Alternating Groups and Moduli space lifting invariants

Alternating groups and moduli space lifting Invariants: Published paper hf-can0611591.pdf

Abstract: The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M_g. Yet, modern applications also require a data variable (function) on the curve. The function gives a morphism from the curve to the Riemann sphere P¹ (projective 1-space). The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. The Main Theorem describes all components of Hurwitz spaces of 3-cycle covers.

It combines with the much easier Clebsch case of 2-cycle covers to give a target result for many connectedness results. This combining – discussed in CFPV-Thm.html – explains much of the nuance of the Conway-Fried-Parker-Voelklein result on connected components. The nuance has two parts:

The appearance of the Schur multiplier of the covering group in distinguishing components.
The necessity of repeating the conjugacy classes sufficiently many times to get a clean result.

The 3-cycle case is simple enough to see both these contingencies, while it is hard enough to illustrate what are likely the full set of issues any like-minded connectedness result will face. The 3-cycle case also implies many other connectedness results, for which §VI is a continuation.

[Fr1, §5 and §6] concentrates on applications directly related to the 3-cycle case, since this result touches on issues – under the heading of half-canonical classes – that hark back to Riemann's generalization of Abel's Theorem. We continue Riemann's program, by seeing how the combination of works of Artibani and Pirola [AP], Fay [Fa], Serre [Se], and Shimura [Sh] fill in aspects of producing automorphic functions on these Hurwitz spaces.

Table of Contents of this Page:
I. A correspondence between Hurwitz spaces and braid orbits:
II. The M(ain)R(esult) on a Specific Nielsen Class of Covers and Hurwitz space cusps:
III. Illustration of the Inner/Absolute Result of Fried-Voelklein:
IV. Illustration of Conway-Fried-Parker-Voelklein:
V. Inverse Galois and Theta Function applications:
V.1. The structure of G_Q :
V.2. Production of automorphic functions on Hurwitz spaces:
VI. Application to Modular Towers:

I. A correspondence between Hurwitz spaces and braid orbits: Data for a Nielsen class consists of r ≥ 3 conjugacy classes C in the data variable monodromy G. Unadorned notation Ni(G,C) indicates r tuples of generators (g₁,…,g_r) of G lying in any order in the classes comprising C, satisfying the product-one condition: g₁^…g_r =1 (product in order of appearance). Usually, an equivalence relation enters naturally. That interprets an equivalence of covers of P¹= P¹_z , with z denoting the variable from 1st year complex variables, in a given Nielsen class. This paper uses the two equivalences, absolute and inner, standard now for dealing with classical results and also for the inverse Galois problem.

A Nielsen class generalizes the genus as an appropriate invariant for a space of covers of the sphere. For it there are stable-compactification theorems by which to complete the Hurwitz space H(G,C) of covers in the Nielsen class (different spaces for different equivalences). Some Nielsen classes define connected spaces. To detect, however, the components of others requires further subtler invariants, as in #1 above. Identifying components of Hurwitz spaces, for any Nielsen class, is equivalent to computing a braid orbit on the corresponding Nielsen class.

Problems that distinguish between covers in the same connected component are of two types: number theoretic (say, asking about definition fields of covers) or those that tack on an additional geometric property (do some surfaces also appear as representatives of another, related, Nielsen class). Neither can proceed without distinguishing the connected components in a Nielsen class.

II. The M(ain)R(esult) on a Specific Nielsen Class of Covers and Hurwitz space cusps: In [Fr1, Thms 1.2 and 1.3], G=A_n (the alternating group), r counts the data variable branch points and C=C_3^r is r repetitions of the 3-cycle conjugacy class. This Nielsen class defines a space of degree n, genus g=r-(n-1) ≥ 0 covers we call H(A_n,C_3^r)^abs, the absolute Hurwitz space. When g=0, H(A_n,C_3^r)^abs has exactly one component. When g ≥ 1, the parity of the Fried-Serre spin invariant precisely identifies its two components H_±(A_n,C_3^r)^abs.

