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1. Starts with text from Ralph Cicerone (UCI Chancellor in '90s at UCI) 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, Mich. Math. Journal 17 (1970), 41–55. The monodromy method must look peculiar at first. You take a seemingly simple polynomial cover, replacing it by an apparantly complicated branch cycle description for a geometric cover. It works though by translating functional properties into group theory. Then, it figures what possible branch cycle descriptions could possibly go with such polynomial properties. As permutation manipulations are seriously more efficient than working with algebraic equations, even without a computer, it has solved old conjectures by divining the nature of polynomials that could possibly have such descriptions. Concluding the Schur conjecture, describing polynomials giving one-one mappings over infinitely many residue class fields of a number field, was by-far the easiest unsolved problem. Yet, it as an elementary primer to the method. Harder techniques, including going beyond polynomials appear in the solution of Davenport's problem (for a modern exposition UMStory.html), where the monodromy groups were no longer solvable. SchurConj.pdf

3. 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

4. with Moshe Jarden, Field Arithmetic, Springer Ergebnisse der Mathematik III, 11, Springer Verlag, Heidelberg, 1986, 457 pps; 2nd edition 2004, 780 pps. ISBN 3-540-22811-x. Fr-JTable-of-Contents.html %-%-% Fr-JTable-of-Contents.pdf

5. 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

6. 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

7. 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

8. 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

9. 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

10. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433, arXiv:0910.3331v1 [math.NT]. 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:

exceptTower0910-3331v1.html %-%-% exceptTower0910-3331v1.pdf %-%-% exceptTower0910-3331v1.cor

11. 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

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