Abstract:We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P (x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over F p (x) for almost all primes p. §0. INTRODUCTIONMany attempts have been made to realize finite groups as Galois groups of extensions of Q(x) that are regular over Q (see the end of this introduction for definitions). We call this the "regular inverse Galois problem." We show that to each finite group G with trivial center and integer r ≥ 3 there is canonically associated an algebraic variety, H in r (G), defined over Q (usually reducible) satisfying the following. It is convenient to introduce the following terminology: A group G is called regular over a field k if G is isomorphic to the Galois group of an extension of k(x) that is regular over k. The above has the following immediate corollary for P(seudo)A(lgebraically)C(losed) fields P of characteristic 0: Every finite group is regular over P (Theorem 2). Another corollary (which can be viewed as a special case of the previous one) is that every finite group is regular over the finite prime field F p for almost all primes p (Corollary 2).In §6 we derive an addendum to our main result that is crucial for the preprint [FrVo]. In that paper we prove a long-standing conjecture on Hilbertian PAC-fields P (in the case char(P ) = 0): Every finite embedding problem over P is solvable. For countable P this, combined with a result of Iwasawa, implies that the absolute Galois group of P is ω-free. That is, G(P /P ) is a free profinite group of countably infinite rank, denotedF ω . By a result of [FrJ, 2], every countable Hilbertian field k of characteristic 0 has a Galois extension P with the following properties: P is Hilbertian and PAC, and G(P/k) ∼ = ∞ n=2 S n (where S n is the symmetric group of degree n). From the above, G(k/P ) = G(P /P ) ∼ =Fω, and we get the exact sequenceModuli spaces for branched covers of P 1 were already considered by Hurwitz [Hur] in the special case of simple branching (where the Galois group is S n ). Fulton [Fu] showed -still in the case of simple branching -that the analytic moduli spaces studied by Hurwitz are the sets of complex points of certain schemes. Fried [Fr,1] studied more generally moduli spaces for covers of P 1 with an arbitrary given monodromy group G ⊂ S n and with a fixed number of branch points. The new moduli spaces H in r (G) studied in the present paper are coverings of those previous ones, parametrizing equivalence classes of pairs (χ, h) where χ is a Galois cover of P 1 with r branch points and h is an isomorphism between G and the automorphism group of the cover χ. The extra data of the isomorphism h associated to the points of H in r (G) ensures that a Qrational point corresponds to a cover of P 1 that can be defined over Q such that also all ...

This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realisation of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.

We study genus 2 function fields with elliptic subfields of degree 2. The locus L2 of these fields is a 2-dimensional subvariety of the moduli space M2 of genus 2 fields. An equation for L2 is already in the work of Clebsch and Bolza. We use a birational parameterization of L2 by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parameterized by the j-invariant of these subfields.

We continue our study of genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n. These curves C form an irreducible 2-dimensional subvariety L n of the moduli space M 2 of genus 2 curves. Here we study the case n = 5. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general n.We compute a normal form for the curves in the locus L 5 and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of L 5 as subvarieties of M 2 and classify all curves in these loci which have extra automorphisms.

No abstract