HTML and/or PDF files in the folder paplist-cov
paplist-cov: R(egular)I(nverse)G(alois)P(roblem) and Arithmetic of Covers (outside Modular Towers) For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 28] items below.
An unclickable "Pending" is still in construction. Use [ Comment on ...] buttons to respond to each item, or to the whole page at the bottom.

1. On a Conjecture of Schur, Michigan Math. J. Volume 17, Issue 1 (1970), 41–55 (pdf also on-line at the Michigan Math Journal). My first paper, though not first in print. It gives the classification of exceptional polynomials – those that map one-one on infinitely many residue fields – of a number field. They are up to (very precise) linear change over the algebraic closure compositions of cyclic (like xn) and Chebychev polynomials. Schur's 1921 Conjecture generated much literature: at its solution Charles Wells sent me a bibliography of over 550 papers, most showing certain families of polynomials – given by the form of their coefficients – contained none with the exceptionality property. An essential step was recognizing and using a reduction to polynomials with primitive monodromy group.

Includes the first serious use of R(iemann)'s E(xistence) T(heorem) on a problem of this type, a start of the monodromy method. Tchebychev covering groups are dihedral and easy to characterize. So, RET was quick, but not essential here. Yet, Schur's Conjecture was special within Davenport's problem, and RET has proved essential for that. Further, using RET opened the territory to many other problems (see the html file). Schur's original conjecture was technically easier than Davenport's Problem. Still, by considering its analog for rational functions, the monodromy method connected to Serre's O(pen)I(mage)T(heorem) (UMStoryExc-OIT.html and GCMTAMS78.pdf) and, so, to modular curves. SchurConj70.html %-%-% SchurConj70.pdf

2. The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois Journal of Math. 17, (1973), 128–146. The pdf file is a scan. The paper's center is the solution of Davenport's Problem UMStory.html is a user-friendly guide to this paper and other problems it influenced through the monodromy method. It explains the first serious use of a B(ranch)C(ycle)L(emma) for information on the defining field of an algebraic relation.

Davenport's problem was essentially to classify polynomials over Q by their ranges on almost all residue class fields. The most general results, restricted to polynomials not composable (indecomposable) from lower degree polynomials, gave two very different conclusions:

  1. Over Q two such polynomials with the same range are linearly equivalent: obtainable, one from the other, by a linear change of variables.
  2. For certain number fields polynomials that aren't linearly equivalent could have the same ranges for all residue class fields, though exceptional degrees are understandable and limited.
From Item #2 came the formulation and eventual solution of the Genus zero Problem. Its gist: Monodromy groups of rational functions are severely limited. dav-red.pdf

3. Fields of Definition of Function Fields and Hurwitz Families and; Groups as Galois Groups, Communications in Algebra 5 (1977), 17–82. The Branch-Cycle-Lemma (p. 62) specializes to say: A necessary condition for a sphere cover, with geometric transitive covering group G &le Sn and inertia generators in a set of conjugacy classes in G, to be over Q is that the classes form a rational union. We say this cover is in the absolute Nielsen class defined by G and the classes. This is the same as Q being the definition field of the associated Hurwitz space. A stronger necessary condition is that some component is over Q. Thm. 5.1 says three hypotheses (rational union condition, transitivity of a braid group action on the Nielsen classes, and G has no centralizer in Sn) are equivalent to Q being the intersection of all definition fields of covers in the Nielsen class. An extra condition gives a regular realization criterion.

Prop. 5 translates existence of symmetrized Hurwitz families even without the centralizer condition, using Grothendieck's pointed cohomology exact sequence. The only previous Hurwitz space use was Fulton's (non-arithmetic) for simple-branched covers, a thesis I read in the Princeton archives while I was at IAS, '67-69. HurMonGG.html %-%-% HurMonGG.pdf

4. Galois groups and Complex Multiplication, T.A.M.S. 235 (1978), 141–162. Hurwitz families are families of projective line, P1 covers, modulo a natural equivalence relation. Any one such cover will have a Galois closure over any field of definition. If you take it over some natural field F of definition, the Galois closure may only be defined over a proper extension field. The distinction between the two fields gets scientific if you consider that problem running over the covers in a Hurwitz family. This paper formulates that problem. One precise application identifies rational functions (in a single variable x) having the Schur covering property as in The Schur Conjecture. § 2 shows that for prime degree rational functions identifying those with this covering property is equivalent to the theory of complex multiplication. The essential point of this section is the description of modular curves as natural reduced Hurwitz spaces.

