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

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