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 Ur denote the space of r unordered points on the Riemann sphere P1z, uniformized by a(n inhomogeneous) variable z. The three main results are these:
  1.  Ni(G,C)* determines a Hurwitz space H(G,C)* with a natural map to Ur as a coarse moduli space of r-branch point covers of P1z in Ni(G,C)*.
  2. 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.
  3. 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 rn; 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:
  1. How precisely does you know CFPV for C supported in  C'? 
  2. Can you locate located special components defined over Q on the Corresponding Hurwitz space?
  3. 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 orbit of 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.


12/18/07 Mike Fried