Fields of moduli of three-point $G$-covers with cyclic $p$-Sylow, II

Andrew Obus

Displaying similar documents to “Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II”

Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei (2014)

Publications mathématiques de Besançon

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Fix an integer $ℓ \geq 3$ . Rikuna introduced a polynomial $r (x, t)$ defined over a function field $K (t)$ whose Galois group is cyclic of order $ℓ$ , where $K$ satisfies some mild hypotheses. In this paper we define the family of ${r_{n} (x, t)}_{n \geq 1}$ of degree $ℓ^{n}$ . The $r_{n} (x, t)$ are constructed iteratively from the $r (x, t)$ . We compute the Galois groups of the $r_{n} (x, t)$ for odd $ℓ$ over an arbitrary base field and give applications to arithmetic dynamical systems.

Multiple end solutions to the Allen-Cahn equation in $ℝ^{2}$

Michał Kowalczyk, Yong Liu, Frank Pacard (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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An entire solution of the Allen-Cahn equation $Δ u = f (u)$ , where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$ , e.g. $f (u) = u (u^{2} - 1)$ , is called a $2 k$ end solution if its nodal set is asymptotic to $2 k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $- 1$ ) like the one dimensional, odd, heteroclinic solution $H$ , of $H^{''} = f (H)$ . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

On the compositum of all degree $d$ extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum $k^{[d]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{[d]}$ contains all subextensions of degree less than $d$ if and only if $d \leq 4$ . We prove that for $d > 2$ there is no bound $c = c (d)$ on the degree of elements required to generate finite subextensions of $k^{[d]} / k$ . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$ , but that one can take $c = d$ when $d$ is prime. This question was inspired by work of...

Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott, Bouchaïb Sodaïgui (2013)

Annales de l’institut Fourier

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Given an algebraic number field $k$ and a finite group $Γ$ , we write $ℛ (O_{k} [Γ])$ for the subset of the locally free classgroup $Cl (O_{k} [Γ])$ consisting of the classes of rings of integers $O_{N}$ in tame Galois extensions $N / k$ with $Gal (N / k) ≅ Γ$ . We determine $ℛ (O_{k} [Γ])$ , and show it is a subgroup of $Cl (O_{k} [Γ])$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $Γ = V ⋊ C$ , where $V$ is an elementary abelian group of order $p^{r}$ and $C$ is a cyclic group of order $m > 1$ acting faithfully...

The local lifting problem for actions of finite groups on curves

Ted Chinburg, Robert Guralnick, David Harbater (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be an algebraically closed field of characteristic $p > 0$ . We study obstructions to lifting to characteristic $0$ the faithful continuous action $φ$ of a finite group $G$ on $k [[t]]$ . To each such $φ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$ . We say that the KGB obstruction of $φ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X / H$ and $Y / H$ have the same genus for all subgroups $H \subset G$ . We determine for which $G$ the KGB...

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

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We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(α)} (x)$ is a non-zero square for some integer pair $(n, α)$ , with $n \geq 1$ , if and only if $(n, α)$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is the alternating group $A_{n}$ . For example, we establish that for all but finitely many positive integers $n \equiv 2 (mod 4)$ , the only $α$ for which the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is $A_{n}$ is...

On critical values of twisted Artin $L$ -functions

Peng-Jie Wong (2017)

Czechoslovak Mathematical Journal

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We give a simple proof that critical values of any Artin $L$ -function attached to a representation $ρ$ with character $χ_{ρ}$ are stable under twisting by a totally even character $χ$ , up to the $dim ρ$ -th power of the Gauss sum related to $χ$ and an element in the field generated by the values of $χ_{ρ}$ and $χ$ over $ℚ$ . This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

Displaying similar documents to “Fields of moduli of three-point G -covers with cyclic p -Sylow, II”

Displaying similar documents to “Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II”