Effectivity of Brauer–Manin obstructions on surfaces

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.

Arithmetic of 0-cycles on varieties defined over number fields

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.

We generalize a result of Heath-Brown and Skorobogatov [Rational solutions of certain equations involving norms, Acta Math.189 (2002), 161–177] to show that a certain class of varieties over a number field k satisfies Weak Approximation and the Hasse Principle, provided there is no Brauer–Manin obstruction.

Let be a number field. For a variety over that satisfies weak approximation with Brauer–Manin obstruction, we study the same property for smooth projective models of its symmetric products. Based on the same method, we also explore the property of weak approximation with Brauer–Manin obstruction for norm varieties.

Let ϕ : X → X be a morphism of a variety defined over a number field K , let V ⊂ X be a K -subvariety, and let 𝒪 ϕ ( P ) = { ϕ n ( P ) : n ≥ 0 } be the orbit of a point P ∈ X ( K ) . We describe a local-global principle for the intersection V ∩ 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of V ( K ) are Brauer–Manin unobstructed for power maps on ℙ 2 in two cases: (1) V is a translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key tool in the proofs is the classical Bang–Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the étale Brauer–Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over \({{{\mathbb {Q}}}}\).

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].

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.

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.

Norm forms for arbitrary number fields as products of linear polynomials

Arithmetic purity of strong approximation for homogeneous spaces

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.

We give formulas for calculating the unramified Brauer group of a homogeneous space X X of a semisimple simply connected group G G with finite geometric stabilizer F ¯ \bar F over a wide family of fields of characteristic 0 0 . When k k is a number field, we use these formulas in order to study the Brauer–Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on X ( k v ) X(k_v) for every v v outside of an explicit finite set of nonarchimedean places of k k .