On the structure of signed Selmer groups

Let $F$ be a number field unramified at an odd prime $p$ and $F_{\mathrm{\infty }}$ be the ${\mathbf{Z}}_p$-cyclotomic extension of $F$. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, B"uy"ukboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{F}\mathrm{/}F)$-representations. In particular, their construction applies to abelian varieties defined over $F$ with good supersingular reduction at primes of $F$ dividing $p$. Assuming that these Selmer groups are cotorsion ${\mathbf{Z}}_p[[\mathrm{G}\mathrm{a}\mathrm{l}(F_{\mathrm{\infty }}\mathrm{/}F)]]$-modules, we show that they have no proper sub-${\mathbf{Z}}_p[[\mathrm{G}\mathrm{a}\mathrm{l}(F_{\mathrm{\infty }}\mathrm{/}F)]]$-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler-Poincar'e characteristic of these Selmer groups, we obtain an explicit formula on the size of the Bloch-Kato Selmer group attached to these representations. Furthermore, for two such representations that are isomorphic modulo $p$, we compare the Iwasawa-invariants of their signed Selmer groups.