Helmut Völklein

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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(More)
Let G be a finite group, and g ≥ 2. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g = 3 (including equations for the corresponding curves), and for g ≤ 10 we classify those(More)
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(More)
We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) eld of characteristic 0 is a free proonite group of countably innnite rank (Theorem A). As a consequence, G(Q =Q) is the extension of groups with a fairly simple structure (e.g., Q 1 n=2 S n) by a countably free group. In addition, we characterize those PAC elds(More)
Let K be a finitely generated field with separable closure K s and absolute Galois group G K. By a curve C/K we always understand a smooth geometrically irreducible projective curve. Let F (C) be its function field and let Π(C) be the Galois group of the maximal unramified extension of F (C). We have the exact sequence where Π g (C) is the geometric(More)
Let G be a finite group. By Riemann's 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(More)
– We determine the absolute Galois group G tr Q of the field Q tr of totally real algebraic numbers. This group is the free profinite product of groups of order 2 over the Cantor set. Le groupe de Galois absolu des nombres algébriques totalement réels Résumé – On détermine le groupe de Galois absolu G tr Q du corps Q tr des nombres algébriques totalement(More)
In year 1978, Fried-Jarden [1] showed that a conjecture of Ax, which asserts " there is no proper Galois extension of Q that is PAC, " has counter examples; they proved that every countable Hilbertian field has a Galois extension L which is PAC and Hilbertian. Moreover, Gal(L/K) = ∞ n=2 S n where S n is the symmetric group of order n. Then in 1992(More)
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