Showing papers on "Generic polynomial published in 2009"
TL;DR: In this paper, it was shown that there is a close connection between the action of the Selmer group of E over F ∞, and the global root numbers attached to the twists of the complex L-function of E by Artin representations of G.
Abstract: Let E be an elliptic curve over a number field F, and let F ∞ be a Galois extension of F whose Galois group G is a p-adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E over F ∞ , and the global root numbers attached to the twists of the complex L-function of E by Artin representations of G.
58 citations
TL;DR: In this paper, it was shown that the ring of all endomorphisms of J(Cf, p) coincides with a ring of integers in the pth cyclotomic field.
Abstract: Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.
24 citations
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TL;DR: In this paper, it was shown that trivial t -motifs satisfy a Tannakian duality, without restrictions on the base field, save for that it be of generic characteristic.
23 citations
TL;DR: In this article, an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation was established.
21 citations
TL;DR: For a Picard curve C with endomorphism ring Z[ζ3] the images of these representations are full for all but finitely many primes l in the reduction modulo l as discussed by the authors.
9 citations
TL;DR: Jarden et al. as mentioned in this paper showed that for almost all σ ∈ Gal(K) the absolute Galois group of Ktot,S(σ) is the free product of Fe and a free products of local factors over S. The main result of this paper is that Gal(S) is PAC.
Abstract: For a finite set S of primes of a number field K and for σ1, . . . , σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1, . . . , σe in Ktot,S by Ktot,S(σ). We prove that for almost all σ ∈ Gal(K) the absolute Galois group of Ktot,S(σ) is the free product of Fe and a free product of local factors over S. MR Classification: 12E30 Directory: \Jarden\Diary\HJPd 10 May, 2009 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction The Inverse Galois Problem asks whether every finite group is realizable over Q. Although this has been shown to be true for many finite groups, including the symmetric and alternating groups (Hilbert), we are still very far from the solution of the problem. One could ask, more generally, what is the structure of the absolute Galois group of Q. Here we do not even have a plausible conjecture. However, we do know the structure of the absolute Galois group of certain distinguished algebraic extensions of Q, or, more generally, of a countable Hilbertian field K. We fix a separable closure Ks and an algebraic closure K of K and let Gal(K) = Gal(Ks/K) be the absolute Galois group of K. Our goal is to explore the absolute Galois groups of large algebraic extensions of K having interesting diophantine or arithmetical properties. Our study is motivated by two earlier results. By the free generators theorem Gal(Ks(σ)) is, for almost all σ ∈ Gal(K), the free profinite group Fe on e generators (Jarden [FrJ, Thm. 18.5.6]). On the other hand, if K is a global field and S1 is a finite set of primes of K, then the absolute Galois group Gal(Ktot,S1) of the maximal S1-adic extension of K is a free product of local groups (Pop [Pop4, Thm. 3]). In this work we simultaneously generalize both results and prove that Gal(Ks(σ) ∩ Ktot,S1) is, for almost all σ ∈ Gal(K), the free product of Fe and a free product of local groups. Here is a detailed account of our result. The main theorem. For each e-tuple σ = (σ1, . . . , σe) ∈ Gal(K) we denote the fixed field in Ks (resp. K) of σ1, . . . , σe by Ks(σ) (or K(σ) if char(K) = 0). We know that for almost all σ ∈ Gal(K) the field Ks(σ) is PAC [FrJ, Thm. 18.6.1] and Gal(Ks(σ)) ∼= Fe [FrJ, Thm. 18.5.6]. Here “almost all” is meant in the sense of the Haar measure of Gal(K) and we say that a field M is PAC if every absolutely irreducible variety V defined over M has an M -rational point. The PAC property of the field Ks(σ) implies that if w is a nontrivial valuation of Ks(σ), then the Henselian closure of Ks(σ) at w is Ks [FrJ, Cor. 11.5.5]. To bring valuations into the game we consider a finite set S1 of absolute values
9 citations
TL;DR: Steklov and Skopin this paper proved a Golod-Shafarevich equality for analytic pro-p -groups, where the base field is a quadratic imaginary number field.
7 citations
Journal Article•
TL;DR: In this paper, an asymptotically optimal tame tower over the field with p(2) elements introduced by Garcia-Stichtenoth was studied and its Galois closure was investigated.
