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

holmes, david; pannekoek, ren´e;

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 the density properties
of rational points on the Kummer variety Km(Ar) 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(Ar) 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.