## Algebra & Number Theory

### Étale homotopy equivalence of rational points on algebraic varieties

Ambrus Pál

#### Abstract

It is possible to talk about the étale homotopy equivalence of rational points on algebraic varieties by using a relative version of the étale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are étale-Brauer equivalent. We also show that over the real field rational points on projective varieties are étale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Châtelet surfaces.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 4 (2015), 815-873.

Dates
Revised: 11 February 2015
Accepted: 11 March 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842337

Digital Object Identifier
doi:10.2140/ant.2015.9.815

Mathematical Reviews number (MathSciNet)
MR3352821

Zentralblatt MATH identifier
1368.14034

Subjects
Secondary: 14G05: Rational points

Keywords
étale homotopy rational points

#### Citation

Pál, Ambrus. Étale homotopy equivalence of rational points on algebraic varieties. Algebra Number Theory 9 (2015), no. 4, 815--873. doi:10.2140/ant.2015.9.815. https://projecteuclid.org/euclid.ant/1510842337

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