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 = A [rtimes ] 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[wreath ]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[wreath ]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 étale cohomology are used.