Exposition on "Arithmetic of 3 and 4 branch point covers"

The paper starts by explaining there are generalizations of rigidity for r > 3 by going back to two papers that were written when I was at the Institute for Advanced Study in 1972:
These are respectively papers #6 and #7 in the part of my web site on arithmetic uses of covers: http://www.math.uci.edu/~mfried/paplist-cov.html.

[HMIG] inroduces the branch cycle argument combined with moduli spaces of covers (Hurwitz space; see the short html exposition in HurMonGG.html to which we refer to throughout this exposition). It is dedicated to describing the relation between our first invariant of a cover of the Riemann sphere – called its Nielsen class – and the possible fields of definitions of covers in a given Nielsen class. The approach is unabashedly that of using moduli spaces.

[GGCM] concentrates on the relation between three topics:
  1. The theory of complex multiplication;  
  2. Schur's Conjecture for rational functions; and 
  3. Serre's Open Image Theorem. 
The conclusion is that #2 and #3, when understood properly, are variants of each other (see comments on [SC] below which has a, later, complete exposition). One sees that by taking a Hurwitz space approach to modular curves:  regarding them as parametrizing sphere covers. While we can still see them as parametrizing elliptic curves in the traditional way, the moduli problem is more general, though the spaces are the same.

I trained everyone who worked with me on the prescient §3 in [GGCM] on the bigger implications of working with reduced Hurwitz spaces (see the discussion on [HM] in the section on Modular Towers). The technique to make the path outlined there work eventually came to me as the key device in the Modular Tower program (below). 

In [HMIG] the Hurwitz spaces are those we call absolute. In this paper we need the reduction of covers by linear fractional transformations: reduced Hurwitz spaces (§2 in [GGCM]). The fruition of these ideas for my contributions to the absolute Galois group of Q and a bigger understanding of the significance of the Inverse Galois Problem came in later papers. The beginning of that appears in HurMonGG.html. 

The full developments are epitomized by four distinct directions, for which I give four representative papers, each with its own expositional html.
[SC] and [DP] (with their abstracts) are, respectively, items #17 and #20 in the part of my web site reserved for papers related to finite fields: http://www.math.uci.edu/~mfried/paplist-ff.html. [AG] is # 36 in the list http://www.math.uci.edu/~mfried/paplist-cov.html. Finally [HM] is #5 in my most important project, Modular towers,  at http://www.math.uci.edu/~mfried/paplist-mt.html which contains most of my recent papers. 

I reference each of these very long papers very quickly below, relying that someone with a serious interest in them will  look at the corresponding html files. I am very generous with my time. I have spent much of it explaining things in the html files. Again, all papers have short – not telegraphic – abstracts in the html files I've mentioned above, in addition to the html files attached to them.
Table of Contents
 
1. Basic data for covers.
2. Families for r = 3 and the Hurwitz monodromy group H3
3. Families for r = 4 and the Hurwitz monodromy group H4
4. Modular curves and H4
5. Generalizations of rigidity and examples

This paper notes an extremely important observation the Hurwitz monodromy group braid quotient on 4 strings, H4: It is a natural extension of PSL2(Z) (§3). The fruition of this observation is in [HM, §2.10]. That includes a proof that all reduced Hurwitz spaces of 4 branch point covers are upper half-plane quotients branched over the three classical points of the j-line.

Then, §4 reveals just how special are modular curves among these. The classical division for upper half quotients is between quotients by congruence subgroups (modular curves) and the rest. This paper shows a significant collection of such quotients between these. These are natural  j-line (not ƛ-line) covers, ramified at the expected points, of order (at most) 3 and 2 at the two elliptic j point. Few are modular curves, and we recognize those easily: they are reduced Hurwitz spaces for groups G akin to dihedral groups.

Yet, they also not the general upper half plane covers by a finite index subgroup of PSL2(Z), of which there are many, something known long ago. They deserve a special name. Here we refer to them as Internally Defined (ID) reduced Hurwitz spaces.

