## The inverse Galois problem and explicit computation of families of covers of $$\mathbb{P}^1\mathbb{C}$$ with prescribed ramification

### Das Umkehrproblem der Galoistheorie und explizite Berechnung von Familien von Überlagerungen des $$\mathbb{P}^1\mathbb{C}$$ mit vorgegebener Verzweigung

Please always quote using this URN: urn:nbn:de:bvb:20-opus-100143
• In attempting to solve the regular inverse Galois problem for arbitrary subfields K of C (particularly for K=Q), a very important result by Fried and Völklein reduces the existence of regular Galois extensions F|K(t) with Galois group G to the existence of K-rational points on components of certain moduli spaces for families of covers of the projective line, known as Hurwitz spaces. In some cases, the existence of rational points on Hurwitz spaces has been proven by theoretical criteria. In general, however, the question whether a givenIn attempting to solve the regular inverse Galois problem for arbitrary subfields K of C (particularly for K=Q), a very important result by Fried and Völklein reduces the existence of regular Galois extensions F|K(t) with Galois group G to the existence of K-rational points on components of certain moduli spaces for families of covers of the projective line, known as Hurwitz spaces. In some cases, the existence of rational points on Hurwitz spaces has been proven by theoretical criteria. In general, however, the question whether a given Hurwitz space has any rational point remains a very difficult problem. In concrete cases, it may be tackled by an explicit computation of a Hurwitz space and the corresponding family of covers. The aim of this work is to collect and expand on the various techniques that may be used to solve such computational problems and apply them to tackle several families of Galois theoretic interest. In particular, in Chapter 5, we compute explicit curve equations for Hurwitz spaces for certain families of $$M_{24}$$ and $$M_{23}$$. These are (to my knowledge) the first examples of explicitly computed Hurwitz spaces of such high genus. They might be used to realize $$M_{23}$$ as a regular Galois group over Q if one manages to find suitable points on them. Apart from the calculation of explicit algebraic equations, we produce complex approximations for polynomials with genus zero ramification of several different ramification types in $$M_{24}$$ and $$M_{23}$$. These may be used as starting points for similar computations. The main motivation for these computations is the fact that $$M_{23}$$ is currently the only remaining sporadic group that is not known to occur as a Galois group over Q. We also 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 Q(t). Special attention will be given to reality questions. As an application we compute the first examples of totally real polynomials with Galois groups $$PGL_2(11)$$ and $$PSL_3(3)$$ over Q. As a suggestion for further research, we describe an explicit algorithmic version of "Algebraic Patching", following the theory described e.g. by M. Jarden. This could be used to conquer some problems regarding families of covers of genus g>0. Finally, we present explicit Magma implementations for several of the most important algorithms involved in our computations.
• Das Umkehrproblem der Galoistheorie und explizite Berechnung von Familien von Überlagerungen des $$\mathbb{P}^1\mathbb{C}$$ mit vorgegebener Verzweigung