Journal of the Mathematical Society of Japan

Non-constant Teichmüller level structures and an application to the Inverse Galois Problem


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In this paper, we generalize the Hurwitz space which is defined by Fried and Völklein by replacing constant Teichmüller level structures with non-constant Teichmüller level structures defined by finite étale group schemes. As an application, we give some examples of projective general symplectic groups over finite fields which occur as quotients of the absolute Galois group of the field of rational numbers $\mathbb Q$.

Article information

J. Math. Soc. Japan Volume 68, Number 3 (2016), 1189-1218.

First available in Project Euclid: 19 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 12F12: Inverse Galois theory
Secondary: 14D23: Stacks and moduli problems

Inverse Galois Problem Hurwitz space Hurwitz stack Teichmüller level structure


SAKUGAWA, Kenji. Non-constant Teichmüller level structures and an application to the Inverse Galois Problem. J. Math. Soc. Japan 68 (2016), no. 3, 1189--1218. doi:10.2969/jmsj/06831189.

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  • S. Arias-de-Reyna and N. Vila, Tame Galois realizations of $\mathit{GSp}_4(\mathbb F_\ell)$ over $\mathbb Q$, Int. Math. Res. Not. IMRN, 2011, no.,9, 2028–2046.
  • S. Arias-de-Reyna, C. Armana, V. Karemaker, M. Rebolledo, L. Thomas and N. Vila, Galois representations and Galois realizations over $\mathbb Q$, preprint, arXiv:1407.5802v1 (2014).
  • J. Bertin and M. Romagny, Champs de Hurwitz, Mém. Soc. Math. Fr. (N.S.) 125–126 (2011).
  • I. Bouw and S. Wewers, Reduction of covers and Hurwitz spaces, J. Reine Angew. Math., 574 (2004), 1–49.
  • P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 75–109.
  • M. Dettweiler and S. Reiter, An algorithm of Katz and its application to the Inverse Galois Problem, J. Symbolic Comput., 30 (2000), 761–798.
  • M. Fried, Fields of definition of function fields and Hurwitz families, groups as Galois groups, Comm. Algebra, 5 (1977), no.,1, 17–82.
  • W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2), 90 (1969), 542–575.
  • M. Fried and H. Völklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann., 290 (1991), 771–800.
  • C. Hall, An open-image theorem for a general class of abelian varieties with an appendix by Emmanuel Kowalski, Bull. Lond. Math. Soc., 43 (2011), no.,4, 703–711.
  • P. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, 129, Cambridge University Press, Cambridge, 1990.
  • G. Malle and B. H. Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999.
  • S. Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci., 31 (1995), no.,3, 355–441.
  • M. Romagny, Group actions on stacks and applications, Michigan Math. J., 53 (2005), no.,1, 209–236.
  • J. G. Thompson, Some finite groups which appear as $\mathrm{Gal}(L/K)$ where $K\subset\mathbb{Q}(\mu _n)$, J. Algebra, 89 (1984), no.,2, 437–499.
  • H. Völklein, Groups as Galois groups: An introduction, Cambridge Studies in Advanced Mathematics, 53, Cambridge University Press, Cambridge, 1996.
  • A. Wagner, Groups generated by elations, Abh. Math. Sem. Univ. Hamburg, 41 (1974), 190–205.
  • S. Wewers, Construction of Hurwitz spaces, Thesis, preprint no.,21 of IEM, Essen (1998).
  • Rêvetements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA1), dirigé par A. Grothendieck, augmenté de deux exposés de M. Raynaud, Springer Lecture Notes in Math., 224, Springer-Verlag, Berlin-New York, 1971.