Showing papers on "Abelian extension published in 2008"
TL;DR: In this paper, it was shown that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ell, the Galois group of the splitting field of the ell-torsion of A is GSp 2g(Z/ell).
Abstract: Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ell, the Galois group of the splitting field of the ell-torsion of A is GSp_{2g}(Z/ell).
46 citations
TL;DR: In this paper, the authors compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show that these groups are isomorphic over suitable fields.
Abstract: We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of Picard-Vessiot extensions over fields with not necessarily algebraically closed subfields of constants.
41 citations
TL;DR: In this paper, a further refinement of the Brumer-stark conjecture is presented, where a conjectural formula for the exact value of u in Fp×/E∧ is given, where Fp denotes the completion of F at p and E denotes the topological closure of the group of totally positive units E of F.
Abstract: Let F be a totally real number field, and let p be a finite prime of F such that p splits completely in the finite abelian extension H of F. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a p-unit u in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to H/F. This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of u in Fp×/E∧, where Fp denotes the completion of F at p and E∧ denotes the topological closure of the group of totally positive units E of F. We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of u in Fp×
38 citations
TL;DR: A general description of the Galois group of a “pointed” normal extension in categorical Galois theory is examined and the connection betweenGalois theory and group homology is clarified.
Abstract: A general description of the Galois group of a “pointed” normal extension in categorical Galois theory is examined under the presence of a suitable commutator operation. In particular, using the Hopf formula for the second homology group of a group, the connection between Galois theory and group homology is clarified.
36 citations
TL;DR: In this paper, the algebraic theory of log abelian varieties was developed, which is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of the log-abelian variety.
Abstract: We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.
34 citations
TL;DR: In this article, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z ≥ 1,000 is shown to be in fact an Iwasawa module.
Abstract: Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z
p
-extensions of Q(μ
p
) and the Galois group $${\mathfrak{G}}$$
of the maximal unramified pro-p extension of Q
$${(\mu_{p^{\infty}})}$$
. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for $${\mathfrak{G}}$$
to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p < 1,000, Greenberg’s conjecture that X is pseudo-null holds and $${\mathfrak{G}}$$
is in fact abelian.
28 citations
TL;DR: In this article, the inner and outer Galois points with respect to a smooth plane curve C of degree d ≥ 4 in characteristic p > 0 were determined for the case when p ≠ 2.
Abstract: We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(pe + 1) of degree pe + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \({\mathbb{F}_{2}}\) -rational points in \({\mathbb{P}^{2}}\).
26 citations
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TL;DR: In this paper, a Picard-Vessiot theory for differential fields with non-algebraically closed fields of constants is described. And the main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over C(z)
Abstract: We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{R}$ occurs as a differential Galois group over $\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{C}(z)$.
25 citations
01 Jan 2008
TL;DR: In this paper, it was shown that for any m > 1, every finite solvable group that is a union of conjugates of m proper subgroups occurs as the Galois group of such a polynomial, and that the same result holds for all Frobenius groups.
Abstract: Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Qp 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 a union of conjugates of m proper subgroups. We prove that for any m > 1, every finite solvable group that is a union of conjugates of m proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with m = 2) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of Q(t).
23 citations
TL;DR: In this paper, it was shown that E nr n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes.
Abstract: Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E nr n whose coecients are built from the coecients of En and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E nr n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E nr with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.
20 citations
TL;DR: In this paper, it was shown that there exist curves where the above map has a section over each completion of a number field, but not over all the fields in the field. But this is not a generalization of Grothendieck's Section Conjecture.
Abstract: Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2.
Our result is a consequence of a more general investigation of the existence of sections for the projection of the \'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\'el\`ene and Sansuc.
TL;DR: In this article, the authors describe a construction which takes as input a profinite group, which when applied the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. They also discuss properties of the construction, including relationships with the representation theory of infinite discrete groups.
Abstract: We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a topologically finitely generated abelian absolute Galois group, and conjecture that it agrees for all geometric fields. We also discuss properties of the construction, including relationships with the representation theory of infinite discrete groups.
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TL;DR: In this paper, the authors compute all octic extensions of Q 2 and find that there are 1823 of them up to isomorphism, and compute the associated Galois group of each field, slopes measuring wild ramification, and other quantities.
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TL;DR: In this paper, a generalization of the large sieve to situations where the underlying groups are nonabelian is described, and several applications to the arithmetic of abelian varieties are given.
Abstract: We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.
TL;DR: In this paper, the fundamental group of the Galois cover of the torus was computed in ℂℙ8, where 𝕋 is the one-dimensional torus.