Describing prime-squared degree exceptional rational functions is equivalent to the GL2-case of Serre's Open Image Theorem, as in §6.1–6.3 of exceptTowYFFTA_519.html. This also documents the result of Guralnick-Müller-Saxl: All other degrees of indecomposable exceptional rational functions are sporadic. § 3 considers how the extension of constants problem relates to possible descriptions of the absolute Galois group of Q. A precise fruition of that is in a – and still only – presentation of GQ: GQpresentation.html. GCMTAMS78.pdf

5. Exposition on an Arithmetic-Group Theoretic Connection via Riemann's Existence Theorem, Proceed. of Symp. in Pure Math: Santa Cruz Conf. on Finite Groups, A.M.S. Pub. 37 (1980), 571–601. Collated the monodromy method from ingredients in solutions of Schur's Conjecture, generalizing Schur's conjecture to Serre's Open Image Theorem, Davenport and Schinzel Problems, and Hilbert-Siegel problems. Also, the source of data I presented to John Thompson in Sept. 1986, at U. of Florida that initiated the Genus 0 Problem: limiting Monodromy groups of rational functions (genus 0 covers; statement in Genus0.html, framework recounted in § VII.1. of UMStory.html). SantaCruz80.pdf

6. with H. Voelklein, Unramified abelian extensions of Galois covers, Proceedings of Symposia in Pure Mathematics, Part 1 49 (1989), 675–693. For a Galois cover X→P1 of the sphere with group G, the main theorem gives the exact criterion for all unramified abelian extensions of the cover to group theoretically split. To whit: That Pic1(X) is G-isomorphic to Pic0(X), the Jacobian of X. The paper produces from this examples where X has G-invariant divisor classes, but no G-invariant divisor. The html file lays out the complementarity of this with the Modular Tower program that started in 1995. Pending as of Saturday, December 29, 2007. frvoUnramAbExts.html %-%-% frvoUnramAbExts.pdf

7. Combinatorial computation of moduli dimension of Nielsen classes of covers, Contemporary Mathematics 89 (1989), 61–79. This paper was one of the first to take for granted the value of computing properties of Hurwitz spaces attached to a Nielsen class. It references the then newly formulated Genus 0 Problem. UMStory is the short version of Genus 0 Problem history – that most genus 0 covers have monodromy group closely related to particular representations of dihedral and alternating groups – which shows it was the exceptional cases that arose in Davenport's problem that pointed the way. In this paper, we approach the moduli dimension – dimension of the range of a Hurwitz space to moduli of curves of the appropriate genus – by showing how to compute the following quantities for curves in a Hurwitz family: 1. The endomorphism ring of their Jacobians. 2. The monodromy action on the cohomology of a fiber. combcomp-moduli.html %-%-% combcomp-moduli.pdf

8. Arithmetic of 3 and 4 branch point covers: a bridge provided by noncongruence subgroups of SL2(Z), Progress in Math. Birkhauser 81 (1990), 77–117. An exposition of the monodromy method applied to the Inverse Galois Problem, though one telling example: Covers of the sphere with 3-cycle ramification. The html file details some of the developments that came from this, including works of Mestre and Serre, who were present at the Seminar Delange-Pisot-Poiteu in the Spring of 1989 when I delivered this talk. Pending Saturday, December 29, 2007. Arith3-4brptcovers.html %-%-% Arith3-4brptcovers.pdf

9. with P. Debes, Rigidity and real residue class fields, Acta. Arith. 56 (1990), 13–45. The results come in two-steps: 1. An effective criterion for a compact Riemann surface cover of the projective line to contain real points. 2. A precise description of the real points on a Hurwitz space defined by an absolute or inner Nielsen class. #2 interprets as describing all covers (in the given Nielsen class) whose field of moduli is contained in the reals. Then, given that, describing from those all covers with the reals containing a field of definition. The paper concludes by answering related questions from a paper of Serre. Pending as of 01/18/07: rigRealResclass.html %-%-% rigRealResclass.pdf

