We study the asymptotic behaviour of the Bloch–Kato–Shafarevich–Tate group of a modular form $f$ over the cyclotomic ${{\mathbb{Z}}_{p}}$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$ , both defined over $Q$. Let $x$ be an element of ${{H}^{1}}\left( Q,\,A \right)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $Q$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $ {{H}^{1}}\left( Q,\,A \right)\,\to \,{{H}^{1}}\left( \text{Q},\,J \right)$. In this article, we clarify the relation between the two definitions.

Let ${E}$ be a nonconstant elliptic curve, over a global field ${K}$ of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of ${E}$, we show that the order of the Shafarevich-Tate group of ${E}$, is given by${O}({N}^{1/2+6\,\log(2)/\log({q})})$, where ${N}$ is the conductor of ${E}, {q}$ is the cardinality of the finite field of constants of ${K}$, and where the constant in the bound depends only on ${K}$. The method of proof is to work with the geometric analog of the Birch-Swinnerton Dyer conjecture for the corresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.