The
inverse Galois problem and rational points on moduli spaces: pdf file of original paper inv_gal.pdf
This paper reduces the R(egular)
version of the I(nverse) G(alois) P(roblem) to a practical search
for rational points on refined versions of Hurwitz spaces. The
essential data for a Hurwitz space is a Nielsen class consisting of a
finite group G, an ordered
collection of conjugacy classes C={C1,…,
Cr}, and an equivalence relation * on these collections.
Denote the Nielsen
class as Ni(G,C)*. Also, let Urdenote the space of r unordered points on the Riemann sphere P1z, uniformized by a(n inhomogeneous)
variable z. The three main
results are these:
Ni(G,C)* determines a Hurwitz spaceH(G,C)* with a natural map to Uras a coarse moduli space of r-branch point covers of P1z
in Ni(G,C)*.
If * is absolute equivalence (so G comes with an embedding in
Sn), then H(G,C)abs is a fine moduli
space exactly when the centralizer of G
in Sn is
trivial.
If * is inner equivalence, then H(G,C)in is a fine moduli
space exactly when the center of Gis trivial.
The Branch
Cycle
Lemma gives a formula that combinatorially determines
the definition field of the Hurwitz space, with its natural map
to Ur.
The most well-known special case is the criterion for inner spaces to
be over Q: The conjugacy class
is a rational union.
Assuming
high
multiplicity of appearance of each conjugacy class in C, the corresponding Hurwitz space
has exactly one connected component.
Extra
Comment
on #1:
Fine moduli has a clear meaning classically. Suppose you have any
smooth connected family of covers of P1z
– with extra structure – in the Nielsen class, parametrized by P. Assume each point of P indicates –
by that extra structure – a unique element of the equivalence class of
structures. Then there is a uniquely
defined map from P to H(G,C)* (its image will hit only one
connected component) inducing the family over P by pullback. The phrase stack was essentially invented – by Deligne and Mumford – to treat spaces such as Hurwitz spaces as objects for
which a phrase like "as a coarse moduli space over Q" would have meaning.
Hurwitz space
discusses this, the structure of "stable
compactication" and the additional equivalence, called reduced, in more detail.
Extra
Comment on #2:
In item #3 we see there are extra conditions on C that guarantee the Hurwitz space
attached to H(G,C)*
has components over
Q. Without a doubt the most successful of these conditions is
not the C(onway)F(ried)P(arker)V(ölklein)
theorem, though when that applies it is quite powerful, as it uses
group theory to label all
components of an applicable H(G,C)*. Rather, the best criterion
stipulates that Ni(G,C)* contains a H(arbater)-M(umford)
representative, and there is only one braid orbit containing all
H-M reps. Further, there is a very effective criterion for this to
happen coming from the first M(odular)
T(ower) paper [Thm. 3.21,
modtowbeg.pdf]. Comment #3 compares the two methods, and
generalizations of the H-M method to consider g-p' representatives.
Extra
Comment on #3:
Conway and Parker outlined a proof of #3, with
that completed in the Appendix of the paper attached to this html file.
We say C has high multiplicity if each conjugacy
classes appearing in C appears
with high multiplicity. The more definitive C(onway)F(ried)P(arker)V(ölklein)
theorem gives a precise statement of the number of components and their
definition fields under the condition C
has high multiplicity.
Using the 3-cycle case to see the
bigger goals of the paper:
Classical archetypes are the case of either an even number of 2-cycles
in Sn, or an even
number of involutions in Dpk+1,
the dihedral group of
order 2(pk+1), with p
an odd prime. In all these cases there is just one
component. A non-classical archetype is the case of r 3-cycles acting
transitively on {1,2,…,n} (so G=An),
explained in hf-can0611591.html.
Here there are two connected components (corresponding respectively to
the trivial and nontrivial elements of the Schur multiplier of An),
defined over Q, if r ≥n;
just
one if r=n-1; and the Nielsen class is empty
if r < n-1.
The paper aimed at how one could achieve results on the RIGP toward
achieving G using
conjugacy classes supported among specific distinct conjugacy classes C'={C'1,…,
C'r'}. For example,
Mestre (as noted in hf-can0611591.pdf) used the case r=n-1 for n odd, to acchieve the non-split
degree 2 cover of An (called
Spinn). For n even points of don't work to make
such an achievement, and that is precisely explained by the nature of
the Spin-lifting invariant, depends on how
definitively one knows these points:
How precisely does you know CFPV for C supported in C'?
Can you locate located special components
defined over Q on the
Corresponding Hurwitz space?
On particular components from #b, can you locate rational points
using appropriate structure information (especially cusps) of the
component?
The 3-cycle is where r'=1
and C' consists
of the conjugacy class of 3-cycles. Conveniently we regard that one
conjugacy class as being in a direct limit of the collection of all
alternating groups. For this case, we know CFPV for C supported in C' perfectly, and all components
have definition field Q. By
using the word perfectly, I
mean there is no need for assumption #3 (high multiplicity). What makes
this example telling, is that the Main Theorem of hf-can0611591.pdf shows each component
with lifting invariant +1
supports an orbitof
universal g-p' reps.
The continuing story of these special representatives is that they
define special cusps on the Hurwitz spaces, and those cusps have
already played a big role in works of Debes and Emsalem for deciding
when the Hurwitz spaces and MTs
over them, have Qp points for all (or almost
all) primes p.