Hilbert specialization results with local conditions

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with specified local behavior at any given finite set of primes of $k$. First, we give a full non-Galois analog of a result with a ramified type conclusion from a preceding paper and next we prove a unifying statement which combines our results and previous work devoted to the unramified part of the problem in the case $k$ is a number field

Specialization results and ramification conditions

Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of $\mathbb{Q}$ of various finite groups with specified local behavior - ramified or unramified - at finitely many given primes. Secondly, in the case $k$ is a number field, we provide criteria for the extension $E/k(T)$ to satisfy this property: at least one Galois extension $F/k$ of group $G$ is not a specialization of $E/k(T)$

Twists of superelliptic curves without rational points

Let $n\geq 2$ be an integer, $F$ a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $N$ a positive multiple of $n$. The paper deals with degree $N$ polynomials $P(T) \in O_F[T]$ such that the superelliptic curve $Y^n=P(T)$ has twists $Y^n=d\cdot P(T)$ without $F$-rational points. We show that this condition holds if the Galois group of $P(T)$ over $F$ has an element which fixes no root of $P(T)$. Two applications are given. Firstly, we prove that the proportion of degree $N$ polynomials $P(T) \in O_F[T]$ with height bounded by $H$ and such that the associated curve satisfies the desired condition tends to 1 as $H$ tends to $\infty$. Secondly, we connect the problem with the recent notion of non-parametric extensions and give new examples of such extensions with cyclic Galois groups

A note on prime divisors of polynomials P(Tk),k≥1P(T^k), k \geq 1

Let $F$ be a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $P(T) \in O_F[T]$ a monic separable polynomial such that $P(0) \not=0$ and $P(1) \not=0$. We give precise sufficient conditions on a given positive integer $k$ for the following condition to hold: there exist infinitely many non-zero prime ideals $\mathcal{P}$ of $O_F$ such that the reduction modulo $\mathcal{P}$ of $P(T)$ has a root in the residue field $O_F/\mathcal{P}$, but the reduction modulo $\mathcal{P}$ of $P(T^k)$ has no root in $O_F/\mathcal{P}$. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers $k$ more precise.Comment: arXiv admin note: text overlap with arXiv:1602.0670

Automorphism groups over Hilbertian fields

We show that every finite group occurs as the automorphism group of infinitely many finite (field) extensions of any given Hilbertian field. This extends and unifies previous results of M. Fried and Takahashi on the global field case

Specialization results in Galois theory

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If f:X\rightarrow \Pp^1 is a degree $n$ \Qq-cover with monodromy group $S_n$ over \bar\Qq, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points t_0\in \Qq that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an \'etale algebra $\prod_lE_l/k$ over a field $k$ to be the specialization of a $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$. The question is reduced to finding $k$-rational points on a certain $k$-variety, and then studied over the various fields $k$ of our applications

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.Comment: 37 page

Twisted covers and specializations

The central topic is this question: is a given $k$-\'etale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields

On the number of ramified primes in specializations of function fields over Q\mathbb{Q}

We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some Galois theoretical applications.Comment: To appear in the New York Journal of Mathematic

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.Comment: 27 page