The classical Hilbert Irreducibility Theorem can be viewed as saying the following: if \(\pi: Y \to {\mathbb A}^n_k\) is a dominant rational map of finite degree, with \(Y\) irreducible, then there is a rational point \(x \in {\mathbb A}^n_k\) such that the fibre \(\pi^{-1}(x)\) is irreducible. Replacing \({\mathbb A}^n\) with an arbitrary irreducible variety \(X\) over the field \(k\), we obtain a more general notion of Hilbert Irreducibility Theorem. This article concerns the case where \(X\) is an algebraic group, and we would like to restrict \(x\) to lie in a specified Zariski-dense subgroup of \(X(k)\).
To state the principal results of this article, we need to define the condition (PB) on on the cover \(Y \to X\), as follows.
Definition. The cover \(Y \to X\) satisfies condition (PB) (pullback) if, for any integer \(m>0\), the pullback \([m]^* Y := X \times_{[m],\pi} Y\) is irreducible.
The two main theorems cover different types of algebraic groups \(X\).
Theorem 1. For \(i = 1, \ldots, h\), let \(\pi_i : Y_i \to X := {\mathbb G}_{\roman m}^r \times {\mathbb G}_{\roman a}\) be a cover satisfying (PB). Then, if \(\Omega\) is a cyclic Zariski-dense subgroup of \(X(k)\), there exists a coset \(C\) of finite index in \(\Omega\) such that for all \(x \in C\) and for all \(i = 1, \ldots, h\) the fibre \(\pi_i^{-1}(x)\) is irreducible over \(k\).
In the second case, \(X\) is a power of an elliptic curve \(E\) without complex multiplication.
Theorem 2. For \(i = 1, \ldots, h\), let \(\pi_i : Y_i \to E^n\) be a cover satisfying (PB). Then, if \(\Omega\) is a Zariski-dense cyclic subgroup of \(E^n(k)\), there exists a coset \(C\) of finite index in \(\Omega\) such that for all \(x \in C\) and for all \(i = 1, \ldots, h\) the fibre \(\pi_i^{-1}(x)\) is irreducible over \(k\).
Another theorem, an application of the same method, is the following “toric analogue of the Bertini theorem”.
Theorem 3. Let \(\pi: Y \to {\mathbb G}_{\roman m}^n\) be a cover defined over \(\kappa\), satisfying (PB). Then there is a finite union \({\mathcal E}\) of proper connected algebraic subgroups of \({\mathbb G}_{\roman m}^n\) such that if a connected algebraic subgroup \(G\) is not contained in \({\mathcal E}\), then \(\pi^{-1}(\theta G)\) is irreducible (over \(\kappa\)) for every torsion point \(\theta\).
This paper has an extensive introduction describing the motivation and background to these results, their connection to previous results in this area, and several further applications.