相关论文: Cubic surfaces failing the integral Hasse principl…

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic…

Following recent works by E. Fuchs et al. and by the author, we study rational and integral points on Markoff-type K3 (MK3) surfaces, i.e., Wehler K3 surfaces of Markoff type. In particular, we construct a family of MK3 surfaces which have…

In this paper, we construct three kinds of Ch\^atelet surfaces, which have some given arithmetic properties with respect to field extensions of number fields. We then use these constructions to study the properties of weak approximation…

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle…

We show that the Brauer-Manin obstruction is the only obstruction to strong approximation for all stacky curves over global fields with finite abelian fundamental groups. This includes all stacky curves of genus $g = \frac{1}{2}$, thus…

We construct an Enriques surface X over Q with empty \'etale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on…

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This covers the vast majority of the cases when this group is non-zero. Our approach is…

For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, we show that the Brauer-Manin obstruction controls weak approximation. The proof is based on the Hasse…

We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a…

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if…

In 2011, V\`arilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an \'etale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the…

We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class A that is unramified at…

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive…

Fix $k,s,n\in \mathbb N$, and consider non-zero integers $c_1,\ldots ,c_s$, not all of the same sign. Provided that $s\ge k(k+1)$, we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine…

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the…

We study Brauer-Manin obstructions to integral points on open subsets of the projective plane.

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular…