Variation of Shafarevich-Tate groups in Hida families

Let $\mathbf{f}$ be a Hida family of cusp forms over a standard (power series) Iwasawa algebra. We determine the pseudo-isomorphism class of the big Shafarevich-Tate group attached to $\mathbf{f}$ over a suitable imaginary quadratic field by describing all its higher Fitting ideals; this description is given in terms of the universal Kolyvagin system of Howard's big Heegner points. As a second contribution, we study the variation of the structure of the Shafarevich-Tate groups of the classical specializations of $\mathbf{f}$.