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 U5 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 representations. The
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-1, i=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:
- That is six braid orbits on Ni(U4,
C±32)
(all lying over the unique
unobstructed component at level 0).
- Two of them are H-M components.
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:
- At least one of the lifting invariants sR(i) is
nontrivial on each of Ni0,± and Ni3,±.
- The two components corresponding to Ni1,± are
identical to the two
H-M components that turn up at level 1 on the (A5,C34, p=2) tower.
[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: