Uniform bounds and ultraproducts of cycles

TL;DR

This paper investigates whether algebraic cycles that are algebraic over almost all finite fields are also algebraic over the rationals, using ultraproducts and nonstandard techniques to reformulate the problem in terms of uniform bounds.

Contribution

It introduces a new approach using ultraproducts and nonstandard analysis to study algebraic cycles across different fields, providing a reformulation of the problem.

Findings

Reformulation of the algebraicity question via uniform bounds

Application of ultraproducts and nonstandard techniques in algebraic geometry

Potential framework for proving algebraicity over the rationals

Abstract

This paper is about the question whether a cycle in the l-adic cohomology of a smooth projective variety over the rational numbers, which is algebraic over almost all finite fields, is also algebraic over the rationals. We use ultraproducts respectively nonstandard techniques in the sense of A. Robinson, which the authors applied systematically to algebraic geometry. We give a reformulation of the question in form of uniform bounds for the complexity of algebraic cycles over finite fields.