10. with P. Debes, Arithmetic variation of fibers in families: Hurwitz monodromy criteria for rational points on all members of the family, Crelles J. 409 (1990), 106–137. A fundamental question is to decide when in a given family of genus 0 covers you can be certain, whenever a member of the family has definition field a number field K, then the covering curve also has a K point. Most interesting are cases where K is the rationals and the family has a dense subset of members defined over the rationals. Results for Hurwitz families give general expections. The main result here is a Hurwitz monodromy criterion for this rational point outcome. A special case: Each cover over K has a degree one K divisor supported in the ramified point locus. We give examples to show our criterion is more general than this. Pending as of 09/13/08: ArithVarFibFam.html %-%-% ArithVarFibFam.pdf

11. with H. Voelklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290, (1991) 771–800. The 3-cycle Nielsen class case appears in the html file to explain the interplay between three theorems.
  1. How the Branch Cycle Lemma determines the precise definition field of Hurwitz spaces defined by a Nielsen class.
  2. The effect of the C(onway)F(ried)P(arker)V(ölklein) Theorem – with its high appearance hypothesis on the Nielsen class – to decide the precise definition of Hurwitz space components.
  3. Effectiveness of the H(arbator)-M(umford) representative method, as an alternate to CFPV in #2.
inv_gal.html %-%-% inv_gal.pdf

12. with H. Voelklein, The embedding problem over an Hilbertian PAC field, Annals of Math. 135 (1992), 469–481. It is really generalizations of the RIGP that have so many applications. Many problems require producing covers defined over one field whose Galois closures are defined over (possibly) much larger fields. That is true of Serre's Open Image Theorem, and of the construction of exceptional covers – truely down to earth examples of the monodromy method discussed in UMStory.html. More abstractly, it is the heart of the GQ presentations in this paper. GQpresentation.html %-%-% GQpresentation.pdf

13. with D. Haran and H. Völklein, Absolute Galois group of the totally real numbers, C.R. Acad. Sci. Paris, t. 317 (1993), 95–99. Pending as of 02/02/09 QTotallyReal.html %-%-% QTotallyReal.pdf

14. with D. Haran and H. Völklein, Real Hilbertianity and the field of totally real numbers, Cont. Math., proceedings of Arizona conf. in Arith. Geom. 174 (1994), 1–34. Pending as of 02/02/09 TotallyRealAbsGG.html %-%-% TotallyRealAbsGG.pdf

15. with P. Debes, Nonrigid situations in constructive Galois theory, Pacific Journal 163 (1994), 81–122. This uses the results of "Rigidity and Real Residue Classes" on several problems among which are these.
  1. Show every finite group is realized regularly over the totally real (all conjugates real) numbers.
  2. How to construct Sn covers with four branch points with the covers also having real points (can't be done with three branch points).
  3. Based on Mazur's Theorem on torsion points on elliptic curves over the rationals, if m is a prime larger than 7, then the dihedral group of order 2m isn't regularly realized over the rationals with fewer than 6 branch points.
#3 amounts to the formulation of the M(odular) T(ower) program just for dihedral groups (see the html file for URLs to the MT Time Line), a statement equivalent to finding certain types of torsion points on hyperelliptic Jacobians. NonRigidGT.html %-%-% NonRigidGT.pdf

16. Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture, Finite Fields and their applications, Carlitz volume 1 (1995), 326–359: Shows how to use the B(ranch) C(ycle) L(emma) and the monodromy method to disprove two well-known conjectures about polynomial maps (covers). The author of the article's Math Review – John Swallow – calls it a service to the community. The html file explains the two problems. It is also a primer on handling nontrivial points on explicit families – of polynomial covers – in arithmetic geometry. From this we see exactly when the conjectures do hold. chow-coh-zass-conjs.html %-%-% chow-coh-zass-conjs.pdf

17. Enhanced review of J.-P. Serre's Topics in Galois Theory, with examples illustrating braid rigidity, Recent Developments in the Galois Problem, Cont. Math., proceedings of AMS-NSF Summer Conference, Seattle 186 (1995), 15–32. Briefer review---Topics in Galois Theory, J.-P. Serre, 1992, Bartlett and Jones Publishers, BAMS 30 #1 (1994), 124–135. ISBN 0-86720-210-6. ser_gal.html %-%-% ser_gal.pdf