Abstract: In this paper we study an asymptotically optimal tame tower over the field with p(2) elements introduced by Garcia-Stichtenoth. This tower is related with a modular tower, for which explicit equations were given by Elkies. We use this relation to investigate its Galois closure. Along the way, we obtain information about the structure of the Galois closure of X-0(p(n)) over X-0(p(r)), for integers 1 < r < n and prime p and the Galois closure of other modular towers (X-0(p(n)))n.
6 citations
Journal Article•
TL;DR: In this article, the Galois symbol map from T_F(E x E)/p to H^2 (F,E[P] ⊗ E[P]), for E/F ordinary, without requiring that the p-torsion points are F-rational or semisimple.
Abstract: This article studies the Albanese kernel T_F(E x E), for an
elliptic curve E over a p-adic field F. The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from T_F(E x E)/p to the Galois cohomology group H^2 (F,E[P] ⊗ E[P]), for E/F ordinary, without requiring that the p-torsion points are F-rational, or even that the Galois module E[P] is semisimple. A key step is to show that the image is zero when the finite Galois module E[P] is acted on non-trivially by the pro-p-inertia group I_p. Non-trivial classes in the image are also constructed when
E[P] is suitably unramified. A forthcoming sequel will deal with global questions.
6 citations
28 Jul 2009
TL;DR: New results about the computation of a general shape of a triangular basis generating the splitting ideal of an irreducible polynomial given with the permutation representation of its Galois group G are presented.
Abstract: In this article, we present new results about the computation of a general shape of a triangular basis generating the splitting ideal of an irreducible polynomial given with the permutation representation of its Galois group G. We provide some theoretical results and a new general algorithm based on the study of the non redundant bases of permutation groups. These new results deeply increase the efficiency of the computation of the splitting field of a polynomial.
4 citations
TL;DR: The Galois group of the trinomial Xp+aXp−1+a over the field Q of rational numbers is the full symmetric group as soon as it is transitive, namely when a≠±1 or p≢2(mod3) as discussed by the authors.
TL;DR: An algorithm is provided that gives all m-near solutions of a given polynomial F(x,y) over K, and this algorithm isPolynomial time reducible to solving one variable equations over K.
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TL;DR: In this paper, it was shown that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and fibers over singular points.
Abstract: We confirm a conjecture of Bernstein-Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.
01 Jan 2009
TL;DR: In this paper, the authors studied the arithmetic of the generic polynomial for the cyclic group of order n and obtained a generalized Kummer theory, which is useful under the condition that ζ ∈ k and ω ∈ ω = ζ + ζ -1.
Abstract: In this report we study the arithmetic of Rikuna's generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ ∉ k and ω ∈ k where ζ is a primitive n-th root of unity and ω = ζ + ζ -1 . In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.
TL;DR: In this article, the singular locus of the Galois closure of a smooth quasi-projective surface over an algebraically closed field has been determined, using a method suggested by M. Artin.
Abstract: Let k be an algebraically closed field. Let P(X11 ,..., Xnn, T ) be the characteristic polynomial of the generic matrix (X ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.
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22 Oct 2009
TL;DR: In this article, the authors studied the Teichmuller modular group of a compact Riemann surface of genus G and showed that the set of profinite Dehn twists of G is the closure of a set of Dehn twist of G. The main technical result of this paper is a parametrization of the set.
Abstract: For $2g-2+n>0$, the Teichmuller modular group $\Gamma_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hat{\Pi}_{g,n}$ its profinite completion. There is then a natural faithful representation $\Gamma_{g,n}\hookrightarrow Out(\hat{\Pi}_{g,n})$. The procongruence completion $\check{\Gamma}_{g,n}$ of the Teichmuller group is defined to be the closure of the Teichmuller group $\Gamma_{g,n}$ inside the profinite group $Out(\hat{\Pi}_{g,n})$. In this paper, we begin a systematic study of the procongruence completion $\check{\Gamma}_{g,n}$. The set of profinite Dehn twists of $\check{\Gamma}_{g,n}$ is the closure, inside this group, of the set of Dehn twists of $\GG_{g,n}$. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of $\check{\Gamma}_{g,n}$ and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichmuller Lego with procongruence Teichmuller groups as building blocks. As an application, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful.
