Testing local-global divisibility at a stable set

TL;DR

This paper demonstrates that local-global divisibility in algebraic groups over number fields can be tested on small-density prime sets, introduces a new cohomological obstruction description, and provides new examples of stable sets.

Contribution

It introduces a method to test local-global divisibility on small-density prime sets and offers a new cohomological perspective and examples of stable sets.

Findings

Testing on small-density prime sets is effective.

A new cohomological obstruction is described.

New examples of stable sets are provided.

Abstract

We show that the local-global divisibility in commutative algebraic groups defined over number fields can be tested on sets of primes of arbitrary small density, i.e. stable and persistent sets. We also give a new description of the cohomological group giving an obstruction to the problem. In addition, we show new examples of stable sets.