On signed Mordell-Weil groups for abelian varieties

In this short note, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by B\"uy\"ukboduk--Lei, we define some objects called ``the multi-signed Mordell-Weil groups" for supersingular abelian varieties, make comments on the structure of the dual of these groups as an Iwasawa module and show a (weak) control theorem. This recovers the case of elliptic curves over $\mathbb{Q}$ non-ordinary at the prime $p$ with $a_p=0$ studied by Antonio Lei. Using the multi-signed Mordell-Weil groups we define what we call ``the multi-signed Tate-Shafarevich groups" along the cyclotomic tower of $\mathbb{Q}$. Finally we pose some open questions related to our newly defined objects and make a remark on the asymptotic growth of these multi-signed Tate-Shafarevich groups along the cyclotomic tower using an idea of Meng Fai Lim.