Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes

TL;DR

This paper extends anticyclotomic Iwasawa theory results for elliptic curves of L_2-type at non-ordinary primes, establishing new inclusion results for Selmer groups in split and inert prime cases.

Contribution

It generalizes previous plus/minus Iwasawa main conjectures to L_2-type abelian varieties and inert prime scenarios, proving analogous inclusion results.

Findings

Proved upper bounds for Selmer groups in split prime case without assuming a_p(E)=0.

Formulated and proved inclusion results for inert prime case based on recent theoretical developments.

Extended Iwasawa theory to broader classes of abelian varieties and primes.

Abstract

Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and Iovita formulated the plus and minus Iwasawa main conjectures for $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved one-sided inclusion: an upper bound for plus and minus Selmer groups in terms of the associated $p$-adic $L$-functions. We generalize their results to two new settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type Iwasawa main conjectures for abelian varieties of $\mathrm{GL}_2$-type, and prove an analogous inclusion. 2. We formulate, relying on the recent work of the first named author with Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves when $p$ is inert in $K$, and prove an analogous inclusion.