Weak approximation versus the Hasse principle for subvarieties of abelian varieties

2023/09/08 by Brendan Creutz, Creutz, Brendan · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2309.04111

openalex publication_date 2023/09/08 · openalex created_date 2023/09/12 · openalex updated_date 2026/07/28

Abstract

For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational points should be dense in the set of (modified) adelic points surviving finite descent. We show that this follows from the a priori weaker assumption that finite descent is the only obstruction to the existence of rational points. We also prove a similar statement for the obstruction coming from the Mordell-Weil sieve.

Cited by

• Adelic Mordell-Lang and the Brauer-Manin obstruction

Related

• Adelic Mordell-Lang and the Brauer-Manin obstruction
• Approximation forte sur un produit de variétés abéliennes épointé en des points de torsion
• A local-global principle for weak approximation on varieties over function fields
• Non-invariance of weak approximation properties under extension of the ground field
• A Note On a Theorem of Heath-Brown and Skorobogatov