Obstructions to the Hasse principle in families

Bright, Martin

doi:10.1007/s00229-018-1009-0

Cited by 8 publications

(6 citation statements)

References 16 publications

(61 reference statements)

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“…General methods were developed in [BBL16] and [Bri18] which, when they apply, show that almost all of the varieties in the family fail weak approximation and almost all also have no Brauer-Manin obstruction to the Hasse principle. These results do not apply here as they concern smooth projective varieties.…”

Section: Introductionmentioning

confidence: 99%

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

Loughran

Mitankin

2020

International Mathematics Research Notices

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We study the failure of the integral Hasse principle and strong approximation for Markoff surfaces, as studied by Ghosh and Sarnak, using the Brauer-Manin obstruction. Contents1. Introduction 1 2. Geometry of affine cubic surfaces 4 3. Geometry of projective Markoff surfaces 8 4. Brauer group of affine Markoff surfaces 9 5. The Brauer-Manin obstruction 13 References 27 Theorem 1.3. Assume that m ∈ Z is such that U m has a Brauer-Manin obstruction to the integral Hasse principle. ThenThis qualitative statement shows that there is a Brauer-Manin obstruction to the integral Hasse principle for at most O(B 1/2 ) of the surfaces U m for m ∈ Z with |m| ≤ B. A more in-depth analysis of the Brauer-Manin obstruction allows us to prove that not only is m − 4 a product of small primes times a square, but all the prime divisors of m − 4 must satisfy very strong congruence conditions. This allows us to show the improved upper bound O(B 1/2 /(log B) 1/2 ), which is sharp by the following theorem.Theorem 1.4. We have Geometry of affine cubic surfacesBy an affine cubic surface, we mean an affine surface of the formwhere f is a polynomial of degree of 3. The closure of U in P 3 is a cubic surface S. The complement H = S \ U is a hyperplane section on S. Much of the geometry of U can be understood in terms of the geometry of S and H. We begin with some basic remarks.2.1. Basic geometry.Lemma 2.1. Let S be a smooth cubic surface over a field k, let H be a hyperplane section and set U = S \ H. Then O(U)

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Section: Introductionmentioning

confidence: 99%

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

Loughran

Mitankin

2020

International Mathematics Research Notices

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“…Proof of Theorem 1.2. The statement follows from [4,Thm. 1.4], provided that the following hypotheses are met:…”

Section: The Family Of Cubic Surfaces and Local Solubilitymentioning

confidence: 99%

“…However, work of Swinnerton-Dyer [21] confirms it for a special family of diagonal cubic surfaces, conditionally under the assumption that the Tate-Shafarevich group of elliptic curves is finite. A recent investigation of Bright [4] has focused on families of varieties over number fields that have no Brauer-Manin obstruction to the Hasse principle. In §2.1 we shall check that the conditions of his main result are satisfied for the family of cubic surfaces (1.1), thereby leading to the following conclusion.…”

Section: Introductionmentioning

confidence: 99%

Many Cubic Surfaces Contain Rational Points

Browning

2017

Mathematika

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Abstract. Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points. §1. Introduction. This paper is concerned with the Hasse principle for the family of projective cubic surfaceswhere f, g ∈ Q [u, v] are binary cubic forms. Our main result shows that a positive proportion of these surfaces, when ordered by height, possess a Qrational point. First, we discuss the question of local solubility. The following result will be addressed in §2.2. THEOREM 1.1. Approximately 99% of the cubic surfaces (1.1), when ordered by height, are everywhere locally soluble.It has long been known that the Hasse principle does not always hold for cubic surfaces and so local solubility is not enough to ensure that the surface (1.1) has a Q-rational point. In the special case that f and g are diagonal there are many known counter-examples to the Hasse principle, the most famous being the surface 5x A recent investigation of Bright [4] has focused on families of varieties over number fields that have no Brauer-Manin obstruction to the Hasse principle. In §2.1 we shall check that the conditions of his main result are satisfied for the family of cubic surfaces (1.1), thereby leading to the following conclusion.

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“…Proposition 3.4 can be interpreted as saying that N Br (T ) is a subset of the set of weak Campana points of the Campana orbifold (P 3 , 1 2 D) where D = 3 i=0 1 2 {a i = 0} as defined in [PSTVA21,Definition 3.3]. This is also what one should expect from arguments like in [Bri18]. There is as of yet no Manin-type conjecture for the asymptotic behavior of weak Campana points.…”

Section: Introductionmentioning

confidence: 96%

Diagonal quartic surfaces with a Brauer-Manin obstruction

Santens¹

2022

Preprint

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In this paper we investigate the quantity of diagonal quartic surfaces a 0 X 4 0 + a 1 X 4 1 + a 2 X 4 2 + a 3 X 4 3 = 0 which have a Brauer-Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

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Obstructions to the Hasse principle in families

Abstract: Abstract. For a family of varieties over a number field, we give conditions under which 100% of members have no Brauer-Manin obstruction to the Hasse principle.

Cited by 8 publications

References 16 publications

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

Many Cubic Surfaces Contain Rational Points

Diagonal quartic surfaces with a Brauer-Manin obstruction

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