ID: 2003.11324

Symmetric Galois Groups Under Specialization

March 25, 2020

View on ArXiv
Tali Monderer, Danny Neftin
Mathematics
Number Theory
Group Theory

Given an irreducible bivariate polynomial $f(t,x)\in \mathbb{Q}[t,x]$, what groups $H$ appear as the Galois group of $f(t_0,x)$ for infinitely many $t_0\in \mathbb{Q}$? How often does a group $H$ as above appear as the Galois group of $f(t_0,x)$, $t_0\in \mathbb{Q}$? We give an answer for $f$ of large $x$-degree with alternating or symmetric Galois group over $\mathbb{Q}(t)$. This is done by determining the low genus subcovers of coverings $\tilde{X}\rightarrow \mathbb{P}^1_{\mathbb{C}}$ with alternating or symmetric monodromy groups.

Similar papers 1

Specialization results in Galois theory

June 30, 2011

89% Match
Pierre Dèbes, François Legrand
Number Theory
Algebraic Geometry

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 $p$. Th...

Find SimilarView on arXiv

Galois specialization to symmetric points and the inverse Galois problem up to $S_n$

July 27, 2022

88% Match
Borys Kadets
Number Theory
Algebraic Geometry

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in \pi^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to mod...

Find SimilarView on arXiv

Galois groups over rational function fields and explicit Hilbert irreducibility

August 15, 2017

86% Match
David Krumm, Nicole Sutherland
Number Theory

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i....

Find SimilarView on arXiv

Specialization results and ramification conditions

October 8, 2013

84% Match
François Legrand
Number Theory

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 loc...

Find SimilarView on arXiv

On Galois realizations of the 2-coverable symmetric and alternating groups

August 18, 2010

84% Match
Daniel Rabayev, Jack Sonn
Number Theory

Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be "m-coverable", i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inver...

Find SimilarView on arXiv

On the Galois groups of the dualizing coverings for plane curves

March 6, 2014

84% Match
Vik. S. Kulikov
Algebraic Geometry

Let $C_1$ be an irreducible component of a reduced projective curve $C\subset \mathbb P^2$ defined over the field $\mathbb C$, $\mathrm{deg} C_1\geq 2$, and let $T$ be the set of lines $l\subset \mathbb P^2$ meeting $C$ transversally. In the article, we prove that for a line $l_0\in T$ and any two points $P_1,P_2\in C_1\cap l_0$ there is a loop $l_t\subset T$, $t\in [0,1]$, such that the movement of the line $l_0$ along the loop $l_t$ induces the transposition of the points $...

Find SimilarView on arXiv

Density results for specialization sets of Galois covers

April 10, 2019

84% Match
Joachim König, François Legrand
Number Theory

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 an...

Find SimilarView on arXiv

Twisted covers and specializations

June 30, 2011

84% Match
Pierre Dèbes, François Legrand
Number Theory
Algebraic Geometry

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 ...

Find SimilarView on arXiv

Linearized Polynomials, Galois Groups and Symmetric Power Modules

June 3, 2022

84% Match
Rod Gow, Gary McGuire
Number Theory

We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.

Find SimilarView on arXiv

Specializations of one-parameter families of polynomials

May 7, 2004

83% Match
Farshid Hajir, Siman Wong
Number Theory

Let K be a number field, and let lambda(x,t)\in K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha\in K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this to show that for any fixed n>=10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial are K-irreducible and have Galois group S_n. In conjunction with the theory of complex multi...

Find SimilarView on arXiv