Hilbert irreducibility above algebraic groups

Abstract

This article concerns Hilbert irreducibility for covers of algebraic groups, with results which appear to be difficult to treat by existing techniques. The present method works by first studying irreducibility above “torsion” specializations (e.g., over cyclotomic extensions) and then descending the field (by Chebotarev theorem). Among the results, we offer an irreducibility theorem for the fibers, above a cyclic dense subgroup, of a cover of ${\mathbb{G}}_m^n$ (Theorem 1) and of a power $E^n$ of an elliptic curve without CM (Theorem 2); this had not been treated before for $n>1$. As a further application, in the function field context, we obtain a kind of Bertini's theorem for algebraic subgroups of ${\mathbb{G}}_m^n$ in place of linear spaces (Theorem 3). Along the way we shall prove other results, as a general lifting theorem above tori (Theorem 3.1)