2. with R.E. Macrae, On Curves with Separated Variables, Math. Ann. 180 (1969), 220–226. Given any (nonconstant) polynomial a(x,z) in two variables, we introduce the idea of a minimal separation: A separated variables polynomial f(x)-g(z) that a(x,z) divides and which is minimal with this property. Thm. 4.2 says that if a(x,z) divides some separated polynomial, then it has a minimal separation. The paper InvChainsFields69.pdf has several proofs, for polynomials f of degree prime to the characteristic of k, the following holds. In the lattice of fields between L=k(x) and K=k(f(x), mapping a field to its degree over K maps the intersection (resp. the composite) of two fields to the gcd (resp. lcm) of the degrees. § 5 gives counterexamples when either f is a rational function or the characteristic assumption doesn't hold. These are also counter to the extension of k to its algebraic closure preserving the lattice of fields. Yet, for the general polynomial, maximal chains all have the same length and relative degrees (in possibly different order). SepVarbsCurves69.pdf

3. On the Diophantine equation f(y)-x = 0: Acta Arithmetica, XIX (1971), 79–87. This as part of 3 one-hour lectures at the 1969 Summer conference at Stony Brook just before my eight years as a tenured faculty there. My relation with Ax – a controversial figure – was a personal difficulty for me. This paper's virtue by comparison with dav-exc.pdf.: It applied the monodromy method – in a limited situation – without assuming covers had primitive monodromy groups. The Main Result: For a cover f: X → P^{1} assume the monodromy group G has two permutation representations T_{i}, i=1,2 for which the stablizer G(1,T_{1}) of 1 in T_{1} is intransitive in T_{2}. Also, that some point of X ramifies of index p^{u}, u>0, but for no other point of X does p^{u} divide its ramification index. Then, every definition field of f intersects nontrivially the field generated by the p^{u} roots of 1. It applied to show there are no Schinzel pairs, f,g ∈Q[x] – f(x) -g(y) reducible – with the degree of f an odd prime-power and f affine inequivalent to g. Equationfy-x71.pdf

4. A theorem of Ritt and related diophantine problems, Crelles J. 264, (1973), 40–55. I use the notation that arises in a later paper with Ivica, Gusic, Genus 0 (or 1) components of variables separated equations preprint February, 2016. After preliminaries Problem 1 asks, with f, g rational functions of respective degrees m and n, to characterize those for which:

5. with M. Jarden, Stable Extensions and Fields with the Global Density Property, Can. J. Math., Vol. XXVIII, No. 4 (1976), 774–787. This paper extends basic ideas of P(seudo)A(lgebraically)C(losed) fields to fields with a valuation ω. A field M with valuation ω has the ω-density property if for every absolutely irreducible variety V over M, its M points are dense in all points on V over the algebraic closure of M. Take K to be a countable Hilbertian field, and G_{K} its absolute Galois group. Main Theorem: For any integer d < ∞ , and almost all (Haar measure) d elements σ in G_{K}, the field K( σ) fixed by σ in the algebraic closure, K^{–}, is ω-dense for each absolute value of K^{–}. Main Idea: As with PRC fields, reduce to where V is a curve over K. Then, show that the projective normalization of V has a non-constant K-map (cover) to the projective line P^{1} so the Galois closure of the cover is a regular extension of K. This is an arithmetic version of a Lefschetz pencil. FrJ76StableExtsGlobDensProp.pdf

6. 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 S_{n} 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 S_{n}) 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

7. Galois groups and Complex Multiplication, T.A.M.S. 235 (1978), 141–162. Hurwitz families are families of projective line, P^{1} 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 GL_{2}-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 G_{Q}: GQpresentation.html. GCMTAMS78.pdf

8. PonceletCorr78.pdf

9. with M. Jarden, Diophantine Properties of Subfields of Q^{_}: Amer. J. Math. Vol 100 (1978), 653–666. An algebraic P(seudo)A(lgebraically)C(losed) field F is one for which every absolutely irreducible Q variety has an F point. Ax found that the algebraic closure of Q isn't the only such: almost (but not) all of the fields of algebraic numbers of a non-trivial ultraproduct of finite fields are PAC. Based on (resp.) Chevalley's Theorem and the Lang-Weil result, Ax made two conjectures:

