Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II (2310.06813v1)

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 $K$, Castella and Wan formulated the plus and minus Heegner point main conjecture for $E$ along the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved it for semistable curves. We generalize their results to two settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type main conjectures for $\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove it under some conditions. 2. We formulate, relying on the recent work of the first-named author with Kobayashi and Ota, plus and minus Heegner point main conjectures for elliptic curves when $p$ is inert in $K$, and prove the minus main conjecture for semistable elliptic curves. These results yield a $p$-converse to the Gross--Zagier and Kolyvagin theorem for $E$. Our method relies on Howard's framework of bipartite Euler system, Zhang's resolution of Kolyvagin's conjecture and the recent proof of Kobayashi's main conjecture.