Polynomials with AGL-monodromy
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Abstract:
Let $p$ be a prime and $r,e$ positive integers. In this paper we prove that the full affine group $\mathrm {AGL}(r,p^e)$ of dimension $r$ over the field with $p^e$ elements can be realized as the geometric monodromy group of a polynomial in characteristic $p$. We also determine the arithmetic monodromy group of this polynomial and the ramification structure it induces.References
- Shreeram S. Abhyankar, Fundamental group of the affine line in positive characteristic, Geometry and analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 1995, pp. 1–26. MR 1351500
- Shreeram S. Abhyankar, Galois theory on the line in nonzero characteristic, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 68–133. MR 1118002, DOI 10.1090/S0273-0979-1992-00270-7
- Shreeram S. Abhyankar, Mathieu group coverings and linear group coverings, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 293–319. MR 1352279, DOI 10.1090/conm/186/02188
- Shreeram S. Abhyankar, Nice equations for nice groups, Israel J. Math. 88 (1994), no. 1-3, 1–23. MR 1303488, DOI 10.1007/BF02937504
- Shreeram S. Abhyankar, Resolution of singularities and modular Galois theory, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 131–169. MR 1816069, DOI 10.1090/S0273-0979-00-00892-2
- Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1–22. MR 599634, DOI 10.1112/blms/13.1.1
- P. J. Cameron and W. M. Kantor, $2$-transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra 60 (1979), no. 2, 384–422. MR 549937, DOI 10.1016/0021-8693(79)90090-5
- P. J. Cameron, W. M. Kantor, Antiflag transitive collineation groups revisited, http://www.maths.qmul.ac.uk/$\sim$pjc/odds/antiflag.pdf, unpublished and incomplete draft, February 2002.
- John Conway, John McKay, and Allan Trojan, Galois groups over function fields of positive characteristic, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1205–1212. MR 2578514, DOI 10.1090/S0002-9939-09-10130-2
- M. R. Darafsheh, Order of elements in the groups related to the general linear group, Finite Fields Appl. 11 (2005), no. 4, 738–747. MR 2181417, DOI 10.1016/j.ffa.2004.12.003
- Leonard Eugene Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), no. 1, 75–98. MR 1500882, DOI 10.1090/S0002-9947-1911-1500882-4
- Leonard Eugene Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. of Math. 11 (1896/97), no. 1-6, 65–120. MR 1502214, DOI 10.2307/1967217
- Noam D. Elkies, Linearized algebra and finite groups of Lie type. I. Linear and symplectic groups, Applications of curves over finite fields (Seattle, WA, 1997) Contemp. Math., vol. 245, Amer. Math. Soc., Providence, RI, 1999, pp. 77–107. MR 1732230, DOI 10.1090/conm/245/03725
- Michael D. Fried, Robert Guralnick, and Jan Saxl, Schur covers and Carlitz’s conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225. MR 1239049, DOI 10.1007/BF02808112
- Robert M. Guralnick and Peter Müller, Exceptional polynomials of affine type, J. Algebra 194 (1997), no. 2, 429–454. MR 1467161, DOI 10.1006/jabr.1997.7028
- Robert M. Guralnick, Joel Rosenberg, and Michael E. Zieve, A new family of exceptional polynomials in characteristic two, Ann. of Math. (2) 172 (2010), no. 2, 1361–1390. MR 2680493, DOI 10.4007/annals.2010.172.1367
- Robert M. Guralnick and Jan Saxl, Monodromy groups of polynomials, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 125–150. MR 1320519, DOI 10.1017/CBO9780511565823.012
- Robert M. Guralnick and Michael E. Zieve, Polynomials with $\textrm {PSL}(2)$ monodromy, Ann. of Math. (2) 172 (2010), no. 2, 1315–1359. MR 2680492, DOI 10.4007/annals.2010.172.1321
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1237403
- Peter M. Neumann, Some primitive permutation groups, Proc. London Math. Soc. (3) 50 (1985), no. 2, 265–281. MR 772713, DOI 10.1112/plms/s3-50.2.265
- Moshe Roitman, On Zsigmondy primes, Proc. Amer. Math. Soc. 125 (1997), no. 7, 1913–1919. MR 1402885, DOI 10.1090/S0002-9939-97-03981-6
- Henning Stichtenoth, Algebraic function fields and codes, 2nd ed., Graduate Texts in Mathematics, vol. 254, Springer-Verlag, Berlin, 2009. MR 2464941
- B. L. van der Waerden, Die Zerlegungs-und Trägheitsgruppe als Permutationsgruppen, Math. Ann. 111 (1935), no. 1, 731–733 (German). MR 1513024, DOI 10.1007/BF01472249
Additional Information
- Florian Möller
- Affiliation: Institut für Mathematik, Universität Würzburg, Hubland Nord, 97074 Würzburg, Germany
- Email: fmoeller@mathematik.uni-wuerzburg.de
- Received by editor(s): August 11, 2009
- Received by editor(s) in revised form: December 8, 2010, and March 23, 2011
- Published electronically: January 12, 2012
- Communicated by: Ted Chinburg
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2967-2980
- MSC (2010): Primary 12F05, 12F10
- DOI: https://doi.org/10.1090/S0002-9939-2012-11199-2
- MathSciNet review: 2917070