HTML and/or PDF files in the folder
paplist-cov: R(egular)I(nverse)G(alois)P(roblem) and Arithmetic of Covers (outside Modular Towers) For an html and pdf (or rtf, or ppt) file with the same name, the html is an exposition. Click on any of the [ 43] 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. with R.E. MacRae, On the Invariance of Chains of Fields, Ill. J. Math. Vol. 13 #1, March 1969, 165–171. We start with a general hypothesis about a field extension L/K having a valuation. Then, Thm. 3.6 guarantees, for the lattice of fields between L and K, the function taking a field to its degree over K maps the intersection (resp. composite) of the two fields to the gcd (resp. lcm) of the degrees. Applying this to K=k(x) and L=k(f(x)) when f is a polynomial of degree prime to char(k) quickly gives a theorem of Ritt over any allowed k. Also, going to the algebraic closure of k preserves the lattice of fields (Thm. 3.5), a result used in the proof of the Schur Conjecture. We conclude with a separated variable characterization of two polynomials having a common composition factor. InvChainsFields69.pdf

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 P1 assume the monodromy group G has two permutation representations Ti, i=1,2 for which the stablizer G(1,T1) of 1 in T1 is intransitive in T2. Also, that some point of X ramifies of index pu, u>0, but for no other point of X does pu divide its ramification index. Then, every definition field of f intersects nontrivially the field generated by the pu 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:

  1. f(x)-g(y) is irreducible; and
  2. its projective normalization, C~f,g defines a curve of genus 0.
Prop. 1 gives two genus formulas based on knowing the precise nature of the singularities of the fiber products of the covers given by f: P1xP1z and g: P1yP1z.
  1. Assuming #2: For the genus of C~f,g, by applying Riemann-Hurwitz (RH) to it as a cover of P1y.
  2. When f = g (so quite contrary to #2) a formula for the genus of C~f,f minus its diagonal component.
The application of these formulas is to the two cases gcd(m,n) is either 1 or 2. The case when gcd = 1 gives a quick proof of the harder of Ritt's two theorems on compositions of polynomials. While the case gcd = 2, based on Siegel's famous theorem, is the complete solution of a series of papers doing particular searches for curves with infinitely many quasi-integral points on separated variable equations. The Fried-Gusic paper drops assumption #1. ThmofRitt-RelatedDioph73.pdf

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 GK its absolute Galois group.

Main Theorem: For any integer d < ∞ , and almost all (Haar measure) d elements σ in GK, 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 P1 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 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

7. 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

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:

  1. A Q form of degree d in projective d space has an F point;
  2. No such non-algebraically closed F could be Galois over Q.
He was significantly right about the first as Kollar showed. He was significantly wrong about the second – not only could be it Galois, it could also be Hilbertian – as this paper shows. Many natural fields of algebraic numbers have now been shown to be PAC, though there are still fascinating problems. Example: Is the solvable closure of Q PAC (Prob. 1 a, p. 748 [FrJ86]). FrJ78PACGaloisQExt.pdf

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. A Note on Automorphism Groups of Algebraic Numbers fields, Proceedings of the Amer. Math. Soc. Vol. 80, No. 3, Nov. 1980, 386–388. In the process of reviewing for Math. Reviews a paper of E. Fried and J. Kollar, I found a correctable error in weakened form of the celebrated Hilbert-Noether conjecture sometimes called the Inverse Galois Problem. For each finite group G, the paper gives an explict and simple construction of a (not necessarily Galois) extension of Q having its full automorphism group equal to G. AutomorphismGroupsAlgNoFields80.pdf

12. FrattCovsProjGps80.pdf

13. 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

14. 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:

BiggersFried1982.html %-%-% BiggersFried1982.pdf

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

16. 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

  1. Primitive monodromy groups of polynomials, Recent developments in the Inverse Galois Problem conference, vol. 86, 1995, AMS Cont. Math series, pp. 385 –401.
  2. Reducibility behavior of polynomials with varying coefficients, Israel J. 94 (1996), 59 –91
for which he is well known. Hilb-Sieg86.pdf

17. 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

18. 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:

The former problem was the domain of heavy group theory, especially the classification in the hands of Guralnick and Thompson, but it was of a different nature entirely from that this paper considers. The latter problem called for the development of new techniques in the moduli and arithmetic of Hurwitz spaces. Between the two, the main result of this paper was forgotten. Denote the moduli space of curves of genus g by Mg. Denote the curve associated to a point pMg by Xp.

The use of the word generic in the title means that on the moduli space, Mg, for g > 6 the only points p for which Xp is uniformized by radicals lie in a proper Zariski closed subset of the quasi-projective variety Mg. Being uniformized by radicals means that Xp has a presentation φ: XpP1z as a (ramified) cover of the Riemann sphere uniformized by z whose monodromy group is solvable. We may immediately assume that φ is an indecomposable cover. So, the monodromy group of the cover, in its natural deg(φ) permutation representation, is primitive. This reduction is key in managing the genus 0 problem (and its variants). The html file contains the pieces of the paper divided into 4 parts (Chaps. 1 and 2, Chap. 3, Chap. 4 and Chap. 5) with tables of contents showing what they accomplish. gencurveslist-notrads.html

19. 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 XP1 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. frvoUnramAbExts.html %-%-% frvoUnramAbExts.pdf

20. 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 P1 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:

  1. The endomorphism ring of their Jacobians.
  2. The monodromy action on the cohomology of a fiber.
combcomp-moduli.html %-%-% combcomp-moduli.pdf

21. 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. Arith3-4brptcovers.html %-%-% Arith3-4brptcovers.pdf

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. Topics in Galois Theory, review of J.-P. Serre, 1992, Bartlett and Jones Publishers, BAMS 30 #1 (1994), 124–135. ISBN 0-86720-210-6. We seamlessly added introductions to Serre's book developments he did not include:

Followed by this quote from Serre to the Author: ''The inverse Galois problem gives us excuses for learning a lot of new Mathematics [Se3].'' SerreGalTh-OriginalRev92.pdf

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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 indecomposable polynomials over Q whose ranges on Z/p are the same for almost all p.
  2. Showing that the monodromy groups of indecomposable 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

39. 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. rn). 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 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 implies 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

40. 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 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 applications, and complications, 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

41. 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,gC[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

42. with I. Gusi'c, Genus 0 (or 1) components of variables separated equations, preprint 10/17/16. Many applications start by describing those curves of form Cf,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 Cf,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 Cf,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. 4.12 (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 Cf,g have a genus 0 component, including all cases when that – in not obvious ways – happens. We have a collective of three results – Cors. 4.4, 4.7 and 4.8 – that handles the first nontrivial Neilsen class that arises when f is a polynomial, the degree 7 case of Davenport's problem. This has specific versions of all pieces for the general case. ReducibilityHypothesis-VarSep.pdf

43. 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