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Mathematics of Computation

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Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups

Author: Joachim König
Journal: Math. Comp.
MSC (2010): Primary 11R32, 12Y05
Published electronically: June 2, 2016
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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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Joachim König
Affiliation: Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

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
Article copyright: © Copyright 2016 American Mathematical Society