On signed Mordell-Weil groups for abelian varieties

Abstract

In this short note, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Büyükboduk--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.

To appear in Conference proceedings of ICCGNFRT (International Conference on Class Groups of Number Fields and Related Topics)