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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  (   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.

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