They also have appropriate modular curve-like conjectures attached to them, even though they are not defined by congruence subgroups. Indeed, that is their greatest virtue, for each of them are – like modular curves – part of  natural sequences that exhibit modular curve-like properties, though proofs so far require new ideas. Also, these ideas aren't restricted to just 4 (and 3) branch point covers.

Significance of the §5 examples

§5 intends to show the direct value of these intermediate covers. It has a little story in itself. Serre came to this talk, with two of his minions, in a somewhat belligerant mood. He had initiated previous correspondence with me on several occasions; on other topics. This Paris seminar talk went well in the sense that I handled his belligerance, and I figured it would be left there.

Shortly after, however, when Pierre Dèbes was visiting me at Irvine, a letter came from Serre with a question on generalizing the alternating group (A5) example in §5. The example was an improvement on Hilbert's regular realization of alternating groups as Galois groups over Q. Serre's question was whether the natural degree 2 cover, called Spinn, of An could be realized as an unramified extension of the generalization of the covers in §5.

In §5 the Nielsen class has either the notation Ni(A5,C34)abs or Ni(A5,C34)in  for its inner Hurtwitz space generalization.  Here abs refers to the permutation representation is the natural degree 5 representation of A5, and C34 is 4 repetitions of the conjugacy class of 3-cycles. Serre thought the answer was "No!" for n = 5 and wondered about the other values of n with 3-cycle covers.

By return mail I noted the answer was "Yes!" for n = 5. That followed immediately from my Paris talk. That also pointed to this answer for the generalization of 3-cycle covers, Nielsen class Ni(A5,C34)abs: "Yes" (resp. "No") if (-1)n-1 is even (resp. odd). It was because the kernel, <±1> of the Spinn → An cover can be identified as a calculable lift invariant.

Further, it didn't matter what cover you took, because the lift invariant was also a braid invariant. My talk had shown that there was only one braid component. So the answer being independent of the cover for Ni(A5,C34)abs was immediate. Serre wrote a paper on this considering any Nielsen class of the form Ni(An,C)abs where all conjugacy classes in An were of odd order elements.

The result was a formula computing the lift invariant in all these Nielsen class cases. He used the braid group as I had shown him. He did not show the stronger statement that there was only one braid orbit. 

[AG] has a list of results related to this example for all (not just genus 0) cases.
Many more specific results came out about these Nielsen classes because there is a rich classical tie-in to theta functions varying canonically and analytically on the reduced Hurwitz spaces [AG, §5]. Serre developed a formula that I interpret as detecting whether the thetas on a given Hurwitz space component are even or odd. When they are even, their corresponding (canonically defined) theta-nulls act like automorphic functions on the corresponding Hurwitz space component, though the Hurwitz spaces are not homogeneous spaces.

Modular Towers were the eventual result from these Alternating Group examples

To see that the ID j-line covers defined by a Nielsen class Ni(G,C)  (C consisting of r conjugacy classes in G) seed a natural structure similar to the modular curve sequences like {X0(pk+1)}k=0∞ requires knowing something that isn't classical about finite groups. It is that there is a universal p-Frattini cover of any G whose order is divisible by p, but it has no Z/p quotient. We call k the level.

Then, so long as the conjugacy classes C have no element divisible by p, the construction of the sequence of spaces {H(G,C,pk+1)}k=0∞ is canonical depending on what version of the (reduced) Hurwitz spaces you decide to use. The theory is well-developed now, with [HM] the source book for all further developments.

http://www.math.uci.edu/~mfried/paplist-mt/mt-overview.html, called the Modular Tower Overview, connects all these pieces. When C has exactly 4 conjugacy classes, then the Modular Towers are the ID j-line covers discussed above. Trust me, if you are thinking you know what this universal p-Frattini cover is, you are wrong, unless you have heard me lecture about it. Also, try and trust me, even though you don't know it, it is not an esoteric object, or unfathomable in any way. References to examples are in the first Modular Tower paper, to which you can go by way of http://www.math.uci.edu/~mfried/paplist-mt/mt-overview.html.

Still, one need not be an aficianado of modular curves to find connections to these topics. [SC] and [DP] connect to some of the oldest and most classical problems in diophantine geometry through these ideas, often through zeta functions attached to these problems.