18. 1998 response to a request from an NSF program officer to describe progress on the Inverse Galois Problem. In lieu of two Fried-Voelklein papers: Translated the regular Inverse Galois as a statement on Rational points on Inner Hurwitz spaces, and its corollary presentation of the absolute Galois group GQ as an extension of products of symmetric groups by a pro-free group. gal_prog98.pdf

19. In response to questions of Stefan Wewers, we describe the space of Frey-Kani covers, as a variant on the space of dihedral covers. The reduced Hurwitz space of the latter is a modular curve. In this approach the Frey-Kani covers also show up their modular curve nature, with a different slant than in the original description by Frey and Kani (01/04/1998). frey-kani.html %-%-% frey-kani.pdf

20. Variables Separated Polynomials and Moduli Spaces, No. Theory in Progress, eds. K. Gyory, H. Iwaniec, J. Urbanowicz, proceedings of the Schinzel Festschrift, Summer 1997 Zakopane, Walter de Gruyter, Berlin-New York (Feb. 1999), 169–228. This paper came from the lead talk I gave at Schinzel's elaborate celebration. It contains a complete outline of the characteristic 0 version of the solution of Davenport's Problem for polynomial pairs over a number field with one of them indecomposable: That only the degrees 7, 11, 13, 15, 21 and 31 are possible. Also, the completely different implications for Davenport pairs over any finite field (characteristic p): That there are infinitely many possible p' degrees of indecomposable Davenport pairs. Includes tie-ins to problems posed by A. Schinzel, R. Abhyankar, R. Guralnick (implications for the genus 0 problem in positive characteristic) and P. Mueller. varseppolynoms.html %-%-% varseppolynoms.pdf

21. with P. Debes, Integral Specialization of families of rational functions: PJM 190, 1999, 45–85. Siegel's Theorem says an affine curve covering the affine z line has but finitely many quasi-integral points unless there are at most two points on the curve's (nonsingular) compactification over z=∞. We call a cover satisfying this hypothesis a Siegel cover. This condition defines a Nielsen class of covers. This paper goes after a converse. Suppose you have a Siegel-Type Nielsen class, and its parameter space has a dense set of Q points. When can you prove there is a cover over Q in the Nielsen class for which there are infinitely many Z[1/a] for some integer 1/a? The paper gives an affirmative answer to one definitive case: That the only possible violation of the Hilbert-Siegel problem -- special degree 5 polynomials -- have a dense subset of them that are counterexamples to it. It also shows how to produce many Nielsen classes that pose similar challenges. From its practical applications, this problem is a big challenge to modern Inverse Galois techniques. dfr-deg5.html %-%-% dfr-deg5.pdf

22. with E. Klassen and Y. Kopeliovic, Realizing alternating groups as monodromy groups of genus one covers, PAMS 129 (2000), 111–119. Precisely: The full family of 3-branch point covers of genus 1 curves have nonconstant maps to the moduli space of genus 1 curves. Other results supercede this in one way: They produce families of 3-branch point covers of arbitrary genus g>0 whose map to the moduli of curves of genus g is dominant. Yet, present applications could use the explicitness of this case in those higher genus results. moddimAn.html %-%-% moddimAn.pdf

23. with A. Mezard, Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. of the American Math. Society (2002), 223–247. The paper is arranged to show how it generalizes the first half of Grothendieck's famous theorem on deforming tame covers to wildly ramified covers, and how it allows practical attacks on the 2nd half. Significantly, the first business is to generalize to dealing with non-Galois extensions of local fields. The html file exposits on these topics:
  1. Motivation from Hurwitz spaces
  2. Local Ramification Data
  3. Global Configuration Spaces
  4. The Major Unsolved Problem
fr-mez.html %-%-% fr-mez.pdf

