Existence of rational points as a homotopy limit problem

Existence of Rational Points as a Homotopy Limit Problem

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of étale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck’s Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected...

Distribution of Rational Points: A Survey

Rational Points, R-equivalence and Étale Homotopy of Algebraic Varieties

We study a generalisation of the anabelian section conjecture of Grothendieck by substituting the arithmetic fundamental group with a relative version of the étale homotopy type. We show that the map associating homotopy fixed points to rational points factors though R-equivalence for projective varieties defied over fields of characteristic zero. We show that over p-adic fields rational points...

Deciding Existence of Rational Points on Curves: An Experiment

The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to the question whether there exists an algorithm that is able to perform this task for any given variety. This is probably too much to ask for: we know that Hilbert’s Tenth Problem, which asks the sam...

Existence of Rational Points on Smooth Projective Varieties

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of ...