How to look at the analysis below: We analyze components at level 1 of the (A4, C±32, p=2)  MT from Paul's Thesis that we refer to in other places. This is meant to be a more complete explanation even if the analysis here is slightly telegraphic. We expect to publish this analysis at some time,  though not until we remove the use of GAP. In our recounting below, we say explicitly where GAP  has entered.

Over the rest of 2007 we expect to document the discovery of Paul that connects A4 and A5, each with four 3-cycles as conjugacy classes defining their MTs for p=2 by including more detailed comments in this HTML file. From this point we refer to [pBaileyThesis2002.pdf] as [BT] and our page numbering is that from what acrobat gives at the bottom of acrobat frame (not the printed numbering in the pdf file). Much of what leads into our conclusions is taken from [h4-0104289.pdf]. We refer to that as [BF02]. Notation from [BT, p. 104]: Un=G1(An), n=4 or 5.

1. Characteristic 2-Frattini Modules: Recall: M0(A4)=ker(U4→A4) is (Z/2)5. The action of A4 on this module appears in [BF02, Prop. 5.6]. It identifies M0(A4) with the Z/2[A4] module generated by the (6) cosets of Z/2 in A4, modulo the sum of all the cosets. For the action of A5 on the same set to get M0(A5), replace cosets of Z/2 by cosets of D5 (in A5). Regard Z/2 as D5∩A4.

[BF02, Cor. 5.7] (or [BT, p. 102, Prop. 34]) lists three conjugacy classes of (nontrivial) elements in M0(A5) represented respectively by a sum of one, or a sum of three (mod the sum of all cosets),  or a sum of two cosets of D5 in A5.  Call these respective sets M5', M3' and M2'.  As in [BT, p. 104, Prop. 39]: M2' and M3'  each break up into two conjugacy classes under A4 (M5' does not), distinguished by the sizes of their centralizers in U4. In Paul's notation, the bifurcation is represented by M2' → two orbits J1 and J2, M3' → J3 and J4 and M5' is now called J5.

The Schur multiplier of A4 is Z/2, while for U4 it is Z/2×Z/2. By contrast the Schur multiplier for G1(A5)=U5 is just Z/2. In particular there are three different central Frattini extensions H→ U4 with kernel Z/2. We label these R(1), R(2) and R(3), with R(1) the antecedent to the Schur multiplier of A4. Use R(0) →A4 to denote the representation cover of A4, so R(0)  is another name for Spin4. Then, you get a generator of ker(R(1) → U4) by  lifting a generator of ker(R(0) →A4) to R(1) and squaring it. You do the same thing with A5 replacing A4, to get the generator of the Schur multiplier of U5.

2. Spin Representations: We can make R(1) more explicit using [BF02, Cor. 9.16] (reported on in [BT, §3.1 and §3.2]. That lists the ways to embed U in AN for some integer N, so that the representation cover of it is just its pullback to SpinN. The possible values of N for transitive representations are 40, 60 and 120. We use this below. These are examples of spin representationsThe meaning of the phrase for any group G is an embedding in some AN for which its pullback, G^, to SpinN doesn't split over G. We also call G^ G a spin cover.

[BT, p. 108, Prop. 46 ] says that if h1 and h2 are any conjugate, order 3 generators of U4, then there is a unique automorphism of U4 that takes hi to hi-1i=1,2, allowing the conclusion in Prop. 47 that the outer automorphism group Out(U4) of U4 has order 16. We know the centralizer in M0(A4) of hi is cyclic, generated by some ci whose image in R(0) is the nontrivial element of ker(R(0) →A4).

3. Schur multiplier analysis of the Modular Tower defined by (A4, C±32, p=2) at level 1: We can capture each of the covers R(i) → U4,  i=1,2 and 3, as a spin representation. Here is an outline. The following corollary to [BF02, Cor. 9.16]  helps us characterize R(1) as a spin representation.

Prop [BT, Prop. 53]: If T: U4→ AN is a transitive spin representation, then its pullback to SpinN is not R(1) (nor restriction of a spin rep. of U5).

Now, for each T compute and each conjugacy class
Ji , i=1, …,5 as in §2 above, let xi be the order of the lift of an element of Ji to U4^ (pullback to SpinN from T): T → (x1, …, x5) is the pattern of T. Then, [BT, Prop. 54] lists three spin representations of U4 in the style of those given by [BF02, Cor. 9.16] for which the patterns are easy to compute. The patterns are respectively (2,2,4,4,4), (2,4,2,4,4) and (2,4,4,2,2) with the spin cover clearly R(1), and the other two respectively to be identified with R(2) and R(3). In this way we identify these three extensions with spin covers. So far, no need for GAP.

4. Fundamental Conclusion on Patterns of obstruction at level 1: This is what must be explained. There are six components at level 1 of the (A4, C±32, p=2) MT:
We label the corresponding braid orbits on Ni(G1(A4,±C32), as Ni0,±, Ni1,±, Ni3,±, with the subscripts indicating two components each whose respective reduced Hurwitz spaces have genus respectively 0, 1 and 3. In each case the ± has a different reason for being there. GAP enters in showing two things:
  [lum-fried0611594pap.pdf, §6.4] notes the implications of these last two, for regularly realizing G1(A5) over Q. Given this data, we continue by noting the implications to the Main Conjecture for this MT. 

5. Conclusions about orbits of H-M reps: Now take h=(h1, h1-1,h2, h2-1)  be the H-M rep. form from h1 and h2. The conditions imply the middle product h1-1h2 has order 4 (the lift of an element of A4 of order 2). All those automorphisms assure there is one orbit of H-M reps. among absolute classes at level 1.


To be continued 02/18/07: