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Handbook of magma functions

This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt construction for p-groups, p != 2, as well as Hilbert's irreducibility theorem and the large sieve inequality, are presented. The second half is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T). While proofs are not carried out in full detail, the book contains a number of examples, exercises, and open problems.

Topics in Galois Theory

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on varieties. As applications, we give a positive answer to a long standing conjecture which originates in an unpublished note of Roquette, and asserts that the absolute Galois group of a countable, hilbertian, PAC field is profinite free; see [F-J, Problem 24.41]. Secondly, we give new evidence for the conjecture of Shafarevich which asserts that the absolute Galois group of Qab is profinite free. Finally, one of the most interesting applications of the theory we develop here is the insight in the Galois structure of the totally EG-adic numbers.

Embedding problems over large fields

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 Fp(x) for almost all primes p.

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The inverse Galois problem and rational points on moduli spaces

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π1(Σ)→Q. If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

Finite covers of random 3-manifolds

Part 1. The Basic Rigidity Criteria: 1. Hilbert's irreducibility theorem 2. Finite Galois extensions of C (x) 3. Descent of base field and the rigidity criterion 4. Covering spaces and the fundamental group 5. Riemann surfaces and their functional fields 6. The analytic version of Riemann's existence theorem Part II. Further Directions: 7. The descent from C to k 8. Embedding problems: braiding action and weak rigidity Moduli spaces for covers of the Riemann sphere Patching over complete valued fields.

Groups as Galois groups : an introduction

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 M 2 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.

Elliptic Subfields and Automorphisms of Genus 2 Function Fields

We show that the absolute Galois group of a countable Hilbertian P(seudo)- A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., � ∞=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RG-Hilbertian property

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Helmut Völklein

Papers

Groups as Galois groups : an introduction

The inverse Galois problem and rational points on moduli spaces

Elliptic Subfields and Automorphisms of Genus 2 Function Fields

The embedding problem over a Hilbertian PAC-field

On linearly rigid tuples