arXiv subjects
Publications and source records attributed to Joachim König.
The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.
We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field of $g(X)-t$ over $K(t)$ is non-$G$-parametric, i.e.\ not all Galois extensions of $K$ with group $G$ arise as specializations of $F(s)|K(s)$.
We prove a conjecture by Stefan Kohl on the existence of triples of permutations of bounded degree with prescribed orders and product 1. This result leads to an existence result for covers of the complex projective line with bounded degree and prescribed ramification indices.
We prove a result about reducibility behaviour of Thue polynomials over the rationals that was conjectured by Müller. More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many reducible integer specializations. Special cases have been proved e.g. by Müller and Langmann. The proof uses ramification theory to reduce the assertion to a statement about permutation groups containing an $n$-cycle. This statement is finally proven with the help of the classification of primitive permutation groups containing an $n$-cycle (a result which rests on the classification of finite simple groups).
We compute the first explicit polynomials with Galois groups $G=PΓL_3(4)$, $PGL_3(4)$, $PSL_3(4)$ 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, Couveignes, Granboulan and Hallouin. 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.
We compute new polynomials with Galois group $M_{11}$ over $\mathbb{Q}(t)$. These polynomials stem from various families of covers of $\mathbb{P}^1\mathbb{C}$ ramified over at least 4 points. Each of these families has features that make a detailed study interesting. Some of the polynomials lead, via specialization, to number fields with very small discriminant or few ramified primes.
In this paper we prove that a generic rational equation of degree $7$ is solvable by 2-fold origami. In particular we show how to septisect an arbitrary angle. This extends the work of Alperin & Lang and Nishimura on 2-fold origami. Furthermore we give exact crease patterns for folding polynomials with Galois groups $A_7$ resp. $PSL_3\mathbb{F}_2$.
We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.
We compute a family of coverings with four ramification points, defined over $\mathbb{Q}$, with regular Galois group $PSL_5(2)$. On the one hand, this is (to my knowledge) the first explicit polynomial with group $PSL_5(2)$ over $\mathbb{Q}(t)$. On the other hand, it also positively answers the question whether $PSL_5(2)$ is the monodromy group of a rational function over $\mathbb{Q}$. At least this does not follow from considering class triples in $PSL_5(2)$, as there are no rigid, rational genus-zero triples. Also, for 4-tuples, our family is the only one with a Hurwitz curve of genus zero (however it does not seem immediately clear without explicit computations whether this curve can be defined as a rational curve over $\mathbb{Q}$). There are also genus zero families with five branch points, and maybe their Hurwitz spaces can be shown to have rational points; however, so far I have not seen such arguments.
The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character phi of U with solvable factor group U/ker(phi). Using the classification of finite simple groups, we prove that these groups are also solvable. This means in particular that the mentioned properties do not enable one to deduce a proof of the famous Artin conjecture for any non-solvable group from a possible proof for solvable groups.