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

TL;DR

This paper extends anticyclotomic Iwasawa theory results for abelian varieties of GL2-type at non-ordinary primes, proving main conjectures and establishing a p-converse to the Gross–Zagier and Kolyvagin theorems.

Contribution

It formulates and proves new main conjectures for GL2-type abelian varieties at non-ordinary primes in both split and inert cases, generalizing prior results.

Findings

Proved Sprung-type main conjectures for GL2-type abelian varieties at non-ordinary primes.

Established a p-converse to the Gross–Zagier and Kolyvagin theorems for semistable elliptic curves.

Extended the framework of bipartite Euler systems to new settings.

Abstract

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the plus/minus Heegner point main conjectures for $E$ along the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved them for semistable curves. We generalize their results to two settings: 1. For $p$ split in $K$, we formulate Sprung-type main conjectures for $\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove them under some conditions. 2. For $p$ inert in $K$, we formulate, relying on the work of the first-named author with Kobayashi and Ota, plus/minus Heegner point main conjectures for elliptic curves, and prove the minus main conjecture for semistable curves. The latter yields a $p$-converse to the Gross--Zagier and Kolyvagin theorem for semistable elliptic curves $E$ at supersingular primes $p\geq 5$, complementing the pioneering $p$-converse theorems of Skinner and Zhang. Our method relies on Howard's framework of bipartite Euler systems, Zhang's resolution of Kolyvagin's conjecture and the recent proof of cyclotomic main conjecture at non-ordinary primes.