HTML and/or PDF files in the folder http://www.math.uci.edu/~mfried/paplist-ff
For an html and pdf (or rtf, or ppt) file with the same name, the html is an exposition. Click on any of the [ 20] 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. Starts with text from Ralph Cicerone (UCI Chancellor in '90s) announcement of the cryptography implications of the Fried-Guralnick-Saxl classification of exceptional polynomials over a finite field. This was the launching of non-abelian cryptology. The easiest case, dihedral groups, interprets as variants on Serre's Open Image Theorem. This has a list of many papers not yet in electronic form: nonabel-cryptology.html

2. 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. Chebychev covering groups are dihedral and easy to characterize. So, RET was quick, but not essential here. Yet, Schur's Conjecture was special, and much easier, within Davenport's problem, and RET has proved essential for that. 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. Further, using RET opened the territory to many other problems (especially see the complete story of Davenport's problem in UMStory.pdf). SchurConj70.html %-%-% SchurConj70.pdf

3. 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. Variables separated Equations, contrasts the contributions of the simple group classification and of the Branch Cycle Lemma (for figuring the defining field of an algebraic relation) and the other problems these techniques influenced through the monodromy method.

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. Yet, there are only finitely many exceptional degrees, and we understand them.
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

4. On the Diophantine equation f(x) -y = 0: Acta Arith. XIX (1971), 79–87. On the interface between Davenport's (comparing value sets of polynomials on residue classes) and Schinzel's (on factorizations of f(x)-g(y)) problems, and related to the discovery of their near equivalence when f is indecomposable. Though without the depth or breadth of dav-red.pdf as related in UMStory.pdf (describing the many consequences of it in the work of others), it has one big virtue: Without any restriction on indecomposability it treats the case of odd prime-power degrees for polynomials over Q. This was a rare case of circumventing assuming a cover has a primitive monodromy group. ActArith71f-x.pdf

5. On a Theorem of MacCluer, XXV Acta Arith.~(1974), 121–126. The Cebotarev density theorem applied to a cover over a finite field is often applied to translate diophantine statements into statements about the monodromy group of the cover. Usually it is somewhat clumsy and you can't conclude the monodromy statement implies the original diophantine condition except off the ramification locus. Such a Monodromy Converse is very valuable. MacCluer's thesis in 1967 answered the main question of a Davenport-Lewis 1966 paper by showing exceptional, tamely ramified, polynomial covers satisfy the Monodromy Converse. This paper generalizes this to drop the tame and dimension 1 conditions. Eventually, a far more general condition called p(ossibly)r(educible)-exceptional was shown to satisfy the monodromy converse: Cor. 3.6 of exceptTower0910-3331v1.pdf. As a corollary this applied also to the generalization of Davenport pairs. MacTheom74.pdf

6. Arithmetical properties of function fields (II): The generalized Schur problem: Acta Arith. XXV (1974), 225–258. Using the analog of Schur's conjecture over a finite field, this paper introduced a new approach to understanding wildly ramified covers of the projective line. The outstanding point about the monodromy version of the Schur property is the significance of distinguishing between the arithmetic and geometric monodromy of a cover. The main topic of § 1 is the introduction of ramification data for wild ramification, extending the notion of higher ramification groups for not necessarily Galois covers. While not so general (and with limited use of families) as the treatment in fr-mez.pdf, it has many helpful examples. Again, with many examples, the § 2 is a forerunner of GCMTAMS78. genSchur74.pdf

7. with G. Sacerdote, Solving diophantine problems over all residue class fields of a number field ..., Annals Math. 104 (1976), 203–233. Picked up (with Ken Ribet's help) from JStor http://links.jstor.org/sici?sici=0003-486X%28197609%292%3A104%3A2%3C203%3ASDPOAR%3E2.0.CO%3B2-P. Introduces the Galois Stratification procedure in its Original, geometric form. Corresponds roughly to the non-geometric approach of Chap. 25 of the Fried Jarden book (1986 edition; Chap. 30 in 2005 edition). Chap. 26 (resp. Chap. 31) includes the start of zeta function applications developed in an untexed preprint, "L-series on a Galois Stratification," I spoke on in Spring, 1979 Lectures at Yale. annals76.html %-%-% annals76.pdf

8. LSeriesGalSt86.pdf

9. with R. Lidl, On Dickson Polynomials and R&eacut;dei Functions, Contributions to General Algebra 5, Proceedings of the Salzburg Conference, Mai 29 - June 1,1986 Verlag Holder-Plchler-Tempsky, Wien 1987 - Verlag B. G. Teubner, Stutgart, 139–149. Dickson-Redei87.pdf

10. with R. Guralnick and J. Saxl, Schur Covers and Carlitz's Conjecture, Israel J.; Thompson Volume 82 (1993), 157–225: sch-carlitz.html %-%-% sch-carlitz.pdf

11. with S. Cohen, The Carlitz-Lenstra-Wan conjecture on Expectional Polynomials: An Elementary Version: Finite Fields and their applications, Carlitz volume 1 (1995), 372–375. If you want to be able to algebraically scramble data embedded as an element in an arbitrarily large finite field while fixing the scrambling function, then you must use an exceptional rational function as scrambler. Finding exceptional polynomials (they fix the point at ∞) is a piece of that, and [FGS] comes close to it. The much weaker Lentra-Wan Statement – proved here – says exceptional polynomials Fq have degrees prime to q - 1. The html file explains just how weak is that statement. carlitz-quick.html %-%-% carlitz-quick.pdf

