret-Tamagawa base their formulation on considering 1-dimensional characters that do not appear from complex multiplication as in [Sh64].

The usual conditions for forming a MT:  p is any prime  and you have the Nielsen class defined by r conjugacy classes C=C1,…,Cr, all p' (elements of order prime to p) in a finite p-perfect group G. 

By considering the (1-dimensional) family of abelian varieties attached to a reduced Hurwitz space when r=4, Cadoret-Tamagawa conclude the Main Conjecture – no rational points at high tower levels – for any MT defined by the usual conditions when r=4. The discussion compares this result with the more explicit – but at the moment less general – method of [Fr09b].

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II. Covers over the reals and real points on Hurwitz spaces:

Serre developed a formula for 3 branch point covers, basically a special case of the Branch-Cycle-lemma over the reals. It showed that, excluding a small set of obvious groups, you could not regularly realize groups over the reals with just three branch points, all of them real.

Given r points on the Riemann sphere, defined as a set over the reals, its  configuration type  indicates how many real points, and how many complex conjugate pairs, of points there are.

The paper [DeFr90] decides, for any Nielsen class Ni(G,C), what are the possible configuration types so that it is possible to find a real point on the corresponding Hurwitz space (absolute or inner) whose associated cover has branch points of that configuration type.

For achieving results there are several considerations.

Thm. 1.1a gives an example result of a general type. When does a finite group G have a regular realization over the the reals with all branch points real. Answer: If and only if involutions (elements of order 2) generate G.

In contrast to Serre's result, the paper notes this gives Sn covers for r=4 where the configuration type is four real points. To test the method, §4 considers realizing covers of the same configuration type over Q. This example tests for deciding if the reduced Hurwitz space has genus 0, and it does that by explicitly computing the genus of the reduced Hurwitz spaces for all the examples. Thm. 4.11 shows the cases n=5 and 7 pass this test.  We do not know if you can modify our choice of Nielsen class to give other examples of such realizing covers over Q with r=4.

Other results use the reals to show the tools used in the R(egular) I(nverse) G(alois) P(roblem)  cover much more territory than realization of groups as Galois groups.
  1. Thm. 1.1b: Gives the precise condition for when G is the monodromy group of a degree n cover (not Galois) over the reals with all branch points real.
  2. Cor. 1.2: Any nontrivial finite group G has a Frattini cover H → G for which there is no regular realization of H with only real branch points.
  3. Comment (***) §3.5: Gives the precise condition differentiating between the Hurwitz space having a real point, and that point actually corresponding to a cover over the reals.
  4. §5.3: Each finite group G has regular realizations over the totally real (all conjugates are real) numbers.
We comment further on results #1-#3.

Extending #1: The cases here consider degree n (not Galois) covers in a given Nielsen class which pass a formula criterion that describes all such covers defined over the reals. Then, the formula differentiates among those covers which have their Galois closure  cover defined over the reals versus the complexes.  The case of also assuring real branch points has an especially simple formula.

Extending #2: This was the forerunner of a complete characterization of all the real points on any Modular Tower [BFr02, §6]. That also showed how to apply it to the main example MTs of the paper.  The remainder of this essay lists precise results for that example: the MT for A5, with four repetitions, C34, of the conjugacy class of 3-cycles, and the prime p=2.

As M(odular) T(ower)s Time Line, §II discusses, there is a full MT, and also an abelianized MT attached to this data. They differ in whether we use the full exponent 2k Frattini extension Gk of A5, or the abelianization Gk,ab, given by abelianizing the extension's kernel.

Our example statements apply to either, because in all the cases, the only real points fall on H-M components. That is, their corresponding braid orbits contain Harbater-Mumford representatives: 4-tuples in the Nielsen class of form (g1,g1-1,g2,g2-1). So, we state results using the notation Gk. Still, I can easily tell you inductively more about Gk,ab. Use the natural Frattini covering map Gk+1,ab → Gk,ab: the kernel is the A5  module of order 25 given by A5 acting on its 5-Sylows modulo the submodule generated by summing the 5-Sylows [Fr95, Prop. 2.4]. 