TL;DR: In this paper, a generalization of a theorem of Saltman on the existence of generic extensions with group A⋊G over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups.
DOI•
01 May 2009TL;DR: In this article, it was shown that there exist subgroups K and N of G such that K is a normal subgroup of N and one of the following three cases holds: (i) VB(B K ) is a central Galois algebra over C with Galois group K, (ii) V B K )i s separable C-algebra with an automorphism group induced by and isomorphic with K.
Abstract: Let B be a Hirata separable and Galois extension of B G with Galois group G of order n invertible in B for some integer n, C the center of B, and VB(B G ) the commutator subring of B G in B. It is shown that there exist subgroups K and N of G such that K is a normal subgroup of N and one of the following three cases holds: (i) VB(B K ) is a central Galois algebra over C with Galois group K, (ii) VB(B K )i s separable C-algebra with an automorphism group induced by and isomorphic with K, and (iii) B K is a central algebra over VB(B K ) and a Hirata separable Galois extension of B N with Galois group N/K. More characterizations for a central Galois algebra VB(B K ) are given.
TL;DR: In this article, a proper polynomial map of arbitrary topological degree up to equivalence was investigated. But the results were restricted to the case where the maps are Galois coverings.
Abstract: Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow
\mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$.
We investigate proper polynomial maps of arbitrary topological degree $d \geq 2$ up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case $d=2$.
TL;DR: This work provides a rigorous method for deciding whether there is a polynomial p in P such that f is a factor of p and shows a method for computing a nearest polynomials q to p in a weighted l^~-norm such thatf is a factors of q.
TL;DR: In this paper, an equivariant analog of the Morse formula for a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v was shown.
Abstract: Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula $$ {\rm Ind}^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\partial_{k}^{+}X) $$ which takes its values in A(G). Here IndG(v) denotes the equivariant index of the field v, $\{\partial_{k}^{+}X\}$ the v-induced Morse stratification (see [10]) of the boundary ∂X, and $\chi^G(\partial_{k}^{+}X)$ the class of the (n - k)-manifold $\partial_{k}^{+}X$ in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas ([3, 4]).
01 Jan 2009
TL;DR: In this article, it was shown that Δ is a Galois extension of Δ G with Galois group induced by and isomorphic with G/N where N = {g ∈ G |g(x )= x for all x ∈ Δ}.
Abstract: Let B be a Galois extension of B G with Galois group G, Δ the commutator subring of B G in B, and G|Δ the restriction of G to Δ. Equivalent conditions are given for a Galois extension Δ of Δ G with Galois group G|Δ. It is shown that the following statements are equivalent: (1) Δ is a Galois extension of Δ G with Galois group induced by and isomorphic with G/N where N = {g ∈ G |g(x )= x for all x ∈ Δ}. (2) B G Δ is a Galois extension of B G with Galois group induced by and isomorphic with G/N and Δ is a finitely generated and projective module over Δ G . (3) B is a composition of two Galois extensions: B ⊃ B G Δ with Galois group N and B G Δ ⊃ B G with Galois group induced by and isomorphic with G/N such that Δ is a finitely generated and projective module over Δ G . Consequently, more results can be derived for several well known classes of Galois extensions such as DeMeyer-Kanzaki Galois extensions, Azumaya Galois extensions, and Hirata separable Galois extensions.
TL;DR: In this article, it was shown that the fixed point group-subgroup subfactor N G ⊆ N H is conjugate to the group subfactor n G / H ⊈ N.
Abstract: We review some concepts of Galois theory for subfactors N ⊆ M , computing some Galois groups and correspondences in this framework. Given an outer action of a group G on a II 1 von Neumann factor N and a normal subgroup H of G, we prove that the fixed point group–subgroup subfactor N G ⊆ N H is conjugate to the group–subfactor N G / H ⊆ N .
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TL;DR: In this paper, the field isomorphism problem of cubic generic polynomial over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations was studied.
Abstract: We study the field isomorphism problem of cubic generic polynomial $X^3+sX+s$ over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations $x^3-2mx^2y-9mxy^2-m(2m+27)y^3=\lambda$ where $\lambda^2$ is a divisor of $m^3(4m+27)^5$.