The Brauer group of an affine double plane associated to a hyperelliptic curve

We study the Brauer group of an affine double plane π:X→𝔸² defined by an equation of the form z² = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ?¹×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y²?p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.     

The equivariant Brauer group of a group

We consider the Brauer group of a group (finite or infinite) over a commutative ring with identity. A split exact sequence

is obtained. This generalizes the Fröhlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of to the infinite case of . Here is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.

     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of on the real tensor powers of its natural module, , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for and with the decomposition of into irreducible submodules is considered.

     

Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group

In this article necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the dihedral group.     

The Brauer group of Yetter-Drinfel'd module algebras

Let be a Hopf algebra with bijective antipode. In a previous paper, we introduced -Azumaya Yetter-Drinfel'd module algebras, and the Brauer group classifying them. We continue our study of , and we generalize some properties that were previously known for the Brauer-Long group. We also investigate separability properties for -Azumaya algebras, and this leads to the notion of strongly separable -Azumaya algebra, and to a new subgroup of the Brauer group .

     

The Brauer group of a noncomplete real algebraic surface

The Brauer group of a noncomplete real algebraic surface is calculated. The calculations make use of equivariant cohomology. The resulting formula is similar to the formula for a complete surface, but the proof is substantially different. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 355–359, March, 2000.     

Subfields of Nondegenerate Tame Semiramified Division Algebras

We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of the center (Theorems 2.5, 2.12, and Proposition 2.16).     

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a generic mean zero locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss the relations of this result with the multiresolution wavelet analysis. The text was submitted by the authors in English.     

Nondegenerate Semiramified Valued and Graded Division Algebras

The main goal of this article is to give examples of division p-algebras that are not tensor product of cyclic algebras (Corollary 2.19) and to prove that nondegenerate tame semiramified division algebras of prime power degree over a Henselian valued field are indecomposable (Theorem 3.5). For this, we give new results concerning nicely semiramified division algebras over Henselian valued fields, and we develop a new study for nondegenerate valued and graded division algebras.     

On the homology groups of the Brauer complex for a triquadratic field extension

The homology groups , , and of the Brauer complex for a triquadratic field extension are studied. In particular, given , we find equivalent conditions for the image of D in to be zero. We consider as well the second divided power operation , and show that there are nonstandard elements with respect to γ₂. Further, a natural transformation , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension such that the cohomology group of the Brauer complex contains the images of prescribed elements of , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions with nontrivial are given. As a consequence we show that the homology group can be arbitrarily big.     

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism which splits , where is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over .     

Computing the -Selmer group of an elliptic curve

In this paper we explain how to bound the -Selmer group of an elliptic curve over a number field . Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most . Our method is practical for , but for larger values of it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the -Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of -descent fails.