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

TL;DR

This paper extends the theory of multi-signed Selmer groups for non-ordinary motives to imaginary quadratic fields, proving a control theorem and analyzing their Iwasawa invariants over bf4p extensions.

Contribution

It generalizes existing control theorems for elliptic curves to non-ordinary motives over bf4p extensions of imaginary quadratic fields.

Findings

Established a control theorem for multi-signed Selmer groups over bf4p extensions.

Derived conditions for cotorsionness of Selmer groups over two-variable Iwasawa algebras.

Compared bf4p Iwasawa fcmf6 invariants for congruent Galois representations.

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 $p$.