Comparing descent obstruction and Brauer-Manin obstruction for open varieties

Arithmetic purity of strong approximation for homogeneous spaces

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces X X under a linear group G G (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of X X are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients G / H G/H , where G G and H H are connected algebraic groups and H H is linear, and certain abelianization morphisms associated with these spaces.

Arithmetic of 0-cycles on varieties defined over number fields

We study gerbes \(\mathcal X\) over a field banded by groups of multiplicative type \(A\) and prove the formula \({\text {ed}}_{p}(\mathcal X)={\text {cdim}}_{p}(\mathcal X)+{\text {ed}}_{p}(A)\) when \(A\) is the kernel of a homomorphism of algebraic tori \(Q\rightarrow S\) with \(Q\) invertible and \(S\) split. Here \(p\) is either a prime or \(0\). This result is applied to prove new results on the essential dimension of algebraic groups.

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely accounted for by the Brauer–Manin obstruction. We investigate how often the family of quadric hypersurfaces ax 2 +by 2 +cz 2 =n has a Brauer–Manin obstruction where a,b,c,n are integers. We improve previous bounds of Mitankin [7].

For a homogeneous space with connected or abelian stabilizer of a connected linear algebraic group defined over a number field, a cohomological obstruction to the Hasse principle is defined in terms of Galois hypercohomology with coefficients in a complex of two abelian algebraic groups. This obstruction is proved to be the only obstruction to the Hasse principle. It is proved that up to sign this cohomological obstruction coincides with the Brauer-Manin obstruction.

OF THE DISSERTATION The Brauer-Manin Obstruction on Families of Hyperelliptic Curves by Thom Tyrrell Dissertation Director: Jerrold Tunnell In [19], Manin introduced a way to explain the failure of the Hasse principle for algebraic varieties over a number field. For curves, the problem of whether all such failures can be explained by this method is open. In this thesis, we construct unramified quaternion algebras that obstruct the existence of rational points on families of curves admitting maps to elliptic curves. In Chapter 1, we introduce the Brauer group of a projective variety. For curves, we relate the Brauer group to torsors over the Jacobian of the curve, and establish some properties of the Brauer group over local fields that will simplify later computations. In Chapter 2, we define the Brauer-Manin obstruction and show that it explains all failures of the Hasse principle for genus 1 curves and curves without a degree 1 rational divisor. After reviewing a topological and more computational characterization of the obstruction, we note that it suffices to consider curves that admit maps to positive rank abelian varieties. Having established the obstruction, we introduce the motivating example for this thesis a construction by D.Quan [21] of a genus 11 hyperelliptic curve and an unramified quaternion algebra that obstructs the existence of rational points on it.

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

Consider strong approximation for algebraic varieties defined over a number field k k . Let S S be a finite set of places of k k containing all archimedean places. Let E E be an elliptic curve of positive Mordell–Weil rank and let A A be an abelian variety of positive dimension and of finite Mordell–Weil group. For an arbitrary finite set T \mathfrak {T} of torsion points of E × A E\times A , denote by X X its complement. Supposing the finiteness of ⨿ ⨿ ( E × A ) {\amalg \kern -.25pc\amalg } (E\times A) , we prove that X X satisfies strong approximation with Brauer–Manin obstruction off S S if and only if the projection of T \mathfrak {T} to A A contains no k k -rational points.

We study the failure of the integral Hasse principle and strong approximation for Markoff surfaces, as studied by Ghosh and Sarnak, using the Brauer–Manin obstruction.

On a projective variety defined over a global field, any Brauer–Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.

Let G be a connected linear algebraic group over a number field k. We establish an exact sequence describing the closure of the group G(k) of rational points of G in the group of adelic points of G. This exact sequence describes the defect of strong approximation on G in terms of the algebraic Brauer group of G. In particular, we deduce from those results that the integral Brauer-Manin obstruction on a torsor under the group G is the only obstruction to the existence of an integral point on this torsor. We also obtain a non-abelian Poitou-Tate exact sequence for the Galois cohomology of the linear group G. The main ingredients in the proof of those results are the local and global duality theorems for complexes of k-tori of length two and the abelianization maps in Galois cohomology introduced by Borovoi.

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

It is proved that the algebra of the general covariance group x/sub j/ = f/syb i/ (x/sub i/, x/sub 2/, x/sub 3/, x/sub 4/) turns out to be the closure of algebras of the linear and conformal groups, i.e. that any generator of the general covariance group is representable as some linear combination of repeated commutators of generators SL/sub 4,R/, and C/sub 15/. (LBS)