We consider the problem of counting the number of varieties in a family over Q with a rational point. We obtain lower bounds for this counting problem for some families over P 1, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.

In the case of the representation of an integer by an indefinite ternary quadratic form, the violation of the integral Hasse principle can be explained via the Brauer-Manin obstruction. In this note, we study the occurrences of this phenomenon for several families of non-diagonal ternary quadratic forms. 

We prove a Green-Tao type theorem for multiplicative functions.

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function n bar right arrow delta(omega(n)), where delta is an element of R \ {0} and where omega counts the number of distinct prime factors of n, as well as the function n bar right arrow vertical bar lambda(f)(n)vertical bar, where lambda(f)(n) denotes the Fourier coefficients of a primitive holomorphic cusp form. For this class of functions we show that after applying a W-trick, their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green-Tao-Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalizes work of Green and Tao on the Mobius function.

Given a number field K=Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) D P(t). Combining additive combinatorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.