Diophantine statements over Residue fields: Galois stratification and uniformity

TL;DR

This paper revisits Galois stratification over finite fields using arithmetic homotopy and Chow motives, aiming to improve the efficiency and canonical nature of diophantine statement analysis.

Contribution

It introduces a homotopy-based approach to Galois stratification, connecting it with Chow motives and Frobenius fields for more canonical and efficient analysis.

Findings

Galois stratification over one finite field is as efficient as possible, requiring exponential time in statement length.

The paper links Poincare series to Chow motives and Frobenius fields, enabling canonical representations.

It expands the Galois stratification framework beyond finite fields, incorporating arithmetic homotopy and motives.

Abstract

Using Felgner's problem I revisit a key issue in using the "Galois Stratification Procedure" that first appeared in [FrS76]. The emphasis here is on using arithmetic homotopy to make the production of Poincare; series attached to general diophantine statements canonical. According to work in progress of Michael Benedikt and E. Hrushovski, Galois stratification - over one finite field - is as efficient as is possible: on a statement of length n, it requires time bounded by a stack of exponentials of length linear in n. This doesn't take advantage of problems prepped for using homotopy aspects, Chow Motives, efficiently as in the main example which comes from my paper on the generalization of exceptional covers. That example [FrJ, Chap. 30], simplifies aspects of the original procedure. It combines this with the later theory of Frobenius fields to produce objects over Q whose reductions mod primes give the stratification procedure at the prime. The paper separates two different uses of the Chebotarev non-regular analog. 1. Field crossing to interpret Poincare series coefficients directly from traces on Chow motives (providing valuable statements on variation with the prime p; versus 2. Chebotarev using Lang-Weil to approximate the number of points on an appropriate variety for the Galois stratification procedure. We consider variables taking values in the algebraic closure of Z/p but fixed by respective powers of the Frobenius: we call these Frobenius vectors. For this there is a twisted Chebotarev version stemming from a conjecture of Deligne, and outlined in a preprint of Hrushovski. This paper expands on the work of D. Wan, J. Denef and F. Loeser, J. Nicaise, I. Tomasic and E. Hrushovski, all relevant to taking the Galois stratification procedure beyond the original finite field framework.