Integral points on affine quadric 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.

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.

Arithmetic of 0-cycles on varieties defined over number fields

International audience

We obtain local inequalities for log canonical thresholds and multiplicities of movable log pairs. We prove the non-rationality and birational superrigidity of the following Fano varieties: a double covering of a smooth cubic hypersurface in branched over a nodal divisor that is cut out by a hypersurface of degree ; a cyclic triple covering of a smooth quadric hypersurface in branched over a nodal divisor that is cut out by a hypersurface of degree ; a double covering of a smooth complete intersection of two quadric hypersurfaces in branched over a smooth divisor that is cut out by a hypersurface of degree .

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.

Effectivity of Brauer–Manin obstructions on surfaces

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 \({{{\mathbb {Q}}}}\).

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 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.

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.

The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if S is a standard graded (±1)-skew polynomial algebra in n variables with n ≥ 3 and f = x 21 + ⋯ + x 2n ∈ S, then the derived category Db(qgr S/(f)) of the noncommutative scheme qgr S/(f) has a full strong exceptional sequence. The length of this sequence is given by n−2+2r where r is the nullity of a certain matrix over \({\mathbb{F}_2}\). As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of Db(qgr S/(f)) for n = 3, 4.

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.

Arithmetic purity of strong approximation for homogeneous spaces

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.