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

TL;DR

This paper investigates conditions under which the Mordell-Weil ranks of supersingular abelian varieties remain bounded in cyclotomic extensions, extending understanding of Selmer groups and Iwasawa theory.

Contribution

It provides explicit criteria ensuring bounded Mordell-Weil ranks for supersingular abelian varieties over cyclotomic extensions, based on properties of associated Selmer groups.

Findings

Boundedness of Mordell-Weil ranks under torsion assumptions

Explicit sufficient conditions for rank boundedness

Connection between Selmer groups and Mordell-Weil ranks

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 $p$. B\"uy\"ukboduk 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.