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The automorphism group of a function field

Authors: Manohar Madan and Michael Rosen
Journal: Proc. Amer. Math. Soc. 115 (1992), 923-929
MSC: Primary 12F12; Secondary 11R58
MathSciNet review: 1088443
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Abstract: Let $ k$ be an algebraically closed field, $ K$ a function field in one variable over $ k$ , and $ G$ a nontrivial finite group. It is proven that there exist infinitely many Galois extensions $ L/K$ such that $ \operatorname{Gal} (L/K)$ is isomorphic to $ G$ , and $ \operatorname{Gal} (L/K) = {\operatorname{Aut} _k}(L)$. This extends to arbitrary characteristic, a result first proven in the case $ k = \mathbb{C}$ by Greenberg in 1974.

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  • [D'M-Mad] J. G. D'Mello and M. L. Madan, Algebraic function fields with solvable automorphism group in characteristic $ p$, Comm. Algebra 11 (1983), 1187-1236. MR 696988 (85f:14025)
  • [Gr] L. Greenberg, Maximal groups and signatures, Discontinuous Groups and Riemann Surfaces (L. Greenberg, ed.), Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 207-226. MR 0379835 (52:740)
  • [Ha1] D. Harbater, Deformation theory and the tame fundamental group, Trans. Amer. Math. Soc. 262 (1980), 399-415. MR 586724 (82f:14014)
  • [Ha2] -, Mock covers and group extensions, J. Algebra 91 (1984), 281-293. MR 769574 (86c:13004)
  • [Ha3] -, Formal patching and adding branch points, preprint.
  • [Madd-Va] D. J. Madden and R. C. Valentini, The group of automorphisms of algebraic function fields, J. Reine Angew. Math. 343 (1983), 162-168. MR 705883 (85i:11096)
  • [P] H. Popp, Fundamentalgruppen algebraische Mannigfaltigkeiten, Lecture Notes in Math., vol. 176, Springer-Verlag, Berlin, Heidelberg, and New York, 1970. MR 0277540 (43:3273)
  • [Sch] H. L. Schmid, Uber die Automorphismen eines algebraischen Funktionenkörper von Primzahlcharacteristik, J. Reine Angew. Math. 179 (1938), 5-15.
  • [St1] H. Stichtenoth, Die Ungleichung von Castelnuovo, J. Reine Angew. Math. 348 (1984), 197-202. MR 733931 (85d:14036)
  • [St2] H. Stichtenoth, Zur Realisierbarkeit endlichen Gruppen als Automorphismengruppen algebraischer Funktionenkörper, Math. Z. 187 (1984), 221-225. MR 753434 (85m:14040)
  • [Va-Mad] R. C. Valentini and M. L. Madan, Automorphism groups of algebraic function fields, Math. Z. 176 (1981), 39-52. MR 606170 (83k:12013)

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Article copyright: © Copyright 1992 American Mathematical Society

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