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

Abstract

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

14 pages