Exposition on "Fields of Definition of Function Fields and Hurwitz Families and 
Groups as Galois Groups"

I was given an inexperienced typist, long before there was TeX, when I first arrived at UC Irvine in the middle '70s.  The first paper to succumb was the pre-publication retyping of this paper. Then, despite a five year refereeing process, it went to a journal that had recently decided it was making camera ready copy. The typography remains hard to read.

Still, many have used the paper over the years – including in the books of Vöklein or Matzat-Malle – though few have explicitly written on the Hurwitz space constructions that appear here (only alluding to them).

Table of Contents
§1. Riemann's Existence Theorem: Examples of Covers of P1
§2. Arithmetic theory of the Galois closure of a cover
§3. Families of covers of P1; Hurwitz families and symmetrized Hurwitz families
§4. Basic construction of Hurwitz families, with and without fine moduli conditions; the Hurwitz number
§5. The minimum field of definition of (symmetrized) Hurwitz spaces

 Exposition on what was accomplished in this paper

The nomenclature for general Hurwitz spaces was nonexistent. Therefore the terminology changed later. A Nielsen class of r-tuples:
was introduced here. Any (nonsingular, but ramified) cover f : X → P1z of the z-sphere P1z has an attached Nielsen class. If the cover has degree n then its Galois group G has an attached permutation representation Tf.

The main equivalence here is called absolute equivalence, which derives from Tf.  Then, the Nielsen class has the notation Ni(G,C)abs. The paper introduced the language of a "cover in the Nielsen class." 

The only fore runner was Hurwitz's use of 2-cycles in Sn to prove the connectedness of the moduli of curves of genus g, and an attempt by Fulton to imitate this for the same moduli space in positive characteristic based on Grothendieck's theory of the fundamental group. Fulton, however, does not recognize the braid group as the key object.

What is now a Hurwitz space – corresponding to an orbit of the braid group Br – on a Nielsen class – was called here a symmetrized Hurwitz space. The spaces corresponding to pure braid orbits – the natural kernel of Br → Sr – on Nielsen classes are called here Hurwitz spaces. There is no general name for those now; most applications to number theory would fail using them. Early '90s lectures of Manin at Berkeley used the spaces corresponding to pure braids.

There are five results in this paper that have continued to influence all later results. The field of moduli of a cover in Ni(G,C)abs was introduced here based on Weil's coycle condition. A forerunner of this and also of the branch cycle argument (below) already had appeared in the proof of Davenport's Problem. There a special case showed that no pairs of indecomposable polynomials over Q could have the same ranges modulo primes p, for almost all primes p, unless they defined equivalent covers. 
The Goal of all this 

If braids are transitive on the Nielsen class Ni(G,C)abs, so the Hurwitz space has only one absolutely irreducible component, then the minimal possible definition field of any cover in a given Nielsen class is the field computed from Thm. 5.1. Always the collection of Hurwitz components has this field as definition field. The classes in C form a rational untion of conjugacy classes if and only this field is Q.  

If braids are not transitive, then different components can have different definition fields, but there are invariants beyond the Nielsen class that also give us a theory that applies to pin down the definition field of components.

Finally, all of this was done to figure the possibility of answering questions whose conclusion would be affirmative if you could indicate precisely fields which would be definition fields for covers solving a particular arithmetic question. There are two places that served as archetypes of this, in situation where the braid action was shown to be transitive. 

Davenport's Problem: None of the Hurwitz spaces that arise in Davenport's problem had definition field Q, but by showing that those spaces were unirational, the final conclusion was that we knew precisely the fields K over which their were indecomposable polynomials pairs have the Davenport property (generalizing the above).

The Inverse Galois Problem: The results here applied to this famous problem, but they became much more universal with the addition the extension of the results above to Inner Nielsen Classes with their corresponding equivalence classes. The main result of a Mathematische Annalen paper was the relation between Inner and Absolute Hurwitz spaces attached to a given Nielsen class.

That codified the regular version of the Inverse Galois Problem as a problem of finding rational points on Hurwitz spaces. Indeed, to each finite group G it produced an infinite sequence of precisely defined Nielsen class giving absolutely irreducible Hurwitz spaces. Then, any point on any of  these spaces produces a regular realization of G.

The main result of an Annals paper was a precise fruition of the Inverse Galois philosophy started in Main Theorem 5.1, as applied to the large collection of Hilbertian P(seudo)-A(lgebraically)-C(losed subfields of the algebraic closure of Q. One result of this was the first presentation of the absolute Galois group of Q. A more complete set of descriptions of the successes of this philosophy – including the actual realization of groups as Galois groups, like the first serious cases of higher rank Chevalley groups – as a justification of Nielsen class thinking can be found in other html files attached to my papers.