12. Global construction of general exceptional covers, with motivation for applications to coding, G.L. Mullen an P.J. Shiue, Finite Fields: Theory, applications and algorithms, Cont. Math. 168 (1994), 69–100. globConstExcCov.html %-%-% globConstExcCov.pdf

13. Applications of Curves over finite fields, in Curves over Finite Fields Cont. Math., proceedings of AMS-NSF Summer Conf. 1997, Editor M. Fried, Seattle 245 (1999), ix–xxxiii: This is an exposition on the themes in the papers presented at the conference. Starting from the role of Deligne's proof of the Weil Conjectures and the classification of finite simple groups, the sections divide the conference papers into practical tools. As was the last conference attended by Bernie Dwork, the final section includes comments on his work that complement an article of Katz and Tate.
1. Beyond Weil bounds; curves with many rational points: The moduli space approach; The Drinfeld module approach when q is not a square; More on Explicit use of Drinfeld modules; One curve with many points and fiber products; Approach from classical curves.
2. Monodromy groups of characteristic p covers: What to expect of monodromy groups from genus 0 covers; Abhyankar's approach; Reflection on classical invariant theory; Reduction mod p and field of moduli of covers; Refined abelian covers; Good reduction of covers; Explicit computation of monodromy grops over finite fields.
3. Zeta Functions and Trace Formulas: Unit root L-functions; Zeta functions of complete intersections; Properties of a modular curve quotient; Appearance of rank 1 representations in L-functions; Eigenvalues of a Laplacian; Average value of Zeta-functions and elliptic surfaces
4. A Dedication to the Work of Bernie Dwork: Michael Rosen: Dwork's relation to his students; Pierre Dèbes: Dwork's role in G-functions; Alan Adolphson: Dwork's final Conjecture.
curvesFFields.pdf

14. with W. Aitken and L. Holt, Davenport Pairs over finite fields, PJM 216, No. 1 (2004) 1–38. davpairs07-22-04PJ.html %-%-% davpairs07-22-04PJ.pdf

15. 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

16. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433, arXiv:0910.3331v1 [math.NT]; short list of corrections at exceptTower0910-3331v1-cor.html. A cover of normal varieties is exceptional over a finite field if the map on points over infinitely many extensions of the field is one-one. A cover over a number field is exceptional if it is exceptional over infinitely many residue class fields. The first result: The category of exceptional covers of a normal variety Z over a finite field, Fq, has fiber products, and therefore a natural Galois group (with permutation representation) limit. This has many applications to considering Poincare series attached to diophantine questions. The paper follows three lines:

• The historical role of the Galois Theoretic property of exceptionality, first considered by Davenport and Lewis.
• How the tower structure on the category of exceptional covers of a pair (Z,Fq) allows forming subtowers that separate known results from unknown territory.
• The use of Serre's OIT, especially the GL2 case, to consider cryptology periods and functional composition aspects of exceptionality.
exceptTower0910-3331v1.html %-%-% exceptTower0910-3331v1.pdf

17. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433. This is the original journal article, with pencil corrections that have been incorporated in arXiv:0910.3331v1 [math.NT]. exceptTowYFFTA_519.pdf

18. 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

19. Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the Finite Simple Group Classification: arXiv:1012.5297v5 [math.NT] (DOI 10.1007/s11425-011-4324-4). Science China Mathematics, vol. 55, January 2012, 1–72 (Journal article is here UMStory022011-70-120001.pdf.) Davenport's problem is to figure out the nature of two polynomials over a number field having the same ranges on almost all residue class fields of the number field. Solving this problem initiated the monodromy method. That 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 attend to these general questions:
1. What allows us to produce branch cycles from covers, 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 using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?
Each phrase addresses formulating problems based on equations. We seem to need explicit algebraic equations. Yet why, and how much do we lose/gain in using more easily manipulated surrogates for them? To get this straight we consider more than previously the surprising parallel problem of Schinzel on reducible variables separated equations, but this time without the indecomposability hypothesis allowed solving them both. We also enhance the original form of the BCL – especially for non-Galois covers –. This got lost in a reformulation for the Inverse Galois Problem in inv_gal.pdf. UMStoryarXiv1012-5297v5.html %-%-% UMStoryarXiv1012-5297v5.pdf

20. Schur's Conjecture and Exceptional Covers, in Handbook of Finite Fields, (2012), 243–255, editors G. Mullen and D.~Panario. Serre's O(pen) I(mage) T(heorem) gets much credit as a sophisticated piece of mathematics, but its practical uses are little known. This paper puts in one compact place a system of observations over the years that rational functions in one variable are the heart-and-soul of the problems that massively arise in the work of many mathematicians. We specifically clarify the connection between the OIT and the complete description of the following covers, revealed through the monodromy method:
• rational function exceptional covers over number fields; and
• rational functions over a number field or finite field that are indecomposable, but decompose over an algebraic closure.
HFFsec97.pdf