The spin invariant is a special case of a general lift invariant. It arises whenever you have a central extension H→G where the kernel is prime to the orders of entries in the r-tuple g=(g₁,…,g_r) in some Nielsen class. Then, each entry g_i lifts uniquely to g_i^* in H having the same order as g_i. The lift invariant is the product (in the same order) of these g_i^*s. Since the corresponding g_is have product 1, the lift invariant is in the kernel of H→G. The case we use here is when H is the unique non-split extension of A_n with kernel Z/2.

The notation for Clebsh's (1872) 2-cycle case would be H(S_n,C_2^r)^abs (r even): It has one component, consisting of genus g=r/2-(n-1) ≥ 0 covers of P¹_z. Much complication in the 3-cycle case appears in the long [Fr1, §3.3] where n = 4. The two distinct conjugacy classes of 3-cycles in A₄ is part of the trouble. Yet, even here there is something conceptual at the heart of the Hurwitz space approach to classical spaces.

The compactification of reduced Hurwitz spaces has a boundary consisting of cusps. There is a natural pairing – called sh-incidence – on these cusps from which we can read off much data about the spaces. It reveals the components, cusp ramification and the nature of elliptic fixed points. In particular, [Fr1, Prop. 3.5] produces this pairing for n=r=4, and both the absolute and inner spaces. It uses this information to compute their genuses (they are zero) and to prove the spaces aren't modular curves, despite their close resemblance to them.

III. Illustration of the Inner/Absolute Result of Fried-Voelklein: The other half of [Fr1, Thms 1.2 and 1.3] is similar to §II, except it changes the degree n covers to their Galois closures equipped with an isomorphism of the covering group to A_n. These are the inner Hurwitz spaces of covers H(A_n,C_3^r)ⁱⁿ. [inv_gal, App.] says there is a general relation between inner and absolute spaces defined by (G, C): Each component of the former is a natural Galois cover of a component of the latter.

Further, the covering group is a subgroup of the outer automorphism group of G. That covering group may change as the components change. The outer automorphism group for A_nis S_n/A_n and degree 2 is the largest expected covering degree. [Fr1, Thms 1.2 and 1.3] says that is exactly what you get for all components of H(A_n,C_3^r)ⁱⁿany n or allowable r.

This same result about inner/absolute space coverings applies also to reduced Hurwitz spaces (mod out by a natural PGL₂(₢) action – ₢ denoting the complex numbers). This allows comparison with modular curves: Replace A_n by the dihedral group D_n of order 2n (take n odd), and replace C_3^rby C_2^r with r even, denoting r repetitions of the involution conjugacy class. The result in this case is that H(D_n,C_2^r)ⁱⁿ → H(D_n,C_2^r)^abs is a Galois cover with group (Z/n)^*/{±1}, * indicating invertible integers modulo n. For r=4, the reduced spaces then give the modular curve cover X₁(n) → X₀(n).

Here is an example – [Prop. 1.5, twoorbit], described in Ex. 1.5 of this paper – that shows why figuring the connected components of the inner spaces is a stronger result. For n≡1 mod 4, and d(n)=(n+1)/2, consider the spaces H(A_n,C_d(n)⁴)^abs and H(A_n,C_d(n)⁴)ⁱⁿ: the conjugacy classes consist of 4 repetitions of a d(n)-cycle. In all cases, the absolute spaces are connected. The inner spaces, however, have two components, defined over a quadratic extension of Q, for n≡1 mod 8, because you can't braid the outer automorphism of A_n. There is but one component for n≡5 mod 8. The result for absolute spaces is a special case of [Thm. 5.5, LO].

IV. Illustration of Conway-Fried-Parker-Voelklein: [CFPV-Thm] explains, and gives examples of, a result that describes exactly the components of the Hurwitz space, inner or absolute, so long as this condition holds:

(*) Each conjugacy class appearing in C appears suitably often (condition #2 above). [CFPV-Thm] also has a brief history of the result.

