On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Abstract

Let F F be a number field unramified at an odd prime p p and let F ∞ F_\infty be the Z p \mathbf {Z}_p -cyclotomic extension of F F . Let A A be an abelian variety defined over F F with good supersingular reduction at all primes of F F above p p . Büyükboduk and the first named author have defined modified Selmer groups associated to A A over F ∞ F_\infty . Assuming that the Pontryagin dual of these Selmer groups is a torsion Z p [ [ Gal ( F ∞ / F ) ] ] \mathbf {Z}_p[[\textrm {Gal}(F_\infty /F)]] -module, we give an explicit sufficient condition for the rank of the Mordell-Weil group A ( F n ) A(F_n) to be bounded as n n varies.