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

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.