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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)
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 use moduli spaces for covers of the Riemann sphere to solve regular embedding problems, with prescribed extendability of orderings, over PRC fields. As a corollary we show that the elementary theory of Q tr is decidable. Since the ring of integers of Q tr is undecidable, this gives a natural undecidable ring whose quotient field is decidable.
– 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  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)