Abstract: This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover o...
TL;DR: In this paper, it was shown that R (M ) is a subgroup of Cl (M) when ξ p ∈ k and Γ = V ⋊ ρ C, where V is an F p -vector space of dimension r ⩾ 1, C a cyclic group of order p r − 1, and ρ a faithful representation of C in V; an example is the symmetric group S 3.
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TL;DR: In this paper, the authors give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action.
Abstract: We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action.
Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.
12 Feb 2008
16 Oct 2008
TL;DR: In this article, the authors introduce a class of elementary abelian p-extensions of local function fields of characteristic p, which they call one-dimensional and which should be considered no more complicated than cyclic degree p extensions, and show that they possess a Galois scaffold.
Abstract: A Galois scaffold is defined to be a variant of a normal basis that allows for an easy determination of valuation and thus has implications for the questions of the Galois module structure. We introduce a class of elementary abelian p-extensions of local function fields of characteristic p, which we call one-dimensional and which should be considered no more complicated than cyclic degree p extensions, and show that they, just as cyclic degree p extensions, possess a Galois scaffold.
TL;DR: In this paper, for every odd prime p satisfying certain mild technical hypotheses, the values of Artin L-functions are used to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L.
Abstract: Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L.
TL;DR: In this paper, it was shown that for any positive integer d there exists a Galois extension F/Q with Galois group D 2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p^d.
Abstract: Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p^d.
TL;DR: In this article, the Brauer ring of the trivial extension of a Galois field extension was shown to be isomorphic to the group ring of a Brauer group of the extension.
Abstract: In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.
01 Jan 2008
TL;DR: In this article, the 2-primary Coates-Sinnott Conjecture was adjusted by replacing the K-groups by motivic cohomology groups H 2, where H 2 is an abelian extension of number fields with Galois group G.
Abstract: Suppose that E/F is an abelian extension of number fields with Galois group G. The generalized Coates-Sinnott Conjecture predicts that for n 2 a natural higher Stickelberger ideal, defined using values of the associated L-functions evaluated at 1 n, annihilates the higher K-group K2n 2(oE) of the ring of integers oE in E. We concentrate in this paper on the 2-primary part of the Conjecture. We first use results on the 2-primary information provided by the special L-values to suggest an adjustment of the 2-primary part of the Conjecture by replacing the K-groups by motivic cohomology groups H 2
TL;DR: In this article, it was shown that a Galois extension of a finitely generated field K is not separably closed, and that M is not PAC over K, either.
Abstract: We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K.
TL;DR: The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups.
07 Apr 2008
TL;DR: In this article, an efficient quantum algorithm was proposed to test whether a set Γ is a solvable group, or is far from any solvable groups, with a query complexity polylogarithmic in the size of the set.
Abstract: Testing efficiently whether a finite set Γ with a binary operation ċ over it, given as an oracle, is a group is a well-known open problem in the field of property testing. Recently, Friedl, Ivanyos and Santha have made a significant step in the direction of solving this problem by showing that it it possible to test efficiently whether the input (Γ, ċ) is an Abelian group or is far, with respect to some distance, from any Abelian group. In this paper, we make a step further and construct an efficient quantum algorithm that tests whether (Γ, ċ) is a solvable group, or is far from any solvable group. More precisely, the number of queries used by our algorithm is polylogarithmic in the size of the set Γ.
TL;DR: In this paper, the authors define a Drinfel-defined module defined over a finite extension K of 𝔽q(T) and establish a uniform lower bound for the canonical height of a point of ϕ, rational over the maximal abelian extension of K, and thus solve the so-called Abelian version of the Lehmer problem.
Abstract: Let ϕ be a Drinfel'd module defined over a finite extension K of 𝔽q(T); we establish a uniform lower bound for the canonical height of a point of ϕ, rational over the maximal abelian extension of K, and thus solve the so-called abelian version of the Lehmer problem in this situation. The classical original problem (a non torsion point in 𝔾m(ℚab)) was solved by Amoroso and Dvornicich [1]. Soit ϕ un module de Drinfel'd defini sur une extension finie K de 𝔽q(T); nous demontrons une minoration uniforme pour la hauteur canonique d'un point de ϕ, rationnel sur l'extension abelienne maximale de K, et resolvons ainsi la version dite abelienne du probleme de Lehmer dans cette situation. Dans le cadre classique (un point d'ordre infini de 𝔾m(ℚab)), cette question a ete resolue par Amoroso et Dvornicich dans [1].
TL;DR: In this article, the authors use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers.
Abstract: We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover.