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Patching subfields of division algebras

Authors: David Harbater, Julia Hartmann and Daniel Krashen
Journal: Trans. Amer. Math. Soc. 363 (2011), 3335-3349
MSC (2010): Primary 12F12, 16K20, 14H25; Secondary 16S35, 12E30, 16K50
Published electronically: December 29, 2010
MathSciNet review: 2775810
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Abstract: Given a field $ F$, one may ask which finite groups are Galois groups of field extensions $ E/F$ such that $ E$ is a maximal subfield of a division algebra with center $ F$. This question was originally posed by Schacher, who gave partial results in the case $ F = \mathbb{Q}$. Using patching, we give a complete characterization of such groups in the case that $ F$ is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases.

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  • [Abh69] Shreeram S. Abhyankar, Resolution of singularities of algebraic surfaces. In 1969 Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), pp. 1-11, Oxford Univ. Press, London. MR 0257080 (41:1734)
  • [Bo72] Nicolas Bourbaki, Elements of Mathematics. Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1972. MR 0360549 (50:12997)
  • [CS81] David Chillag and Jack Sonn, Sylow-metacyclic groups and $ \mathbb{Q}$-admissibility, Israel J. Math. 40 (1981), no. 3-4, 307-323 (1982). MR 654586 (83h:20026)
  • [COP02] Jean-Louis Colliot-Thélène, Manuel Ojanguren and Raman Parimala, Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. In: Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry, Tata Inst. Fund. Res. Stud. Math., vol. 16, pp. 185-217, Narosa Publ. Co., 2002. Preprint at $ \langle$$ \sim$colliot/$ \rangle$.
  • [DI71] Frank DeMeyer and Edward Ingraham, Separable algebras over commutative rings, Springer-Verlag, Berlin, 1971. MR 0280479 (43:6199)
  • [dJ04] Aise Johan de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), no. 1, 71-94. MR 2060023 (2005e:14025)
  • [Fei93] Walter Feit, The $ K$-admissibility of $ 2A\sb 6$ and $ 2A\sb 7$, Israel J. Math. 82 (1993), no. 1-3, 141-156. MR 1239048 (94f:12003)
  • [Fei02] -, $ \operatorname{SL}(2,11)$ is $ \mathbb{Q}$-admissible, J. Algebra 257 (2002), no. 2, 244-248. MR 1947322 (2004a:12006)
  • [Fei04] -, $ \operatorname{PSL}\sb 2(11)$ is admissible for all number fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 295-299. MR 2037096 (2004k:12006)
  • [FF90] Paul Feit and Walter Feit, The $ K$-admissibility of $ \operatorname{SL}(2,5)$, Geom. Dedicata 36 (1990), no. 1, 1-13. MR 1065210 (91h:12012)
  • [FV87] Walter Feit and Paul Vojta, Examples of some $ \mathbb{Q}$-admissible groups, J. Number Theory 26 (1987), no. 2, 210-226. MR 889385 (88g:12006)
  • [Gro68a] Alexander Grothendieck, Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 46-66. MR 0244269 (39:5586a)
  • [Gro68b] -, Le groupe de Brauer. II. Théorie cohomologique, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 67-87. MR 0244270 (39:5586b)
  • [GS06] Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528 (2007k:16033)
  • [Ha87] David Harbater, Galois coverings of the arithmetic line, in Number Theory: New York, 1984-85, Springer LNM, vol. 1240 (1987), pp. 165-195. MR 894511 (88h:14020)
  • [HH10] David Harbater and Julia Hartmann, Patching over fields, Israel J. Math. 176 (2010), 231-263. MR 2653187
  • [HHK09] David Harbater, Julia Hartmann, and Daniel Krashen, Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), no. 2, 231-263. MR 2545681
  • [Has02] Helmut Hasse, Number Theory, translated by Horst Zimmer, Springer-Verlag, Berlin-Heidelberg, 2002. MR 1885791
  • [Hoo80] Raymond T. Hoobler, A cohomological interpretation of Brauer groups of rings, Pacific J. Math. 86 (1980), no. 1, 89-92. MR 586870 (81j:13003)
  • [Jac96] Nathan Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996. MR 1439248 (98a:16024)
  • [Lie07] Max Lieblich, Period and index in the Brauer group of an arithmetic surface, with an appendix by Daniel Krashen, 2007 manuscript. To appear in Crelle's J. Available at arXiv:math/0702240.
  • [Lip75] Joseph Lipman, Introduction to resolution of singularities, in Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pages 187-230. Amer. Math. Soc., Providence, R.I., 1975. MR 0389901 (52:10730)
  • [Pie82] Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, 9. Springer-Verlag, New York, 1982. MR 674652 (84c:16001)
  • [Sal90] David J. Saltman, Multiplicative field invariants and the Brauer group, J. Algebra 133 (1990), no. 2, 533-544. MR 1067425 (91h:12016)
  • [Sal99] -, Lectures on division algebras, CBMS Regional Conference Series, no. 94, American Mathematical Society, Providence, RI, 1999. MR 1692654 (2000f:16023)
  • [Sch68] Murray M. Schacher, Subfields of division rings. I, J. Algebra 9 (1968), 451-477. MR 0227224 (37:2809)
  • [Ser79] Jean-Pierre Serre, Local fields, Springer-Verlag, New York, 1979. MR 554237 (82e:12016)
  • [Son83] Jack Sonn, $ \mathbb{Q}$-admissibility of solvable groups, J. Algebra 84 (1983), no. 2, 411-419. MR 723399 (86b:20022)
  • [SS92] Murray M. Schacher and Jack Sonn, $ K$-admissibility of $ A\sb 6$ and $ A\sb 7$, J. Algebra 145 (1992), no. 2, 333-338. MR 1144936 (93a:12003)
  • [Wad02] Adrian R. Wadsworth, Valuation theory on finite dimensional division algebras, in Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), Fields Inst. Commun., vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 385-449. MR 1928379 (2003g:16023)

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Additional Information

David Harbater
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395

Julia Hartmann
Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen University, 52062 Aachen, Germany

Daniel Krashen
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Keywords: Admissibility, patching, division algebras, Brauer groups, Galois groups.
Received by editor(s): April 9, 2009
Received by editor(s) in revised form: October 14, 2009
Published electronically: December 29, 2010
Additional Notes: The authors were respectively supported in part by NSF Grant DMS-0500118, the German National Science Foundation (DFG), and an NSA Young Investigator’s Grant.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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