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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups
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by Joachim König PDF
Math. Comp. 86 (2017), 1473-1498 Request permission

Abstract:

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb {Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over $\mathbb {Q}$. All these examples make use of families of covers of the projective line ramified over four or more points, and therefore use techniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g. by Malle (2000), Couveignes (1999), Granboulan (1996) and Hallouin (2009). Unlike previous examples, however, some of our computations show the existence of rational points on Hurwitz spaces that would not have been obvious from theoretical arguments.
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Additional Information
  • Joachim König
  • Affiliation: Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
  • Email: joachim.koenig@mathematik.uni-wuerzburg.de
  • Received by editor(s): January 29, 2015
  • Received by editor(s) in revised form: April 8, 2015, August 5, 2015, and September 28, 2015
  • Published electronically: June 2, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1473-1498
  • MSC (2010): Primary 11R32, 12Y05
  • DOI: https://doi.org/10.1090/mcom/3116
  • MathSciNet review: 3614024