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Joachim König

Publications and source records attributed to Joachim König.

At least 37 records · Page 2Linked to original sources
arXiv2021

The local dimension of a finite group over a number field

Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely many primes of $K$. If furthermore $G$ has a generic extension over $K$, we show that the extension $E/F$ has the so-called Hilbert-Grunwald property. These results are compared to the notion of essential dimension of $G$ over $K$, and its arithmetic analogue.

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arXiv2021

The Hilbert-Schinzel specialization property

We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables, with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables, we show that if they are irreducible over $\mathbb Z$ and satisfy a necessary "Schinzel condition", then the first $k$ variables can be specialized in a Zariski-dense subset of ${\mathbb Z}^k$ in such a way that irreducibility over ${\mathbb Z}$ is preserved for the polynomials in the remaining $n$ variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first $k$ variables in ${\mathbb Z}^k$, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a "coprime" version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials assume coprime values. We prove our results over many other rings than $\mathbb Z$, e.g. UFDs and Dedekind domains for the last one.

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arXiv2021

On the mod-$p$ distribution of discriminants of $G$-extensions

This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of "local obstructions" to such heuristics, in many cases essentially only under the assumption that $G$ occurs as the Galois group of a Galois cover defined over $\mathbb{Q}$. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes.

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arXiv2021

Rank gain of Jacobians over number field extensions with prescribed Galois groups

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infinitely many $G$-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group $G$ and certain local properties.

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arXiv2021

On Galois extensions with prescribed decomposition groups

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realizations with only cyclic decomposition groups and with nonsolvable Galois group. We also investigate the analogous problem over global function fields.

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arXiv2021

A note on invariable generation of nonsolvable permutation groups

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of $S_n$. As an application, we obtain that for a large degree "random" integer polynomial $f$, reduction modulo two different primes can be expected to suffice to prove the nonsolvability of $Gal(f/\mathbb{Q})$.

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arXiv2021

Rational pullbacks of Galois covers

The finite subgroups of ${\rm PGL}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\rightarrow \mathbb{P}^1_{\mathbb{C}}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps $\mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}}$. For $G\subset {\rm PGL}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\not \subset {\rm PGL}_2({\mathbb{C}})$, letting the branch point number grow provides truly new Galois realizations $F/{\mathbb{C}}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of $\mathbb{P}^1_{\mathbb{C}}$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset {\rm PGL}_2({\mathbb{C}})$.

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arXiv2021

On parametric and generic polynomials with one parameter

Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois extensions of $L$ of group $G$ via specialization of $T$ in $L$, and the latter are those which are $L$-parametric for every field $L \supseteq k$. We show, for example, that being $L$-parametric with $L$ taken to be the single field $\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \in \mathbb{C}[T][Y]$ to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.

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arXiv2020

On unramified solvable extensions of small number fields

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$ possesses an unramified $G$-extension. We improve the best known bounds for the degree of such number fields $K$ for certain classes of solvable groups, in particular nilpotent groups.

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arXiv2019

Density results for specialization sets of Galois covers

We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}^1_\mathbb{Q}$ of group $G$. We also introduce a local-global principle for specializations of Galois covers $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local-global conclusion underscores the "smallness" of the specialization set of a Galois cover of $\mathbb{P}^1_\mathbb{Q}$. On the other hand, it allows to generate conditionally "many" curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

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arXiv2019

An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of $3$-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp(6,2).

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arXiv2019

The Grunwald problem and specialization of families of regular Galois extensions

We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by Dèbes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.

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arXiv2019

Unramified extensions over low degree number fields

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$. For these groups $G$, this gives the existence of number fields of degree $ge(G)$ with an unramified $G$-extension. The existence of such extensions over $\mathbb{Q}$ for all finite groups would imply that, for every finite group $G$, there exists a quadratic number field admitting an unramified $G$-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when $\mathbb{Q}$ is replaced with a function field $k(t)$ where $k$ is an ample field.

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arXiv2018

On number fields with $k$-free discriminants

Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with squarefree discriminant. In particular, we obtain a comprehensive result on number fields with cubefree discriminant. Our main tools are arithmetic-geometric, involving in particular an effective criterion on ramification in specializations of Galois covers.

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arXiv2018

On the local behaviour of specializations of function field extensions

Given a field $k$ of characteristic zero and an indeterminate $T$ over $k$, we investigate the local behaviour at primes of $k$ of finite Galois extensions of $k$ arising as specializations of finite Galois extensions $E/k(T)$ (with $E/k$ regular) at points $t_0 \in \mathbb{P}^1(k)$. We provide a general result about decomposition groups at primes of $k$ in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert--Grunwald property, and finite parametric sets.

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arXiv2017

Non-parametric sets of regular realizations over number fields

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular of a certain type, under a conjectural "uniform Faltings" theorem".

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arXiv2017

Some examples of quadratic fields with finite nonsolvable maximal unramified extensions II

Let $K$ be a number field and $K_{ur}$ be the maximal extension of $K$ that is unramified at all places. In a previous article, the first author found three real quadratic fields $K$ such that $Gal(K_{ur}/K)$ is finite and nonabelian simple under the assumption of the GRH(Generalized Riemann Hypothesis). In this article, we will identify more quadratic number fields $K$ such that $Gal(K_{ur}/K)$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the Generalized Riemann Hypothesis.

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