On the signed Selmer groups for motives at non-ordinary primes in $\mathbb{Z}_p^2$-extensions

Abstract

Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary motives. In particular, their construction applies to abelian varieties over $F$ with good supersingular reduction at all the primes of $F$ above $p$. In this article, we scrutinize the case in which $F$ is imaginary quadratic, and prove a control theorem (that generalizes Kim's control theorem for elliptic curves) of multi-signed Selmer groups of non-ordinary motives over the maximal abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the $\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition when these multi-signed Selmer groups are cotorsion over the corresponding two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa $\mu$-invariants of multi-signed Selmer groups over the $\mathbb{Z}_p^2$-extension for two such representations which are congruent modulo $p$.