Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups

König, Joachim

doi:10.1090/mcom/3116

Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups
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Math. Comp. 86 (2017), 1473-1498 Request permission

Abstract:

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb {Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over $\mathbb {Q}$. All these examples make use of families of covers of the projective line ramified over four or more points, and therefore use techniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g. by Malle (2000), Couveignes (1999), Granboulan (1996) and Hallouin (2009). Unlike previous examples, however, some of our computations show the existence of rational points on Hurwitz spaces that would not have been obvious from theoretical arguments.

References

A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1971, pp. 1–25. MR 0337781
Paul Bailey and Michael D. Fried, Hurwitz monodromy, spin separation and higher levels of a modular tower, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 79–220. MR 1935406, DOI 10.1090/pspum/070/1935406
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Jean-Marc Couveignes, Tools for the computation of families of coverings, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 38–65. MR 1708601
Jean-Marc Couveignes, Boundary of Hurwitz spaces and explicit patching, J. Symbolic Comput. 30 (2000), no. 6, 739–759. Algorithmic methods in Galois theory. MR 1800036, DOI 10.1006/jsco.2000.0381
Jean-Marc Couveignes and Louis Granboulan, Dessins from a geometric point of view, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 79–113. MR 1305394
Michael D. Fried and Pierre Dèbes, Rigidity and real residue class fields, Acta Arith. 56 (1990), no. 4, 291–323. MR 1096344, DOI 10.4064/aa-56-4-291-323
Pierre Dèbes and Michael D. Fried, Nonrigid constructions in Galois theory, Pacific J. Math. 163 (1994), no. 1, 81–122. MR 1256178
M. Dettweiler, Kurven auf Hurwitzräumen und ihre Anwendungen in der Galoistheorie. PhD Thesis, Erlangen (1999).
Michael D. Fried and Helmut Völklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290 (1991), no. 4, 771–800. MR 1119950, DOI 10.1007/BF01459271
L. Gerritzen, F. Herrlich, and M. van der Put, Stable $n$-pointed trees of projective lines, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 2, 131–163. MR 952512
Louis Granboulan, Construction d’une extension régulière de $\textbf {Q}(T)$ de groupe de Galois $M_{24}$, Experiment. Math. 5 (1996), no. 1, 3–14 (French, with English and French summaries). MR 1412950
Emmanuel Hallouin, Computation of a cover of Shimura curves using a Hurwitz space, J. Algebra 321 (2009), no. 2, 558–566. MR 2483281, DOI 10.1016/j.jalgebra.2008.10.019
Emmanuel Hallouin and Emmanuel Riboulet-Deyris, Computation of some moduli spaces of covers and explicit $\scr S_n$ and $\scr A_n$ regular $\Bbb Q(T)$-extensions with totally real fibers, Pacific J. Math. 211 (2003), no. 1, 81–99. MR 2016592, DOI 10.2140/pjm.2003.211.81
A. Hurwitz, Über ternäre diophantische Gleichungen dritten Grades. Vierteljahrschr. d. Naturf. Ges. in Zürich 62 (1917), 207–229.
Moshe Jarden, Algebraic patching, Springer Monographs in Mathematics, Springer, Heidelberg, 2011. MR 2768285, DOI 10.1007/978-3-642-15128-6
Michael Klug, Michael Musty, Sam Schiavone, and John Voight, Numerical calculation of three-point branched covers of the projective line, LMS J. Comput. Math. 17 (2014), no. 1, 379–430. MR 3356040, DOI 10.1112/S1461157014000084
Jürgen Klüners and Gunter Malle, Explicit Galois realization of transitive groups of degree up to 15, J. Symbolic Comput. 30 (2000), no. 6, 675–716. Algorithmic methods in Galois theory. MR 1800033, DOI 10.1006/jsco.2000.0378
Jürgen Klüners and Gunter Malle, A database for field extensions of the rationals, LMS J. Comput. Math. 4 (2001), 182–196. MR 1901356, DOI 10.1112/S1461157000000851
J. König, The inverse Galois problem and explicit computation of families of covers of $\mathbb {P}^1{\mathbb {C}}$ with prescribed ramification. Disseration, Würzburg (2014).
A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
Kay Magaard, Sergey Shpectorov, and Helmut Völklein, A GAP package for braid orbit computation and applications, Experiment. Math. 12 (2003), no. 4, 385–393. MR 2043989
Gunter Malle, Multi-parameter polynomials with given Galois group, J. Symbolic Comput. 30 (2000), no. 6, 717–731. Algorithmic methods in Galois theory. MR 1800034, DOI 10.1006/jsco.2000.0379
Gunter Malle, Polynomials with Galois groups $\textrm {Aut}(M_{22}),\;M_{22},$ and $\textrm {PSL}_3(\textbf {F}_4)\cdot 2_2$ over $\textbf {Q}$, Math. Comp. 51 (1988), no. 184, 761–768. MR 958642, DOI 10.1090/S0025-5718-1988-0958642-3
Gunter Malle, Polynomials for primitive nonsolvable permutation groups of degree $d\leq 15$, J. Symbolic Comput. 4 (1987), no. 1, 83–92. MR 908415, DOI 10.1016/S0747-7171(87)80056-1
Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1711577, DOI 10.1007/978-3-662-12123-8
P. Müller, A one-parameter family of polynomials with Galois group $M_{24}$ over $Q(t)$. Preprint (2012), available at http://arxiv.org/abs/1204.1328.
Matthieu Romagny and Stefan Wewers, Hurwitz spaces, Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, Paris, 2006, pp. 313–341 (English, with English and French summaries). MR 2316356
Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1171452, DOI 10.1007/978-1-4757-4252-7
Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961
Helmut Völklein, Groups as Galois groups, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, Cambridge, 1996. An introduction. MR 1405612, DOI 10.1017/CBO9780511471117
D. Zywina, Inverse Galois problem for small simple groups. Preprint (2013), available at http://www.math.cornell.edu/˜zywina/papers/smallGalois.pdf.