The Brauer-Manin obstruction on Kummer varieties and ranks of twists of abelian varieties

Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of density properties of rational points on the Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence of our results is the following. Fix an abelian variety A over k, and suppose that for some r > 0 the Brauer-Manin obstruction to weak approximation on the Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at least r. Hence if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded. This relates two significant open questions.