COUNTEREXAMPLES TO THE HASSE PRINCIPLE IN FAMILIES

We generalise two quartic surfaces studied by Swinnerton-Dyer to give two infinite families of diagonal quartic surfaces which violate the Hasse principle. Standard calculations of Brauer–Manin obstructions are exhibited.

Thue equations that simultaneously fail the Hasse principle

We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x, y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree ≥ 3 and are ordered by the absolute value of the maximum of the coefficients.

Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

Integral Hasse Principle and Strong Approximation for Markoff Surfaces

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.

Systems of cubic forms in many variables

We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided {n\geq 25R}, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in {25R} variables, previous work having required that {n\gg R^{2}}. One conjectures that {n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.