Principal Homogeneous Spaces Research Articles

Algebraic families of nonzero elements of Shafarevich-Tate groups

Principal homogeneous spaces under an abelian variety defined over a number field k k may have rational points in all completions of the number field without having rational points over k k . Such principal homogeneous spaces represent the nonzero elements of the Shafarevich-Tate group of the abelian variety. We produce nontrivial, one-parameter families of such principal homogeneous spaces. The total space of these families are counterexamples to the Hasse principle. We show these may be accounted for by the Brauer-Manin obstruction.

Divisors on real algebraic varieties without real points

The image of the Picard group of a nonsingular projective variety X over R into the Galois-invariant part of the Picard group of X⊗C is described in terms of cohomological invariants of the analytic manifold X(C) with its antiholomorphic involution. This description gives for example a link between the level of the function field of X and the topological level of (Zariski-open subsets of) X(C); it easily follows that a nontrivial principal homogeneous space over a real abelian variety has a function field of level 2.

Torus quotients of homogeneous spaces — II

We study torus quotients of principal homogeneous spaces. We classify the Grassmannians for which semi-stable=stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring ofT invariants of the homogeneous co-ordinate ring of the GrassmannianG 2,n (n odd) over the ring generated byR 1, the first graded part of the ring ofT invariants.

Projective Module Description of the q -Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern–Connes pairing of cyclic cohomology and K-theory is computed for the winding number −1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf–Galois extensions) their associated covariant derivatives on projective modules.

On p-adic height pairings and locally free classgroups of Hopf orders

Let E be an elliptic curve with complex multiplication by the ring of integers [Ofr ] of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]; we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

Tame Realisable Classes over Hopf Orders

LetObe the ring of algebraic integers in a number fieldKand letGbe a finite abelian group. McCulloh has characterised the classes in the locally free classgroup Cl(OG) which are realisable by rings of integers of tame normal extensions ofKwith Galois groupG. We extend this to certain extensions which in general need be neither tame nor normal by replacingOGwith a Hopf order A and introducing the strongly A-tame orders as an analogue of tame rings of integers. We also describe the classes realised by principal homogeneous spaces over the dual of A.

An elememtary approach to the abelianization of the Hitchin system for arbitrary reductive groups

We consider the moduli space M of stable principal G-bundles over a compact Riemann surface C of genus g ≥ 2, G being any reductive algebraic group and give an explicit description of the generic fibre of the Hitchin map H: T*M → K. If T ⊂ G is a fixed maximal torus with Weyl group W, for each given generic element φ ∈ K one may construct a W-Galois covering ~C of C and consider the generalized Prym variety P=HomW(X(T),J(~C)), where X(T) denotes the group of characters on T and J(C) the Jacobian. The connected component P0⊂ P which contains the trivial element is an abelian variety. In the present paper we use the classical theory of representations of finite groups to compute dim P = dim M. Next, by means of mostly elementary techniques, we explicitly construct a finite map F from each connected component H–1(φ)c of the Hitchin fibre to P0 and study its degree. In case G=PGl(2) one has that the generic fibre of F:H–1()c→ P0 is a principal homogeneous space with respect to a product of (2d-2) copies of Z/2Z where d is the degree of the canonical bundle over C. However if the Dynkin diagram of G does not contain components of type Bl, l≥ 1 or when the commutator subgroup (G,G) is simply connected the map F is injective.

An alternative proof of Scheiderer's theorem on the Hasse principle for principal homogeneous spaces

We give an alternative proof of the Hasse principle for principal homogeneous spaces defined over fields of virtual cohomological dimension at most one which is based on a special decomposition of elements in Chevalley groups.

Remarks on Galois cohomology and definability

In this paper we develop some basic features of Galois cohomology, specifically the connection between first Galois cohomology groups and principal homogeneous spaces, in a model-theoretic context. “Descent theory” also fits into our approach.The model theory involved is elementary, and the reader is referred to [2]. It should be said that we make crucial use of Meq in our analysis. The reader is also referred to Poizat's seminal paper “Une theorie de Galois imaginaire” ([6]). Although our results do not depend on Poizat's work, it is in his paper that the model-theoretic context is suggested for a generalised treatment of Galois theory.Nothing in this paper is particularly deep. We are concerned mainly with translating between the Galois cohomological language and the language of definable sets and definable families of definable sets. We will introduce (in a suitable context) the notion of a definable cocycle (from an automorphism group to a definable group G). The (classical) situation of profinite and continuous cocycles will be a special case. Kolchin's theory of constrained cohomology will be another special case, and our results yield a substantially simpler proof of his Theorem 5 from Chapter VII of [4]. In any case model-theorists will see that definable cocycles correspond to objects with which they are already quite familiar—commuting families of definable bijections.

