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)
We show, conditionally on Schinzel's hypothesis, that the only obstruction to the integral Hasse principle for generalised affine Châtelet surfaces is the Brauer-Manin one.In the current set-up a ∈ Z is non-zero squarefree, P (t) ∈ Z[t] is separable as a polynomial over Q and N Q( √ a) (x, y) = N Q( √ a)/Q (x+ωy) is the norm form of Q( √ a) for the basis 1, ω over Q with ω = −(1 + √ a)/2 when a ≡ 1 mod 4 and ω = √ a when a ≡ 2, 3 mod 4. We shall refer to X as a generalised affine Châtelet surface. Let X be the integral model of X over Z given by the same equation, X(Z) its set of integral points and X(A Z ) = X(R) × p X(Z p ) its set of adeles. One can embed X(Z) diagonally in X(A Z ) and hence a necessary condition for the existence of integral points on X is X(A Z ) = ∅. We say that X satisfies the integral Hasse principle if this is also a sufficient condition. Colliot-Thélène and Xu [CTX09, §1] found that the Brauer-Manin obstruction [Man74] plays an important role when it comes to the existence of integral points on X. More precisely, given a Q-variety X there is a pairing between the Brauer group Br X of X and the set of Q-adeles X(A Q ) of X. This pairing allowed Manin to construct a sequence
Abstract. Quadric hypersurfaces are well-known to satisfy the Hasse principle. However, this is no longer true in the case of the Hasse principle for integral points, where counter-examples are known to exist in dimension 1 and 2. This work explores the frequency that such counter-examples arise in a family of affine quadric surfaces defined over the integers.
We study the distribution of the Brauer group and the frequency of the Brauer-Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals. We also study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical.
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