Deformations of dihedral 2-group extensions of fields

Abstract

Abstract. Given a G-Galois extension of number fields L/K we ask whether it is a specialization of a regular G-Galois cover of P 1 K . This is the "inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral 2-groups under certain assumptions on the base field k. We also show that dihedral groups of order 8 and 16 have generic extensions over any base field k with characteristic different from 2.

Galois covers of \mathbb{P}^1 over \mathbb{Q} with prescribed local or global behavior by specialization

“…It suffices to prove that, for every prime number 5 (resp. p = 5), there exists t E ~ such that the polynomial has non-zero discriminant and splitting field over Q unramified at p (resp.…”

“…[3]) and, more generally, if there exists a generic extension for G over K (cf. [5]). G an arbitrary finite group and K a PAC field (cf.…”

“…Up to rational squares, the discriminant D( f ) of f (X) is bn n. Since 5 (mod 8) and D( f ) is a non-zero square integer, it must be v2(an). This allows us to assume that v2(b) n. Note that v2(a) > 2. Let us first consider the case b even.…”

On a variant of the Beckmann--Black problem

“…As shown by Moret-Bailly [MB01], extending a former result in characteristic zero of Colliot-Thélène [CT00], the answer is Yes for arbitrary G if k is ample. If k = Q, the answer is known to be Yes for only a few groups G, including abelian groups (Beckmann [Bec94]), some dihedral groups (Black [Bla98,Bla99]), symmetric groups, and alternating groups (Mestre [Mes90], Klüners-Malle [KM01]), and no counter-example is known. While this gives support for the geometric approach, it should also be recalled that, if the answer to the Beckmann-Black problem is affirmative for every finite group and every field, then all fields fulfill the regular inverse Galois problem (Dèbes [Dèb99]).…”