Diophantine Questions over all Residue Classes of a Number Field and where the topic is today

Gist of the paper: It has two straightforward points. First: That expanding diophantine equations to elementary statements, while it includes important number theory topics, it isn't sufficiently general to have important conclusions. To wit: Even when over the integers and asking about reductions moduli infinitely many primes, you couldn't eliminate quantifiers, so as to resolve the questions.

Second: When questions about primes for which certain properties hold, we should be able to measure those primes through properties of L-series for which natural groups are attached.

So, the idea of the paper has a clear statement. If you generalize elementary statements to Galois stratifications, you have happy endings to both concerns. The method looks technical because it deals with all possible elementary statements. Statements that had variables quantified by both ∃(exists) and ∀(for all) had previously only ad hoc approaches.

Logicians often don't care about computing anything in such a general context. To them this procedure doesn't matter if it is no better than a non recursive quantifier elimination procedure. That is, they thought this procedure couldn't really compute anything. That is not correct. By involving group theory – the most successul non-linear equation attacker – it revealed the nature of many problems, as I had shown  many cases.

What Price to pay?: Nothing brought more pain – in an otherwise merely unhappy career – than the reception to this paper. One (I think of a Stony Brook colleague, for example)  would say it was just a technical result; another would say schemes was the wrong method (algebraic number theorists with no feeling for geometry; Grothendieck haters – did you know they abounded then?); and still a 3rd [group] would say you can do it in an easier way (usually someone thinking of the pure existential case).

The first direct positive statement about this successful way to get the power of group theory into general diophantine equations occurred many years later at Ihara's 60th Birthday Conference in Tokyo. It came from Francois Loeser.  (He and I were two of the six invited foreign speakers, but I hadn't met him until the event of the next paragraph.)

He had spoken on Motivic Measure (a la Kontsevic), and never said anything about this particular paper. Still, though I had not touched this topic for the previous 20 years, I recognized its connection to my topics, and wondered if he knew also. As my wife and I came down the stairs to the street car station after the morning session of his talk, he looked our way and put up a book in front of him. That was Fried-Jarden "Field Arithmetic" opened to Chap. 25 [FJ]. Said he, "How come no one knows anything about this?" It is common for mathematicians to praise those who are already honored, even on topics they don't know much about. This was essentially the opposite, as I now explain.

Passing from Galois Stratifications to Chow Motives: For some purposes (the category of) Galois Stratifications isn't the right place to stop. Yes, they do give  zeta functions to statements with partially quantified variables, through quantifier elimination.  Yes, the coefficients of the Poincare series are then equivalence classes of Galois stratifications, to be specialized at the various primes. Still, there are two objections.
  1. You don't clearly see how to produce canonical Galois Stratification coefficients – choices are made.
  2. You don't automatically involve the great objects of arithmetic geometry – non-singular projective varieties (satisfying the Weil conjectures).
As I say in in the appendix to [Ha], I showed Jan Denef (and Diane Meuser) [LGS], with its generalization to p-adic topics, in the early 80s (at a Bowdoin College conference). Especially how to formulate the p-adic Chebotarev generalization appropriate for that. Near the time we met in Japan, Loeser and Denef had written a version of [DL]. This used Galois stratifications but produced the Poincare series coefficients as Covering space versions of Chow Motives.

That is, as equivalence classes of virtual algebraic projective nonsingular varieties over Q, with groups and attached projectors that still encoded the Galois Stratification data from group theory. You specialize the coefficients at a particular prime – as did Galois stratifications – by applying the Frobenius at that prime to the coefficients.

Interpret applying the Frobenius to mean you are applying it to something in a category from ètale cohomology groups of varieties. This has a(t least one) virtue and also a weakness by comparison with [LGS].
The subject today and bringing Chow Motive questions down-to-earth: Nicaise [N] does the relative version of [DL] using Voevodsky's relative Chow motives. While there are Galois covers in [DL] and [N], Chow motives are usually not given by covering data. This relation between using group theory and covering data to motivic questions needs some examples that reflect on all Chow motives, yet have simple properties understood from covers.

[EC] gives an example of such, and is intended for readers to grasp the practical value of Chow Motives. It uses exceptional covers.  For example [EC, §8.2.2] asks what are the implications if the Poincare series attached to a projective curve has a certain type of Weil relation with the Poincare series of the projective line. The converse of a known fact would be that there is a chain of exceptional correspondences from the curve to the projective line.

Exceptional covers (over finite fields) have a huge literature. This is not a typical question for the people who work on them, but rather its converse. The gist being that exceptional covers, a serious aspect of cryptology  [EC, §4.3], are related to Chow motives in a big way.

REFERENCES

[DL] J. Denef, F. Loeser, Definable sets, motives and p-adic integrals, J. Amer. Math. Soc. 14 (2001) 424–69.
[EC] M.D. Fried, The place of exceptional covers among all diophantine relations, Finite Fields and their Apps. 11 (2005), 367–433.
[FJ] M.D. Fried, M. Jarden, Field Arithmetic, Ergebnisse der Mathematik III, vol. 11, Springer, Heidelberg, 1986. New edition, 2004, ISBN 3-540-22811-x.
[Ha] T.C. Hales, What is Motivic Measure?, BAMS 42, 119–135. Hales kindly put my appendix – Historical Remarks on Galois Stratification – in after he published the paper.
[LGS] M. Fried, L-series on a Galois Stratification, preprint 1979 (it was never texed).
[N] J. Nicaise, Relative motives and the theory of pseudo-finite fields, International Mathematics Research Papers 2007, 2007:rpm001-69.