Oberseminar Arithmetische Geometrie

By work of Fried, Völklein et al., the regular inverse Galois problem is famously linked to questions about the existence of rational points on certain moduli spaces. In some cases, this existence can be proven by theoretical means; a famous example is Matzat's realization of M₂₄ over Q.

In cases where the theoretical criteria do not work, little is known. I will present techniques for explicit computation of moduli spaces and use them to produce algebraic equations for Hurwitz spaces belonging to the Mathieu group M₂₃, the last remaining sporadic group not yet realized as a Galois group over Q. Suitable rational solutions for these equations would lead to M₂₃-realizations over Q.

Still, finding such points — or proving that they do not exist — remains a difficult problem. I will discuss some approaches that might help.

Similar methods may be used to classify monodromy groups of rational functions over Q. As partial results, I will present new polynomials for several almost simple groups, e.g. the first explicit polynomials with regular Galois groups PSL₅(2) and PSL₃(4) over Q.