10. 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

11. FrattCovsProjGps80.pdf

12. with R. Whitley Effective Branch Cycle Computation: This preprint, sent to Geometrae Dedicata, if we had rewritten it for publication would now need reconsideration for the question below. I have included referee comments. Recall: For a cover of the Riemann sphere, branch cycles consists of an r-tuple of elements in the monodromy group of the cover, one for each branch, that characterize the cover. The paper is an algorithm to compute branch cycles for a given cover, based on rearranging power series along a collection of polygonal paths forming a classical set of generators of the r-punctured sphere. The serious question is whether, based on the degree of the cover, the height of the coefficients, and the distance between branch points there is a polynomial bound for such a computation. EffCompBrCycles.pdf

13. with R. Biggers, Moduli spaces of covers and the Hurwitz monodromy group: Crelle's Journal 335 (1982), 87–121. This pdf file has been scanned, but is searchable. By the late 70s there was motivation to find serious examples of Hurwitz spaces defined by a Nielsen class where there were several components. This paper has the first: an unbraidable outer automorphism. Such examples – and the components identified by Schur multipliers – remain the most interesting, non-accidental identifiers of components. The html file includes these topics:

14. with R.~Biggers, Irreducibility of moduli spaces of cyclic unramified covers of genus g curves, TAMS Vol. 295 (1986), 59–70. CycCovModBiFr86.pdf

15. The Hilbert Siegel Problems and Group Theory solving cases of them, formerly a long preprint from 1986, Rigidity and applications of the classification of simples group to monodromy. The remainder of that material has been placed in other papers, leaving only this remnant for which the group theory finishes off limitation of the Davenport pairs (§ 2), and rational functions with double-degree permutation representation (§ 3) for the first Hilbert-Seigel problem (§ 4). Part of this paper also contained the main result of Alternating groups and moduli spaces lifting inbvariants, but not the applications of that paper. Several papers quote the original, Serre for the alternating group work which led to the Fried-Serre lift invariant, and Peter Müller for two papers

16. Irreducibility Results for Separated Variables Equations, J. Pure and App. Alg. 48 (1987), 9–22. A sum of 3 (or more, nonconstant) rational functions each in a distinct variable is irreducible. This is the Main Theorem. Many applications, however, technically depend on summing two polynomials in distinct variables. The (n,m)-problem stated here is a serious challenge to understanding when the result will be reducible. We know precisely what happens if one of the polynomials is indecomposable; a conclusion embodied in the solution to Davenport's Problem (say, dav-red.pdf or UMStory.pdf). Without indecomposability it requires treating non-primitive monodromy groups of genus 0 covers. Even the simple group classification doesn't help. Prop. 2.10 introduces an integer parameter k into the (2,3)-problem, and gives a negative solution to the problem for k=1 or 2. IrrResultsSepVar87.pdf

17. with R. Guralnick, The Generic Curve of Genus g > 6 is not uniformized by radicals: This paper was typed up in the late '80s. There was much activity in the early '90s around related topics:

18. 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→P^{1} 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 Pic^{1}(X) is G-isomorphic to Pic^{0}(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. frvoUnramAbExts.html %-%-% frvoUnramAbExts.pdf

19. 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. (A fairly complete exposition is in §7.2 of Two Genus Problems of John Thompson.) The roughest statement is that most genus 0 covers have monodromy group closely related to particular representations of dihedral and alternating groups. That shows it was the exceptional cases that arose in Davenport's problem that pointed the way to the finitely many groups that were exceptions to these general genus 0 covers. Finding what monodromy groups can appear as covers of g of P^{1} by the general curves of a given genus g, is a sub-question. This paper studies two approaches to 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:

20. Arithmetic of 3 and 4 branch point covers: a bridge provided by noncongruence subgroups of SL_{2}(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. Arith3-4brptcovers.html %-%-% Arith3-4brptcovers.pdf

21. 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. rigRealResclass.html %-%-% rigRealResclass.pdf

22. 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. ArithVarFibFam.html %-%-% ArithVarFibFam.pdf

23. 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.

24. 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 G_{Q} presentations in this paper. GQpresentation.html %-%-% GQpresentation.pdf

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

26. 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. TotallyRealAbsGG.html %-%-% TotallyRealAbsGG.pdf

27. with H. Voeklein, The Absolute Galois Group of a Hilbertian PRC Field, Israel J. of Math. 85 (1994), 85–101. We show the absolute Galois group of a countable Hilbertian P(seudo) R(eal) C(losed) field is real-free, determined up to isomorphism by the topological space of orderings of the field. Any proper extension of the field of all totally real (all embeddings of the field are into the reals) numbers is an example. A PRC field P is one for which any absolutely irreducible non-singular variety over P has a P-point if and only if it has a point over each real closure of P. This follows the techique of the Fried-Voelklein 1992 Annals paper, The embedding problem over a Hilbertian PAC field, except that we must extend embedding properties of groups to include care with involutions. FrVHilbPRCField1994.pdf

28. 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.

