Semilinear transformations
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- by Shreeram S. Abhyankar PDF
- Proc. Amer. Math. Soc. 127 (1999), 2511-2525 Request permission
Abstract:
In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic $p$ with the projective general group $\mathrm {PGL}(m,q)$ and the general linear group $\mathrm {GL}(m,q)$ as Galois groups where $m>1$ is any integer and $q>1$ is any power of $p$. These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group $\mathrm {P}\Gamma \mathrm {L}(m,q)$ and the general semilinear group $\Gamma \mathrm {L}(m,q)$. We also obtain the semilinear versions of the local coverings considered in previous papers.References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Shreeram Abhyankar, Coverings of algebraic curves, Amer. J. Math. 79 (1957), 825–856. MR 94354, DOI 10.2307/2372438
- 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, Nice equations for nice groups, Israel J. Math. 88 (1994), no. 1-3, 1–23. MR 1303488, DOI 10.1007/BF02937504
- Shreeram S. Abhyankar, Projective polynomials, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1643–1650. MR 1403111, DOI 10.1090/S0002-9939-97-03939-7
- Shreeram S. Abhyankar, Local fundamental groups of algebraic varieties, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1635–1641. MR 1403110, DOI 10.1090/S0002-9939-97-03938-5
- Shreeram S. Abhyankar and Paul A. Loomis, Once more nice equations for nice groups, Proc. Amer. Math. Soc. 126 (1998), no. 7, 1885–1896. MR 1459101, DOI 10.1090/S0002-9939-98-04421-9
- 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
- David Harbater, Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117 (1994), no. 1, 1–25. MR 1269423, DOI 10.1007/BF01232232
- M. Raynaud, Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425–462 (French). MR 1253200, DOI 10.1007/BF01231568
Additional Information
- Shreeram S. Abhyankar
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: ram@cs.purdue.edu
- Received by editor(s): March 5, 1997
- Received by editor(s) in revised form: July 2, 1997
- Published electronically: May 4, 1999
- Additional Notes: This work was partly supported by NSF grant DMS 91-01424 and NSA grant MDA 904-97-1-0010
- Communicated by: Ron Donagi
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2511-2525
- MSC (1991): Primary 12F10, 14H30, 20D06, 20E22
- DOI: https://doi.org/10.1090/S0002-9939-99-05400-3
- MathSciNet review: 1676323