Kolyvagin's conjecture at non-ordinary primes

Abstract

Let $K$ be an imaginary quadratic field and let $p\ge 5$ be a prime that is unramified in $K$. Let ${\mathcal{A}}_f/\mathbb{Q}$ be an abelian variety of ${\mathrm{GL}}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In the case where $p$ is inert in $K$, we further assume that ${\mathcal{A}}_f$ is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic ${\mathbb{Z}}_p$-extensions that applies when the corresponding Selmer modules have arbitrary $\Lambda$-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale--Castella--Grossi--Skinner, Castella--Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.