Related papers: Rational points and derived equivalence

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a…

Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian…

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then…

We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

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…

Let F be a number field, and let F\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively…

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…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We show that even within a class of varieties where the Brauer--Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base…

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…

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…

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if…

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic…

In this paper we propose to use a relative variant of the notion of the \'{e}tale homotopy type of an algebraic variety in order to study the existence of rational points on it. In particular, we use an appropriate notion of homotopy fixed…

We give the first examples of smooth projective varieties $X$ over a finite field $\mathbb{F}$ admitting a non-algebraic torsion $\ell$-adic cohomology class of degree $4$ which vanishes over $\overline{\mathbb{F}}$. We use them to show…

Let X be a smooth, projective variety defined over a local field K. Following Manin, two K-points of X are called R-equivalent if they can be joined by a rational curve defined over K. The main result of this note shows that if there are…

In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor…

Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points.…

We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational…