Integral points for multi-norm tori

Wei, Dasheng; Xu, Fei

doi:10.1112/plms/pdr068

Cited by 22 publications

(46 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The results of [11] were then extended to incorporate more general connected algebraic groups in [3]. Similar results hold when X is a principal homogeneous space of an algebraic group of multiplicative type (see [35], [36]). In [17] Harari and Voloch conjecture that the Brauer-Manin obstruction is the only one for S-integral points on open subsets of P 1 S , but show that this does not hold for open subsets of elliptic curves.…”

Section: Introductionmentioning

confidence: 86%

Integral points on conic log K3 surfaces

Harpaz

2018

J. Eur. Math. Soc.

View full text Add to dashboard Cite

Adapting a powerful method of Swinnerton-Dyer, we give explicit sufficient conditions for the existence of integral points on certain schemes which are fibered into affine conics. This includes, in particular, cases where the scheme is geometrically a smooth log K3 surface. To the knowledge of the author, this is the first family of log K3 surfaces for which such conditions are established. 2-Descent for quadratic norm 1 toriLet k be a number field, S 0 a finite set of places of k and O S 0 the ring of S 0 -integers of k. In this section we will develop a 2-descent formalism whose goal is to yield sufficient conditions for the existence S 0 -integral points on schemes of the form ax 2 + by 2 = 1 (2.0.1)where a, b are a mutually coprime S 0 -integers. To this end we will study certain Tate-Shafarevich groups associated with 2.0.1, and define suitable Selmer groups to compute them. The main result of this section is Corollary 2.3.2 below, which will be used in §3.1 to find S 0 -integral points on schemes which are fibered into affine conics of type 2.0.1. PreliminariesLet d ∈ O S 0 be a non-zero S 0 -integer and let K = k( √ d) be the associated quadratic extension. We will denote by T 0 the set of places of K lying above S 0 . Let us assume that d satisfies the following condition: Assumption 2.1.1. For every v / ∈ S 0 we have val v (d) ≤ 1 and val v (d) = 1 if v lies above 2.The following useful lemma is by no means novel, but we could not find an explicit reference.Lemma 2.1.2. If Assumption 2.1.1 holds then the ring O T 0 is generated, as an O S 0module, by 1 and d.

show abstract

Section: Introductionmentioning

confidence: 86%

Integral points on conic log K3 surfaces

Harpaz

2018

J. Eur. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The reader will find many more examples of Brauer-Manin obstructions to the existence of integral points, or to strong approximation, in [CTX09, §8], [KT08b], [CTW12], [WX12], [WX13], [Gun13], [CTX13,§7], [Wei15], [JS16, § §6-8].…”

Section: What About Integral Points?mentioning

confidence: 99%

“…The quotient Br(X)/Br 0 (X) is finite in Theorem 3.4 (see Remark 2.4 (ii)), but in Theorem 3.5, it need not be (example: X = G m ). In the special case of torsors under tori, refinements of Theorem 3.5 involving only a finite subgroup of Br(X)/Br 0 (X), leading to concrete criteria for the existence of integral points, have been worked out by Wei and Xu, see [WX12], [WX13].…”

Section: Integral Pointsmentioning

confidence: 99%

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

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

View full text Add to dashboard Cite

We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory. p. 174]). Let X be a smooth, proper, geometrically irreducible variety defined over a number field k. If X is rationally connected, then X(k) is dense in X(A k ) Br(X) . Conjecture 2.3By "X is rationally connected", we mean that the variety X ⊗ k k is rationally connected in the sense of Campana, Kollár, Miyaoka and Mori, where k denotes an algebraic closure of k. Equivalently, for any algebraically closed field K containing k, two general K-points of X can be joined by a rational curve defined over K. We recall that examples of rationally connected varieties include geometrically unirational varieties (trivial), Fano varieties (see [Cam92], [KMM92]), and fibrations into rationally connected varieties over a rationally connected base (see [GHS03]).Conjecture 2.3 is wide open even for surfaces.Remarks 2.4. (i) The group Br 0 (X) = Im(Br(k) → Br(X)) does not contribute to the Brauer-Manin set: the global reciprocity law implies the equality X(A k ) B = X(A k ) B+Br 0 (X) for any subgroup B ⊆ Br(X). When Br(X)/Br 0 (X) is finite, the subset X(A k ) Br(X) ⊆ X(A k ) is therefore cut out by finitely many conditions. In particular, in this case, it is an open subset of X(A k ).(ii) When X is rationally connected, or, more generally, when X is simply connected and H 2 (X, O X ) = 0, an analysis of the Hochschild-Serre spectral sequence and of the Brauer group of X ⊗ k k implies that Br(X)/Br 0 (X) is finite (see [CTS13b, Lemma 1.1]). Presumably, this quotient should be finite under the sole assumption that X is simply connected, but this is out of reach of current knowledge. (As follows from [CTS13a] and [CTS13b, §4], the finiteness of the ℓ-primary torsion subgroup of Br(X)/Br 0 (X) is equivalent, when X is simply connected, to the ℓ-adic Tate conjecture for divisors on X.) (iii) Letting v be any archimedean place of k and noting that the image of the projection map X(A k ) Br(X) → X(k v ) is a union of connected components, we see

show abstract

“…On peut alors se demander si l'on peut déterminer de façon effective un sous-groupe fini de Br 1 (X) qui suffit pour décider la question de l'existence d'un point entier sur X . Pour un résultat dans cette direction dans le cas des tores, on renvoie aux travaux récents de Dasheng Wei et Fei Xu dans [35].…”

Section: Avec Ce Lemme L'élémentunclassified

Le défaut d'approximation forte dans les groupes linéaires connexes

Demarche

2010

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

Let G be a connected linear algebraic group over a number field k. We establish an exact sequence describing the closure of the group G(k) of rational points of G in the group of adelic points of G. This exact sequence describes the defect of strong approximation on G in terms of the algebraic Brauer group of G. In particular, we deduce from those results that the integral Brauer-Manin obstruction on a torsor under the group G is the only obstruction to the existence of an integral point on this torsor. We also obtain a non-abelian Poitou-Tate exact sequence for the Galois cohomology of the linear group G. The main ingredients in the proof of those results are the local and global duality theorems for complexes of k-tori of length two and the abelianization maps in Galois cohomology introduced by Borovoi.

show abstract

Integral points for multi-norm tori

Abstract: We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi‐norm tori. Several explicit examples are provided.

Cited by 22 publications

References 18 publications

Integral points on conic log K3 surfaces

Integral points on conic log K3 surfaces

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

Le défaut d'approximation forte dans les groupes linéaires connexes

Contact Info

Product

Resources

About