Helmut Völklein

The inverse Galois problem and rational points on moduli spaces.

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.

The embedding problem over a Hilbertian PAC-field

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

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field.

Elliptic Subfields and Automorphisms of Genus 2 Function Fields

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.

A GAP Package for Braid Orbit Computation and Applications

Let G be a finite group. By Riemanns Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus, many problems on algebraic curves require the computation of braid orbits. In this paper, we describe an implementation of this computation. We discuss several applications, including the classification of irreducible familiesof indecomposable rational functions with exceptional monodromy group.

Invariants of Binary Forms

Basic invariants of binary forms over ℂ up to degree 6 (and lower degrees) were constructed by Clebsch and Bolza in the 19-th century using complicated symbolic calculations. Igusa extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. In this paper a simple proof is supplied that works in characteristic p > 5 and uses some concepts of invariant theory developed by Hilbert (in characteristic 0) and Mumford, Haboush et al. in positive characteristic. Further the analogue for pairs of binary cubics is also treated.

On the geometry of the adjoint representation of a Chevalley group

Abstract We prove that the adjoint module of a Chevalley group (not of type C l ) has a presentation by long root subalgebras, subject to certain relations determined by the minimal parabolic subgroups.

The Braid Group and Linear Rigidity

We introduce the BC-operation (short for braid-companion) on tuples of matrices. It corresponds to Katzs middle convolution operation on local systems, and generalizes it to fields K of arbitrary characteristic. It is based on the Burau–Gassner representation of the braid group. We expect many applications, in Galois theory (for finite K) as well as in Katzs original set-up of local systems and linear differential equations (where K = ℂ).

The monodromy group of a function on a general curve

LetCg be a general curve of genusg≥4. Guralnick and others proved that the monodromy group of a coverCg→ℙ1 of degreen is eitherSn orAn. We show thatAn occurs forn≥2g+1. The corresponding result forSn is classical.

Moduli spaces for covers of the Riemann sphere

Moduli spaces for covers of the Riemann sphere have been constructed in a joint work with M. Fried [FV1]. They were used to realize groups as Galois groups [FV1], [Vö1], and to determine the absolute Galois group of large fields [FV2]. Here we simplify and extend the construction of these moduli spaces.