Principe de Hasse pour les espaces principaux homogènes sous les groupes classiques sur un corps de dimension cohomologique virtuelle au plus 1

Letk be a perfect field of virtual cohomological dimension at most 1. LetG be a semi-simple, simply connected classical group. In this paper it is proved that a principal homogeneous space underG which is trivial over every real closure ofk is trivial.

Hopf Galois Kummer theory of formal groups

Let ƒ : F → G be an isogeny between finite n -dimensional formal groups defined over R , the valuation ring of some field extension K of Q p . Let H be the R -Hopf algebra which arises from this isogeny. For such H , we classify Gal( H ), the group of H -Galois objects. Let M be the maximal ideal of R , and let P ( F, K ) denote the n -tuples of M under the group operation induced by F . Our main result is the construction of an isomorphism from the cokernel of P ( ƒ ) to Gal( H ), where P ( ƒ ) is the induced map from P ( F, K ) to P ( G, K ). In geometric language Gal( H ) describes the group of isomorphism classes of principal homogeneous spaces for Spec( H ) over Spec( R ). Geometric methods have been used by Mazur to establish the above isomorphism, but the proof is nonconstructive. A geometric approach has also provided a formula for the cardinality of Gal( H ). We give an alternative derivation of this result using formal group techniques.

Hopf Orders and a Generalization of a Theorem of L. R. McCulloh

Let K be an algebraic number field with ring of integers O and let G be an elementary abelian group of order l k . Let A be a Hopf order in KG and let B be its dual. An order X in a Galois G-extension of K is a semilocal principal homogeneous space over B if X l is a principal homogeneous space over B l and X is integrally closed away from l. We define a map ψ from the group of such X to the locally free classgroup Cl( A ). Assuming that A admits C ≅ F l k ×⊆ Aut( G), we describe the image of ψ in terms of a Stickelberger ideal in Z C. This generalizes a result of L. R. McCulloh on the classes in Cl( O G) realized by tame rings of integers.

Espaces homogènes principaux, unités elliptiques et fonctions $L $

We study the structure of principal homogeneous spaces associated with elements of the Selmer group of an elliptic curve with complex multiplication. Following ideas of Rubin we use elliptic units to construct principal homogeneous spaces with non trivial Galois module structure. This construction provides a new link between a problem of Galois module structure and p-adic L-functions.

Tate-Shafarevich products in elliptic curves over pseudolocal fields with residue fields of characteristic 3

Let k be a general local field with pseudolocal residue field x, char x=3, and A an elliptic curve defined over k. It is proved that the Tate-Shafarevich product H1(k, A)×Ak→ Q/ℤ of the group H1(k, A) of principal homogeneous spaces of the curve A over k and the group Ak of its k-rational points is left nondegenerate.

A local determination theorem for finite principal homogeneous spaces

A local determination theorem for finite principal homogeneous spaces

Principal homogeneous spaces for arbitrary Hopf algebras

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

Higher dimensional obstruction theory in algebraic categories

In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr-Beck's cotriple cohomogy HG2, to higher dimensions by giving a new interpretation of HGn+1 in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to HGn+1(S,A) in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over HGn(S,A).

Principal homogeneous spaces over Hensel rings

We prove that if ( A , a _ ) (A,\underline {a}) is a Hensel couple and G G is an affine, smooth group scheme over A A then H e t 1 ( A , G ) = H e t 1 ( A / a _ , G / a _ G ) H_{\mathrm {et}}^1 (A, G) = H_{\mathrm {et}}^1 (A/\underline {a}, G/\underline {a} G) .

Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem

On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group SL(3,ℝ), which is automorphic with respect to the discrete group SL(3,ℤ). The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla. From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands. The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions σs(n)=∑d¦n,d>od5 make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as Χ → ∞ of the sum ∑n⩽Χτ3(n)τ3(n+κ), where τ3(n) is the number of solutions of the equation d1d2d3=n in natural numbers. Part II of the present work will be devoted to this binary problem. At the end of the paper properties of special functions used in Theorem 1 are discussed.

The order and index of a principal homogeneous space for an elliptic curve over a general local field

The order and index of a principal homogeneous space for an elliptic curve over a general local field