Universal torsors and values of quadratic polynomials represented by norms

Abstract

Let K /k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N K /k (z). We study the Hasse principle and weak approximation for X in three cases. For [K : k] = 4 and P(t) irreducible over k and split in K , we prove the Hasse principle and weak approximation. For k = Q with arbitrary K , we show that the BrauerManin obstruction to the Hasse principle and weak approximation is the only one. For [K : k] = 4 and P(t) irreducible over k, we determine the Brauer group of smooth proper models of X . In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

Rational points and zero-cycles on rationally connected varieties over number fields

“…Derenthal, Smeets and Wei [DSW15] show that for any number field L and any quadratic polynomial P (t) ∈ Q[t], the set X(Q) is dense in X(A Q ) Br(X) if X denotes a smooth and proper model of the affine variety defined by N L/Q (x) = P (t). Their proof consists in applying Theorem 3.10 and Corollary 3.8 to this affine variety: the universal torsors turn out to be precisely the varieties studied by Browning and Heath-Brown [BHB12] by sieve methods (using ideas from the circle method).…”

On the equation N K /k (Ξ)=P (t )

“…A positive answer was given by Colliot-Thélène, Sansuc and Swinnerton-Dyer in their remarkable paper [16] when P (t) has degree 4 and K/k is a quadratic extension. A positive answer is also known for the following cases: the extension K/k has degree 3 and the polynomial P (t) has degree at most 3 [10]; the polynomial P (t) having just two roots in k and K/k arbitrary [8,23,39]; k = Q, P (t) has degree 2 and K/Q arbitrary [1,20]; k = Q, P (t) is split in In this section, we shall handle three new classes of fibrations over the projective line whose generic fibre is birationally a principal homogeneous space under a torus. In each of these classes one of conditions (a) or (b) is not in general fulfilled.…”

On the fibration method for zero-cycles and rational points

“…, ω n ) of the k-vector space K. The study of the arithmetic of a smooth and proper model X of V has a long history. Conjecture 1.2 was previously known for X only when K/k is cyclic (see [CTSSD98], which builds on [Sal88]), when K/k has prime degree or degree pq for distinct primes p, q (see [Wei14a,Theorem 4.3] and [Lia15, Example 3.1, Remark 3.7]), when K/k is quartic or is abelian with Galois group Z/nZ×Z/nZ under certain assumptions on P (t) (see [Wei14a], [DSW14], [Lia14,Corollary 2.3] and [Lia15, Example 3.9]), and, finally, for arbitrary K/k, when P (t) = ct m (1 − t) n for some c ∈ k * and some integers m, n (see [SJ13], which builds on [HBS02] and [CTHS03] and applies descent theory and the Hardy-Littlewood circle method to establish Conjecture 1.1 for X over every finite extension of k, and see [Lia15, Example 3.10]). By contrast, it follows uniformly from Theorem 1.3 that X satisfies Conjecture 1.2 for any K/k and any P (t).…”