For example, in our case, since the Schur multiplier of A_n is Z/2 and the orders of the elements in C_3^ris prime to 2 (the order of the Schur multiplier), CFPV says, for r large, both H(A_n,C_3^r)ⁱⁿ and H(A_n,C_3^r)^abs would have two components. [Fr1, Thms 1.2 and 1.3] is much more explicit, giving precisely the components for each n and r.

The example of this result, with some provisions on G that don't need to concern us here, appearing in [inv_gal, App.] has all conjugacy classes of G appear in C – under the same suitably-often condition as in #2 above. The conclusion is that the Hurwitz space has but one component. That result gives no bounds to indicate what suitably often would be.

[CFPV-Thm] explains how you can combine the Clebsch and 3-cycle results to get the correct result for any conjugacy classes (modulo condition #2) that includes the 2-cycle and 3-cycle cases where there is just one conjugacy class appearing.

In lieu of #1 above, since S_n has the Klein 4-group as its Schur multiplier, from a bigger viewpoint Clebsch's easy result is surprising. The trick, then, in combining it with the 3-cycle result is to recognize what happened in the 2-cycle case to the Schur multiplier.

V. Inverse Galois and Theta Function applications: We combine [Fr1, Thms 1.2 and 1.3] with [AP]. This gives applications of Hurwitz spaces to properties of M_g, through half-canonical classes.

V.1. The structure of G_Q: We consider howM_g affects a special case of a generalization of a famous conjecture of Shafarevich (see [GQpresentation]). The special case considers the composite Q^alt of all Galois extensions of Q with group some alternating group. If Q^alt is a P(pseudo-)A(lgebraically)C(losed) (the name is due to Frey, who showed that the nilpotent closure of Q is not PAC [FJ, Cor. 11.5.7] field), then [FV]: the canonical map G_Q →G(Q^alt/Q) has pro-free kernel. This would establish a valuable example of the conjecture, where – unlike those in [FV] – the field extension Q^alt has a canonical construction.

A basic result is that a characteristic 0 field K is PAC if and only if each curve over Q has a K point. The result [Fr1, Thm. 6.15] says that "most" curves of any given genus appear in one of the families of [Fr1, Thms 1.2 and 1.3], and further, these families are defined over Q. That seems to bode well for each curve over Q (or at least suitably general such curves) having a cover over Q with odd order branching. That would establish that Q^alt is PAC. Yet, the paper shows such odd order branching covers would also have a half-canonical divisor class over Q. So, that hope is dashed by [Fr1, Prop. 5.11]. Yet, the topic generates these continuing questions:

Can you use other Nielsen classes, not given by odd order branching, to conclude Q^alt is PAC?
Though the moduli space, M_g, of curves of genus g (g >> 0) is known not to be uni-rational, might there still be some Hilbertian property invocable for all of its many Q points relative to a cover of a Zariski open set of the moduli space?

The point of #3 is that the PAC property is the only method that has worked to get such presentations of G_Q and it would be news if it didn't work here. The point of #4 is that M_g must still be considered an odd duck of a space whose many rational subvarieties should be usable for diophantine results without the conniptions that I go through in proving Prop. 5.11.

V.2. Production of automorphic functions on Hurwitz spaces: Based on a generalization of automorphic functions on the Siegel upper half-space, we define a Hurwitz-Torelli automorphic function on a Hurwitz space. The reduced versions of any Hurwitz space defined by odd order conjugacy classes supports the analytic continuation of a canonical ϑ function.

As a corollary of [Se], we can decide when this ϑ is even. That is the case, for example, for the reduced spaces H₊(A_n,C_3^r)^abs,rd (resp. H_-(A_n,C_3^r)^abs,rd) exactly when r (≥n) is even (resp. odd). Here is how it works. For g an odd k-cycle in A_n, define w(g) to be (k²-1)/8. Extend this definition additively to any product of odd order disjoint cycles.

