×

Study and computation of a Hurwitz space and totally real \(\text{PSL}_2(\mathbb F_8)\)-extensions of \(\mathbb Q\). (English) Zbl 1096.12003

In this beautiful paper, the author gives an explicit computation of the Hurwitz space parametrizing the family of \({\mathbb P}_{\mathbb C}^1\)-covers of degree 9 with monodromy group \(\mathrm{PSL}(2,\mathbb{F}_8)\) and with a certain branch point configuration that makes the total space an elliptic curve. The computations follow works by M. D. Fried, H. Völklein [The inverse Galois problem and rational points on modular spaces, Math. Ann. 290, 771–800 (1991; Zbl 0763.12004)], G. Malle, B. H. Matzat [Inverse Galois theory, Springer Monographs in Mathematics. Berlin: Springer (1999; Zbl 0940.12001)], M. Dettweiler [Plane curve complements and curves on Hurwitz spaces, J. Reine Angew. Math. 573, 19–43 (2004; Zbl 1074.14026)], S. Wewers [Construction of Hurwitz spaces. Thesis. Essen: Univ.-GHS Essen (1998; Zbl 0925.14002)]. By specialication, the author obtains totally real polynomials over \({\mathbb Q}\) with Galois group \(\mathrm{PSL}(2,\mathbb{F}_8)\).

MSC:

12Y05 Computational aspects of field theory and polynomials (MSC2010)
68W30 Symbolic computation and algebraic computation
14H30 Coverings of curves, fundamental group

Keywords:

monodromy group
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Couveignes, J.-M., Tools for the computation of families of coverings, (Völkein, H.; Harbater, D.; Müller, P.; Thompson, J. G., Aspects of Galois Theory. Aspects of Galois Theory, London Math. Soc. Lecture Note Ser., vol. 256 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 38-65, The following pages of this book were inadvertently numbered incorrectly: p. 21 should be p. 22, p. 22 should be p. 23 and p. 23 should be p. 21 · Zbl 1016.14012
[2] Couveignes, J.-M., Boundary of Hurwitz spaces and explicit patching, J. Symbolic Comput., 30, 739-759 (2000) · Zbl 0973.12004
[3] Dettweiler, M., Plane curve complements and curves on Hurwitz spaces, J. Reine Angew. Math., 573, 19-43 (2004) · Zbl 1074.14026
[4] Dèbes, P.; Fried, M. D., Nonrigid constructions in Galois theory, Pacific J. Math., 163, 1, 81-122 (1994) · Zbl 0788.12001
[5] Fried, M. D., Introduction to modular towers, (Abhyankar, S. S.; Feit, W.; Ihara, Y.; Völklein, H., Recent Developments in the Inverse Galois Problem. Recent Developments in the Inverse Galois Problem, Contemp. Math., vol. 186 (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 111-171 · Zbl 0957.11047
[6] Fried, M. D.; Völklein, H., The inverse Galois problem and rational points on moduli spaces, Math. Ann., 290, 771-800 (1991) · Zbl 0763.12004
[7] Hallouin, E., available at
[8] Hallouin, E.; Riboulet-Deyris, E., Computation of some moduli spaces of covers and explicit \(s_n\) and \(a_n\) regular \(Q(t)\)-extensions with totally real fibers, Pacific J. Math., 211, 1, 81-99 (2003) · Zbl 1056.14038
[9] Liu, Q., Algebraic Geometry and Arithmetic Curves, vol. 6, Oxf. Grad. Texts Math. (2002), Oxford Univ. Press: Oxford Univ. Press Oxford
[10] Malle, G.; Matzat, B. H., Inverse Galois Theory (1999), Springer: Springer Berlin · Zbl 0940.12001
[11] Silverman, J. H., The Arithmetic of Elliptic Curves, Grad. Texts Math., vol. 106 (1986), Springer: Springer Berlin · Zbl 0585.14026
[12] Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts Math., vol. 151 (1994), Springer: Springer Berlin · Zbl 0911.14015
[13] Völklein, H., Groups as Galois Groups, Cambridge Stud. Adv. Math., vol. 53 (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0868.12003
[14] S. Wewers, Construction of Hurwitz spaces, PhD thesis, Universität-Gesamthochschule, Essen, 1998; S. Wewers, Construction of Hurwitz spaces, PhD thesis, Universität-Gesamthochschule, Essen, 1998 · Zbl 0925.14002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.