Bulletin of the
Korean Mathematical Society BKMS

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime of good supersingular reduction for $E$. Although the Iwasawa theory of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is well known to be fundamentally different from the case of good ordinary reduction at $p$, we are able to combine the method of our earlier paper with the theory of Kobayashi \cite{Kobayashi} and Pollack \cite{Pollack}, to give an explicit upper bound for the number of copies of $\mathbb{Q}_p/\mathbb{Z}_p$ occurring in the $p$-primary part of the Tate-Shafarevich group of $E$ over $\mathbb{Q}$.

Keywords: Iwasawa theory, supersingular prime, elliptic curves, Tate-Shafarevich group

MSC numbers: 11G05