29. 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

30. 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

31. 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 G_{Q} as an extension of products of symmetric groups by a pro-free group. gal_prog98.pdf

32. 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

33. 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

34. 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

35. 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

36. 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:

37. Alternating groups and moduli space lifting Invariants, Arxiv #0611591v4. Israel J. Math. 179 (2010) 57–125 (DOI 10.1007/s11856-010-0073-2). Small correction list at hf-can0611591-cor.html. Main Theorem: Spaces of r-branch point 3-cycle covers, degree n, or their Galois closures of degree n!/2, have one (resp. two) component(s) if r=n-1 (resp. r ≥ n). Improves Fried-Serre on 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: The fibers of the absolute spaces of 3-cycle covers with + (resp. -) lift invariant carry canonical even (resp. odd) theta functions when r is even (resp. odd). A result assures that some of these even thetas produce non-zero even theta-nulls (odd ones are always zero) the Hurwitz-Torelli automorphic functions on the Hurwitz spaces. We still don't know the exact conditions for them being nonzero. For inner spaces the desired even thetas, not induced from lower Hurwitz spaces, appear independent of r. Another application appears in "Connectedness of families of sphere covers of A_{n}-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 implies the Main Conjecture:

38. What Gauss Told Riemann about Abel's Theorem, preprint, up on the Abel Website at the Danish National Academy of Sciences: 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 his 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 Y_{0}(p). Even less known, are early uses made of this:

39. with Ivica Gusiًc, Schinzel's Problem: Imprimitive covers and the monodromy method: ArXiv #1104.1740v3. Acta Arithmetica, Vol 155, (2012), 27–40. The published version, Schinz75-155-1-3acta.pdf. Schinzel's original problem was to describe when an expression f(x)-g(y), with f,g∈ C[x] nonconstant, is reducible. We call such an (f,g) a Schinzel pair if this happens nontrivially: f(x)-g(y) is newly reducible. We accomplished this as a special case of a result in dav-red.pdf, when f is indecomposable. That work featured using primitive permutation representations. Even after 42 years going beyond using primitivity is a challenge to the monodromy method despite many intervening related papers (see UMStory.pdf). Here we develop a formula for branch cycles that characterizes Schinzel pairs satisfying a condition of Avanzi–Gusiًc–Zannier and relate it to this ongoing story. Schinz75Birth1104-1740v3.pdf

40. with I. Gusi^{'}c, Genus 0 (or 1) components of variables separated equations, preprint 04/20/16. Many applications start by describing those curves of form C_{f,g}= {(x,y)| f(x) - g(y)=0} with infinitely many solutions over localizations of the ring of integers of a number field. Immediately you must address finding (f,g) for which the projective normalization of C_{f,g} has a genus 0 (or 1) component. The case when f and g are polynomials and f is indecomposable was essentially solved by distinguishing between when the number, u, of components was 1 versus >1. For u =1, a direct formula for the genus worked: [Fr73b, (1.6) of Prop. 1]. For u >1 the result came from solving Schinzel's problem [Fr73a]: Describe all such components. Here, though, the computation of component genuses was adhoc. Pakovich tried dropping both the indecomposable and polynomial restrictions. Yet, he added a telling assumption: (IC) That C_{f,g} is irreducible (u = 1). Assuming (IC} he showed –for fixed f – unless the Galois closure of f has genus 0 (or 1), the resulting genus grows linearly in the degree of g. We extend [Fr73b, Prop. 1] to when u >1. Thm. 4.7 (Main Theorem) extending Pakovich's result must tend to composition factors of g. We precisely describe for pairs (f,g), as g varies, when infinitely many C_{f,g} have a genus 0 component, describing all cases when that – in not obvious ways – happens. ReducibilityHypothesis-VarSep.pdf

41. with Ivica Gusiًc, Singular points on moduli spaces and Schinzel's Problem:. Latest Version 06/14/12. To solve generalizations of the A(vanzi)G(usiًc)Z(anier) version of Schinzel's problem we must go beyond several limitations in previously successful results. The difficulty in dropping indecomposability of covers lies in dealing with imprimitive groups. That traces to needing group theory beyond the simple group classification. In this paper we considerably generalize the context of Schinzel's problem. We base this on interpreting the solution to the original using Hurwitz spaces of r-branch point covers. Our main formula interprets covers fixed by a Möbius transformation in terms of branch cycles. This describes singular points on reduced Hurwitz spaces when r > 4, and when r=4, it interprets when the mysterious moduli group acts trivially. ImPrimCovers.pdf