On Semidirect Products and the Arithmetic Lifting Property

Let G be a finite group and let K be a hilbertian field. Manyfinite groups have been shown to satisfy the arithmetic liftingproperty over K, that is, every G-Galois extension of K arisesas a specialization of a geometric branched covering of theprojective line defined over K. The paper explores the situationwhen a semidirect product of two groups has this property. Inparticular, it shows that if H is a group that satisfies thearithmetic lifting property over K and A is a finite cyclicgroup then G = A H also satisfies the arithmetic lifting propertyassuming the orders of H and A are relatively prime and thecharacteristic of K does not divide the order of A. In thiscase, an arithmetic lifting for any AH-Galois extension of Kis explicitly constructed and the existence of the arithmeticlifting for any G-Galois extension is deduced. It is also shownthat if A is any abelian group, and H is the group with thearithmetic lifting property then AH satisfies the property aswell, with some assumptions on the ground field K. In the constructionproperties of Hilbert sets in hilbertian fields and spectralsequences in étale cohomology are used.