Manin’s conjecture for forms of additive groups

Abstract

This thesis investigates two topics in the area of arithmetic geometry, particularly in the study of rational points on varieties over global fields. In the first part of the thesis, we study the asymptotic behaviour of rational points of bounded height on a class of equivariant compactifications of commutative unipotent groups over global function fields. More precisely, we prove the Batyrev–Manin conjecture for smooth equivariant compactifications of forms of additive groups, under suitable assumptions on the boundary divisor. To verify that the leading constant agrees with Peyre's prediction, we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. To illustrate new phenomena appearing in the function field setting, we study in detail the case of projective space of dimension p-1, viewed as a compactification of a certain wound group, where p is the characteristic of the base field. The second part of the thesis concerns the study of Brauer groups of surfaces over a field k of characteristic zero, building towards the study of potential Brauer-Manin obstructions. First, we determine the Brauer group of regular conic bundles over elliptic curves, under the assumption that the singular fibres lie above k-points that are divisible by 2 in E(k), where the associated ramification fields are isomorphic. Second, we study affine surfaces obtained as complements of singular hyperplane sections of smooth cubic surfaces. We compute the transcendental Brauer group for the different possibilities of the hyperplane section. For the case when the hyperplane section is geometrically the union of three lines, we give explicit examples where transcendental elements of order 2 and 3 exist over Q.