On Semidirect Products and the Arithmetic Lifting Property

Abstract

Let G be a finite group and let K be a hilbertian field. Many finite groups have been shown to satisfy the arithmetic lifting property over K, that is, every G-Galois extension of K arises as a specialization of a geometric branched covering of the projective line defined over K. The paper explores the situation when a semidirect product of two groups has this property. In particular, it shows that if H is a group that satisfies the arithmetic lifting property over K and A is a finite cyclic group then G l A M H also satisfies the arithmetic lifting property assuming the orders of H and A are relatively prime and the characteristic of K does not divide the order of A. In this case, an arithmetic lifting for any A 1 H-Galois extension of K is explicitly constructed and the existence of the arithmetic lifting for any G-Galois extension is deduced. It is also shown that if A is any abelian group, and H is the group with the arithmetic lifting property then A 1 H satisfies the property as well, with some assumptions on the ground field K. In the construction properties of Hilbert sets in hilbertian fields and spectral sequences in e! tale cohomology are used.

Lifting wreath product extensions

“…As a consequence of our main result here, we show that if G = A is abelian and the orders of G and H are relatively prime, then any semidirect product G H has the arithmetic lifting property. This also strengthens similar results on semidirect products in [Bl2], where G = A must be cyclic.…”

“…The main result of this note is to show that if G has a generic extension over K and H satisfies the arithmetic lifting property over K, then the wreath product G H also satisfies the arithmetic lifting property. In [Bl2] a similar result is obtained but only for G = A an abelian group. As a consequence of our main result here, we show that if G = A is abelian and the orders of G and H are relatively prime, then any semidirect product G H has the arithmetic lifting property.…”

“…The purpose of this note is to strengthen the main result of [Bl2], at the same time providing a much shorter proof which is more geometric and intuitive. The result concerns the arithmetic lifting property, which in some sense can be viewed as an "inverse" of the regular use of the Hilbert Irreducibility Theorem.…”

“…One should note however, that the proof in [Bl2] is constructive, and the arithmetic liftings of Galois extensions are explicitly described. The proof in this note is existential.…”

“…Many groups have been shown to have the arithmetic lifting property (see [Bl1]). In [Bl2] we consider certain semidirect products. The purpose of this note is to remove some hypotheses on the building blocks of these semidirect products thus strengthening the main result of [Bl2].…”