Hilbert Irreducibility above algberaic groups

TL;DR

This paper extends Hilbert's Irreducibility Theorem to algebraic groups, providing new results on lifting points in cyclic subgroups to ramified covers and establishing a related Bertini Theorem.

Contribution

It introduces novel versions of Hilbert's Irreducibility Theorem and Bertini Theorem tailored for algebraic groups and ramified covers.

Findings

New versions of Hilbert's Irreducibility Theorem for algebraic groups

A Bertini Theorem adapted to this context

Results on lifting points in cyclic subgroups

Abstract

The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.