On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

The units-Picard complex and the Brauer group of a product

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under appropriate hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem for, e.g., certain classes of varieties over a field of characteristic zero.     

Brauer groups and cohomology

We explain in an elementary way an example showing that the Brauer group of a scheme X does not always coincide with the torsion of Received: 22 June 2004     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

The relative Brauer group of a cyclic cover of affine space

We study the finite-dimensional central division algebras over the rational function field in several variables over an algebraically closed field. We describe the division algebras that are split by the cyclic covering obtained by adjoining the nth root of a polynomial. The relative Brauer group is described in terms of the Picard group of the cyclic covering and its Galois group. Many examples are given and in most cases division algebras are presented that represent generators of the relative Brauer group.     

The relative Brauer group and generalized cross products for a cyclic covering of affine space

We study algebras and divisors on a normal affine hypersurface defined by an equation of the form zⁿ=f(_x1,…,_xm)$z^n=f(x_1,\dots ,x_m)$. The coordinate ring is T=k[_x1,…,_xm,z]/(zⁿ−f)$T=k[x_1,\dots ,x_m,z]/(z^n-f)$, and if R=k[_x1,…,_xm][f⁻¹]$R=k[x_1,\dots ,x_m]\left[f^{-1}\right]$ and S=R[z]/(zⁿ−f)$S=R\left[z\right]/(z^n-f)$, then S$S$ is a cyclic Galois extension of R$R$. We show that if the Galois group is G$G$, the natural map H¹(G,Cl(T))→H¹(G,Pic(S))$H^1(G,Cl\left(T\right))\to H^1(G,Pic\left(S\right))$ factors through the relative Brauer group B(S/R)$B(S/R)$ and that all of the maps are onto. Sufficient conditions are given for H¹(G,Cl(T))$H^1(G,Cl\left(T\right))$ to be isomorphic to B(S/R)$B(S/R)$. As an example, all of the groups, maps, divisors and algebras are computed for an affine surface defined by an equation of the form zⁿ=(y−_a1x)?(y−_anx)(x−1)$z^n=(y-a_{1}x)?(y-a_{n}x)(x-1)$.     

The Brauer-Manin obstruction for sections of the fundamental group

We introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension and establish Grothendieck’s section conjecture for an open subset of the Reichardt-Lind curve.     

Non-cuspidality outside the middle degree of ?-adic cohomology of the Lubin-Tate tower

In this article, we consider the representations of the general linear group over a non-archimedean local field obtained from the vanishing cycle cohomology of the Lubin-Tate tower. We give an easy and direct proof of the fact that no supercuspidal representation appears as a subquotient of such representations unless they are obtained from the cohomology of the middle degree. Our proof is purely local and does not require Shimura varieties.     

The Witt ring of a Brauer–Severi variety

For a Brauer–Severi variety X over a field k of characteristic not two, every symmetric bilinear space over X up to Witt equivalence is defined over k. Received: 2 February 1998     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

On the F-fundamental group scheme

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let FX be the absolute Frobenius morphism of X. For any vector bundle E→X, and any polynomial g with non-negative integer coefficients, define the vector bundle using the powers of FX and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that . We also investigate the group scheme defined by this neutral Tannakian category.     

Integral models in unramified mixed characteristic (0, 2) of hermitian orthogonal Shimura varieties of PEL type,Part II

We construct relative PEL type embeddings in mixed characteristic (0, 2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0, 2) of hermitian orthogonal Shimura varieties of PEL type.     

A criterion on the semisimple Brauer algebras

In this paper, we give a necessary and sufficient condition for a Brauer algebra to be semisimple.     

Galois module structure of unramified covers

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. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Deformation spaces of one-dimensional formal modules and their cohomology

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over equipped with a level-m-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the ?-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces Mm.