Let $A$ be an abelian variety defined over a number field $k$ and $p$ a prime number. Under some natural and not-too-stringent conditions on $A$ and $p$ we show that certain invariants associated to Iwasawa-theoretic $p$-adic Selmer groups control the Krull-Schmidt decompositions of the $p$-adic completions of the groups of points of $A$ over finite extensions of $k$.
"On the Krull-Schmidt Decomposition of Mordell-Weil Groups." Tokyo J. Math. 40 (2) 353 - 378, December 2017. https://doi.org/10.3836/tjm/1502179233