In [Se, Thm. 2] we take the special case X=P¹_z, so [Se, exp. (17)] applies. The result is that, whether ϑ is even or odd depends only on the Nielsen class and the value of the spin invariant (II for a branch cycle description of the covers in the orbit. In more detail. For g=(g₁,…,g_r) in the braid orbit of a Nielsen class Ni(G,C), we get even for the ϑ exactly when the product of the spin invariant g and (-1)^{Σ_i=1^r w(g_i)} is 1: Serre's formula written multiplicatively. In the 3-cycle absolute case each w(g_i) is 1. For, however, the inner case, each w(g_i) is n!/6, which is even (for n≥ 4). So, the ϑ on the spaces H₊(A_n,C_3^r)^in,rd (resp. H_-(A_n,C_3^r)^in,rd) is always even (resp. odd).

We don't know for sure when the corresponding ϑ -null is nonzero, though Thm. 6.15 shows it is so in many cases; in general if the genus of the covers in the Nielsen class is 1, or if 13n ≥ 12r+16. Whenever it is nonzero, an explicit power of this ϑ -null is a Hurwitz-Torelli automorphic function.

The definitions and results use a combination of [Fa] and [Sh]. The former is closest to where we start, for that works with moduli spaces of curves as do we. Still, we switch to the latter of necessity, for the production of automorphic functions, for that works with global moduli as do we, though our spaces are reduced Hurwitz spaces, not Siegel upper half-spaces.

VI. Application to Modular Towers: A general application of [Fr1, Thms 1.2 and 1.3] appears in [twoorbit]. This uses the Fried-Serre Lifting invariant formula (Invariance Cor. 2.3) and the results of this paper to draw a number of detailed conclusions on Hurwitz spaces for more general – pure-cycle – Nielsen classes.

As a special case of its one-orbit result on pure-cycle absolute Nielsen classes, [LO] suggests there are no distinguishing properties of these spaces
as n varies. Yet, [twoorbit, Prop. 5.15] shows the cusp structure of the Modular Towers for the prime p=2 are strikingly different between the two cases n≡1 mod 8 and n≡5 mod 8 listed in §III. The difference shows in the following statement: In the latter case, starting from level 1, the cusp tree of the tower has a subtree isomorphic to the cusp tree of a modular tower. We have named such a subtree a spire [twoorbit, Cor. 5.17]

It proves the Main Conjecture of Modular Towers – that high tower levels have general type– in these cases. The idea: Connectedness results allow drawing conclusions on the type of cusps on the boundary of the Hurwitz space compactification. The Main Conjecture then follows from a previous result about the types of cusps necessary to prove this.

The sh-incidence cusp pairing mentioned in §II has become the whole story in [twoorbit] where it is applied to an infinite number of distinct Modular Towers, each as rich in its own way as the towers of all modular curves. Since this pairing seems unknown in the case of modular curves, the paper also includes a memorable reconstruction of the modular curve cusp tower using this it.

[AP] M. Artebani and P. Pirola, Algebraic functions with even monodromy, PAMS April 2004.
[Fa] J. Fay, Theta Functions on Riemann Surfaces, Lecture notes in Mathematics 352, Springer Verlag, Heidelberg, 1973.
[LO] F. Liu and B. Osserman, The Irreducibility of Certain Pure-cycle Hurwitz Spaces, Amer. J. Math. # 6, vol. 130 (2008).
[Fr1] M. Fried, Alternating groups and Moduli Space Lifting Invariants, to appear Israel Journal.
[Fr2] M. Fried, The Main Conjecture of Modular Towers and its higher rank generalization, in Groupes de Galois (Lum. 2004; eds. Bertrand and Debes), Congres 13 (2006), 165–233.
[FV] M. Fried and H. Voelklein, The embedding problem over an Hilbertian-PAC field, Annals of Math 135 (1992), 469–481.
[Se] J.-P. Serre, Revêtements a ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris 311 (1990), 547–552.
[Sh] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton U. Press, 1998.