Vanishing of $\mathrm{Sha}(A/K)[{\mathfrak{P}}^{\infty }]$ and its consequences for the anticyclotomic Iwasawa theory of ${\mathrm{GL}}_2$-abelian varieties

Abstract

In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich--Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of ${\mathrm{GL}}_2$-type (defined over a totally real number field $F$) and a CM field $K/F$. Combining this result with the abstract Iwasawa-theoretical argument of [MN19], we show that a similar vanishing holds over the layers of a suitably defined anticyclotomic multi-${\mathbb{Z}}_p$-extension of $K$.

40 pages