Zero-Loci of Brauer Group Elements on Semi-Simple Algebraic Groups

Loughran, Daniel; Takloo-Bighash, Ramin; Tanimoto, Sho

doi:10.1017/s1474748018000440

Cited by 9 publications

(17 citation statements)

References 33 publications

(76 reference statements)

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“…However F 2 \ Q 2 ∼ = A 1 has constant Brauer group, so b in fact extends to all of F 2 . As F 2 meets L transversely, we deduce from [LTBT18,Prop. 4.15] that the evaluation of the residue of b at Q 2 is also trivial, so that f −1 (Q 2 ) consists of exactly 2 rational points.…”

Section: Brauer Group Of Affine Markoff Surfacesmentioning

confidence: 90%

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: Brauer Group Of Affine Markoff Surfacesmentioning

confidence: 90%

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

Loughran

Mitankin

2020

International Mathematics Research Notices

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“…δ S,D (g )H(s, g ) −1 E (g , φ) d g , we will follow and briefly sketch the argument presented in section 4.5 of [19]. At any finite place v we let C ∞ c (G(F v )) denote, as usual, the space of functions on G(F v ) that are locally constant and of compact support.…”

Section: G(a)mentioning

confidence: 99%

“…As in the case of rational points, uniform estimates need to be established. In [24], a technical assumption on local height functions was made to simplify the computations, but a paper of Loughran, Tanimoto, and Takloo-Bighash ( [19]) showed how to remove this restriction. We use their results in this paper.…”

Section: Introductionmentioning

confidence: 99%

The Distribution of Integral Points on the Wonderful Compactification by Height

Chow¹

2019

Preprint

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“…Serre's problem [15] regards the density of elements in a family of varieties defined over Q that have a Q-rational point. Special cases have been considered by Hooley [5,6] Poonen-Voloch [11], Sofos [17], Browning-Loughran [2], and Loughran-Takloo-Bighash-Tanimoto [9]. The recent investigation of Loughran [7] and Loughran-Smeets [8] provides an appropriate formulation of the problem and proves the conjectured upper bound in considerable generality.…”

Section: Introductionmentioning

confidence: 96%

“…• the base of the fibration is a wonderful compactification of an adjoint semi-simple algebraic group (Loughran-Takloo-Bighash-Tanimoto [9]).…”

Section: Introductionmentioning

confidence: 99%