The kth  MT level is the inner Hurwitz space defined by the Nielsen class Ni(Gk,C34). On each H-M component there are two types of real points: Those given by H(arbater)-M(umford) representatives and those by near H-M representatives [BFr02, Prop. 6.8]. As MTs generalize modular curves, this example generalizes characterizing real points on modular curves.

This example has a natural language using the arithmetic behavior of cusps. Any projective system of Harbater-Mumford representatives defines a real branch of the cusp tree on a MT in the language of [Fr06, §3.1.1].  By contrast, in our example, a near H-M representative at level k gives real points, but above them (at level k+1) there are no real points. 

Extending #3: Four different types of Hurwitz spaces see regular duty: Absolute, inner and their reduced versions (modding out by the action of linear fractional transformations on the base cover P1z). For each there is a practical criterion for when the spaces have fine moduli. One major point of having fine moduli is to assure that if you have a K point on such a space, then you get a realizing cover of P1z defined over K.

The inner and absolute criteria, for example, are often sufficiently effective to test all levels of a MT. For example, assuming the usual conditions for forming a MT, the inner Hurwitz spaces at all levels have fine moduli if and only if G is centerless [FrV91, Part of Main Thm.]. The results of this paper therefore combine to give the following example.

Consider the MT for A5, with four repetitions of C34 of the conjugacy class of 3-cycles, with the prime p=2. Then, consider each of the characteric 2-Frattini covering groups Gk → A5. Each Gk (or Gk,ab) has a Frattini cover Rk → Gk,ab  that is a central 2-Frattini extension: meaning it has a center [BFr02, Prop. 3.21, for this case, Ex. 3.20]. In our running example, the kernel is Z/2, arising from what [Fr02, Def. 4.5] calls an antecedent Schur multiplier to the Spin5 → A5 cover (as in the discussion connecting [Se90a] and [Fr09a]). So, each of the inner Hurwitz spaces defined by Ni(Gk,C34) has fine moduli (as stated in [BFr02, Thm. 3.16]).

[BFr02, Prop. 6.8] shows each of the inner Hurwitz spaces attached to the Nielsen class Ni(Rk,C34) has real points p1,k and p2,k, k≥ 1, with the following properties. They both lie on a Harbater-Mumford component: Level k=1 has exactly two components, both defined over Q [BFr02, §9.1], with just one of them an H-M component. Physically, the Hurwitz space components for Ni(Rk,C34) identify with a subset of the components for and Ni(Gk,C34). Indeed, they are exactly the components corresponding to braid orbits of Ni(Gk,C34) for which the lift invariant of elements from Gk to Rk is +1. Any H-M component passes this lift invariant test, and so there are nonempty components corresponding to Rk.

Although the components are exactly the same, the points regarded on each give respective Galois covers with group Rk and Gk. For the group Rk, the cover corresponding to p1,k has definition field the reals,  while the cover corresponding to p2,k does not. That is, there are points exhibiting the conclusion of fine moduli, and points denying it, on the Hurwitz space corresponding to Ni(Rk,C34).

Regarded as giving Gk covers, the cover corresponding to p1,k is defined by an H-M rep.,  while the cover corresponding to p2,k is defined by a near H-M.  So, one group gives an infinite number of examples illustrating the distinction between having fine moduli and not having it.

While the reduced version criterion is more difficult to check, [BFr02, Prop. 4.7] has general results. These apply to our running example in [BFr02, Lem. 7.5] to say that for level 0, the Hurwitz space does not have reduced fine moduli, but are all levels above 0, the MT does have reduced fine moduli.

These and the p-adic version of the real results are important topics to the subject, especially compatible with the results of [DeEm05].

Back to top Mike Fried, Sunday Mar 15, 2009