24. Relating two genus 0 problems of John Thompson, Volume for John Thompson's 70th birthday, in Progress in Galois Theory, H. Voelklein and T. Shaska editors 2005 Springer Science, 51–85. The "relating" entwines three problems:
  1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p.
  2. Showing that the monodromy groups of rational function maps over the complexes are limited to a finite set of groups, outside of groups close to alternating groups (example, symmetric groups) with special representations, and dihedral and cyclic groups.
  3. Relating the genus 0 modular curves to the character group of the Monster simple group, so-called Monstrous Moonshine.
thomp-genus0.html %-%-% thomp-genus0.pdf

25. Alternating groups and moduli space lifting Invariants, Arxiv #0611591v4. Israel J. Math. (2009), 1–68. Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r ≥ n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces of 3-cycle covers with + (resp. -) lift invariant carry canonical even (resp. odd) theta functions when r is even (resp. odd). For inner spaces the result is independent of r. Another use appears in "Connectedness of families of sphere covers of An-Type," This shows the M(odular) T(ower)s for the prime p=2 lying over Hurwitz spaces first studied by Liu and Osserman have 2-cusps. That is sufficient to establish the Main Conjecture:
(*) High tower levels are general-type varieties and have no rational points.
For infinitely many of those MTs, the tree of cusps contains a subtree – a spire – isomorphic to the cusp tree on a modular curve tower. This makes plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs. Establishing these modular curve-like properties opens, to MTs, modular curve-like thinking where modular curves have never gone before. hf-can0611591.html %-%-% hf-can0611591.pdf

26. What Gauss Told Riemann about Abel's Theorem, preprint presented in the Florida Mathematics History Seminar, Spring 2002, as part of John Thompson's 70th birthday celebration. Yes, the well-over 60-year-old Gauss actually did talk to the just 20-year-old Riemann. Abel's explicit production of all analytic functions on a complex torus is well known. Less well-known is development of parameters for all functions of a special type: Those mapping through a prime (p) degree cover of another complex torus. Those parameters describe what we today call the modular curve Y0(p). Even less known, are early uses made of this:
  1. Galois' application of his unsolvability result to show parameters for Y0(p) (p>3) are not "solvable" in the classical j parameter.
  2. Riemann's partial success in finding algebraic parameters for Riemann surface families by dragging, by its branch points, a function on one of them.
The 2nd came from Riemann's conversations with Gauss about the complex torus case. This presented more problems for modern topics than did generalizing the first of Abel's famous theorems. Wh-Gauss-Tld-Riem-ab-Abel.html %-%-% Wh-Gauss-Tld-Riem-ab-Abel.pdf

27. Variables Separated Equations and Finite Simple Groups: (abridged) What I learned from graduate school at University of Michigan, 1964–1967 The story of the monodromy method, as told by recounting the solution of Davenport's Problem. A longer version attached to it discusses its influence on the following projects:
  1. Translation between the Davenport-Lewis conjecture on exceptional covers and Serre's Open Image Theorem.
  2. Applying the simple group classification to the genus 0 problem (conversations with Feit, McLaughlin and Thompson).
  3. the Galois stratification forerunner of Chow motives (from my first Annals paper solving the Ax-Kochen Problem).
The html version is more complete, having a layman's discussion of the connection to the classification and work of Thompson (genus o Problem) and Serre (modular curves and the Open Image Theorem). The pdf version was published in the ContinuUM by the UM Math. Dept. UMStoryShort.html %-%-% UMStoryShort.pdf

28. Variables Separated Equations and Finite Simple Groups: (unabridged) a more complete version of UMStoryShort.html. The initiation of the monodromy method included two new tools: the B(ranch)C(ycle)L(emma) and the Hurwitz monodromy group. By walking through Davenport's problem with hindsight, variables separated equations let us simplify lessons on using these tools. We celebrate that by attending to these general questions:
  1. What allows us to produce branch cycles, and what was their effect on the Genus 0 Problem (of Guralnick/Thompson)?
  2. What is in the kernel of the Chow motive map, and how much is it captured by the using algebraic covers?
  3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?
Each phrase addresses an aspect of formulating problems based on equations. That is, we seem to need algebraic equations. Yet why, and how much do we lose/gain in using more easily manipulated surrogates for them? This issue comes clear by considering the difference in the result for Davenport's Problem and that for its formulation over finite fields, using a technique of R. Abhyankar. UMStory.html

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